STATISTICAL MONITORING AND CLUSTER DETECTION UNDER NATURALLY OCCURRING
HETEROGENEOUS DICHOTOMOUS EVENTS
A Dissertation Presented
by
Aysun Taşeli
to
The Department of Mechanical and Industrial Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in the field of
Industrial Engineering
Northeastern University Boston, Massachusetts
January 2011
i
Abstract
Many processes produce a count statistic that is a sum of multiple non-homogeneous
dichotomous random variables, that is, with different values of the Bernoulli parameter p.
The probability distribution of this count statistic is the convolution of J non-identical
binomial distributions and can significantly differ from its binomial and normal
counterparts. In such cases the homogeneity assumption can result in incorrect
probability calculations and conclusions from statistical procedures such as control
charts, sequential probability ratio tests, and cluster detection via scan statistics. Use of
the exact (J-binomial) distribution, however, can require prohibitively exhausting
calculations as the number (J) of non-identical binomial random variables in the
convolution increases.
Following the above motivations, this dissertation has three foci: The first is testing and
monitoring heterogeneous processes over time. Risk-adjusted sequential probability ratio
tests (SPRTs) and resetting SPRT charts are derived, their accuracy and detection
performances (average run lengths and operating characteristic curves) are compared to
those assuming homogeneity, and shown to be significantly better in some applications.
The second focus area is detection of geographical clusters via scan statistics in the
presence of natural heterogeneity. Two risk-adjusted models of Kulldorff’s Bernoulli
scan statistic, based on the product of risk-adjusted probabilities (J-Bernoulli model) and
the distribution of heterogeneity (J-binomial model) are developed and their comparative
performance versus the conventional method is explored.
Monte Carlo performance analyses show that the risk-adjusted models lead to better
inferences, detection times, and probabilities over a variety of scenarios provide insights
for the selection and use of correct methodologies under the occurrence of heterogeneous
dichotomous events.
ii
The third problem addresses computation issues of J-binomial distributions. Computing
these probabilities is important in many applications, especially since the above
mentioned methods each require tens to thousands of J-binomial probability calculations.
The accuracy of J-binomial probability estimations via a cumulant based expansion that
use orthogonal polynomials and saddle point approximations is explored by comparison
to both exact and Monte Carlo estimations (MCE) of probabilities. A normalized Gram-
Charlier expansion (NGCE) and saddle point approximations are shown to produce the
most accurate results and to be more time-efficient than computing the exact probabilities
or the MCE. The NGCE algorithm is practical, known to produce an estimate under all
scenarios, and of great value to analysts since it easily can be integrated into computer
codes.
iii
To my second parents; my dearest grandmother Fatma Trak,
and
my grandfather Lütfi Trak, whom I lost very early and heartily miss.
iv
Acknowledgements
It is a pleasure for me to thank all of those who supported me in any respect during my
doctoral studies. I feel truly blessed to have so many wonderful people in my life.
First and foremost, I would like to extend my deepest gratitude to my supervisor,
Professor James Benneyan, for his continuous guidance, encouragement, and support
from the preliminary to the concluding level of this dissertation. He provided me with an
excellent atmosphere to do research and shared his vision, wisdom, and expertise which
helped me improve both personally and academically. His passion towards research and
pursuit of perfection has always been the motivation for me during the completion of this
research. I consider myself extremely fortunate to have had such a mentor.
I would like to thank Professor Nasser Fard and Professor Samuel Gutmann for accepting
to be a part of my dissertation committee. I truly appreciate their encouraging words,
sincere feedback, time, and attention. I thank my department chair, Professor Hameed
Metghalchi for his constant endorsement during my PhD program. I also extend my
appreciation to Professor Gülser Köksal who has helped me establish the grounds upon
which this doctoral study is built.
This achievement was also possible because I have a great family who encouraged and
gave unstinting support when I needed it most. I am grateful to my father Hasan Ali
Taşeli for his endless love, caring, and enduring belief in me. I am also thankful to my
mother Ayşe Naile Trak and my sister Gül Başak Taşeli for their unconditional love, the
long phone calls that shortened distances, humor that uplifted my spirits, and always
being my inspiration to continue.
My particular thanks go to my friends Nilüfer Koldan, Aysun Sünnetçi, Mehmet Erkan
Ceyhan, Önder Öndemir, Zeynep Damla Ok, Hande Muşdal and Ayşegül Topçu for their
invaluable friendship and support, all the good times we had together, and making Boston
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a home away from home. I also would like to thank my new friends in Quality and
Productivity Laboratory for the cheer they brought and making one of the busiest times of
this journey more enjoyable.
I warmly thank my dear friend Esen Akyapı for always being there for me, particularly
during the challenging times of this journey. I have to say that I am thankful to her and
also my friends in Türkiye, especially Seyit Mümin Cilasun, Gül Çolak, Figen Çilingir,
and Aleaddin Ertem for their friendship and togetherness in so many things over the long
years. I appreciate always feeling their existence by my side although we are in different
parts of the world, kilometers away.
Last, but certainly not least, I would like to heartily thank Shane Cornell Marshall for his
unwavering support, patience, and encouragement that kept me going. I am deeply
grateful for his love and existence in my life.
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Table of Contents Abstract ............................................................................................................................... i Acknowledgements .......................................................................................................... iv Table of Contents ............................................................................................................. vi List of Figures ................................................................................................................. viii List of Tables .................................................................................................................. xiv Chapter 1 – Introduction.................................................................................................... 1
1.1. Motivation ........................................................................................................... 1
1.1.1. Distribution of Heterogeneity ..................................................................... 3
1.1.2. Examples of J-binomial Data...................................................................... 6
1.1.3. Computation Issues ................................................................................... 12
1.2. Focus of Dissertation ........................................................................................ 13
Chapter 2 – Risk-adjusted Non-resetting and Resetting Sequential Probability Ratio Tests ............................................................................................................. 16
2.1. Background ....................................................................................................... 16
2.2. Methodology ..................................................................................................... 18
2.2.1. SPRTs and Resetting SPRT Charts for Homogeneous Events ................. 18
2.2.2. SPRTs and Resetting SPRT Charts for Non-Homogeneous Dichotomous Events...... .................................................................................................. 20
2.3. Results ............................................................................................................... 31
2.3.1. Risk-adjusted SPRTs ................................................................................ 31
2.3.2. Risk-adjusted RSPRT Charts .................................................................... 47
2.4. Discussion ......................................................................................................... 63
Chapter 3 – Risk-adjusted Bernoulli and Binomial Scan Statistics ................................ 67
3.1. Background ....................................................................................................... 67
3.2. Methodology ..................................................................................................... 70
3.2.1. Kulldorff’s Scan Statistic .......................................................................... 70
3.2.2. Risk-adjusted Bernoulli and Binomial Scan Statistics.............................. 75
3.3. Results ............................................................................................................... 78
3.4. Discussion ......................................................................................................... 86
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Chapter 4 – Approximating J-binomial Distributions ..................................................... 87 4.1. Background ....................................................................................................... 87
4.2. Methodology ..................................................................................................... 90
4.2.1. Cumulant Based Approximation Using Orthogonal Polynomials ............ 90
4.2.2. Saddle Point Approximations ................................................................... 96
4.3. Results ............................................................................................................. 100
4.3.1. Orthogonal Polynomial Expansions ....................................................... 100
4.3.2. Saddle Point Approximations ................................................................. 109
4.4. Discussion ....................................................................................................... 112
Chapter 5 – Conclusions................................................................................................ 115
5.1. Summary of Major Results ............................................................................. 115
5.2. Description of Computer Codes...................................................................... 116
5.3. Future Possible Work ...................................................................................... 125
References ...................................................................................................................... 127
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List of Figures Figure 1-1: K of N system with N units, each having reliability Rj, j = 1, ..., N .............. 10
Figure 1-2: The correct weighted J-binomial distribution of total points scored per game by Boston Celtics player Paul Pierce, and its counterpart under homogeneity assumption .................................................................................................................... 12
Figure 2-1: Graphical illustration of SPRT when Bernoulli data are sampled randomly, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true....................................................................................................................................... 22
Figure 2-2: Graphical illustration of SPRT when binomial samples are gathered with predefined probabilities, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true ............................................................................................ 24
Figure 2-3: Graphical illustration of SPRT when all Bernoulli data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true........................................................................................................... 26
Figure 2-4: Graphical illustration of SPRT when all binomial data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true........................................................................................................... 28
Figure 2-5: Comparison of performance under four sampling scenarios (S: sequentially, R: randomly, WGP: with given probability, JAT: J at a time) on the probability of accepting H0 and the average number of items until a decision is made (ANI). H0: p = (0.02, 0.1, 0.25, 0.4); (i) H1: p = (1.1)p0, (ii) H1: p = (1.5)p0, (iii) H1 : p = (1.9)p0: (a) null hypothesis is true, (b) alternate hypothesis is true ................................................. 33
Figure 2-6: Impact of type I and type II errors on the performance of SPRT (Xm,j terms are unknown), J = 4, nm,j = 1, H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6): (a) P(Accept H0), (b) ANI: Average number of items, ANS: Average number of samples needed until a decision is made....................................................................... 35
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Figure 2-7: Impact of type I and type II errors on the performance of SPRT for J separate binomial events available at a time vs. exact J-binomial data, J = 4, nm,j = 1, H0: p0 = (0.02, 0.1, 0.25, 0.4), H1: p1 = (0.03, 0.15, 0.375, 0.6): (a) P(Accept H0), (b) ANI: Average number of items, ANS: Average number of samples needed until a decision is made .............................................................................................................................. 37
Figure 2-8: Impact of sample size and δ (Xm,j terms are known,1 at a time: 1 Bernoulli or binomial event known at a time, J at a time: J Xm,j terms known simultaneously) on P(Accept H0) and the average number of items needed until a decision is made (ANI) under null hypothesis H0: p = (0.02, 0.1, 0.25, 0.4), and alternate hypotheses (i) H1: p = (1.1)p0, (ii) H1: p = (1.5)p0, (iii) H1: p = (1.9)p0, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true ...................... 38
Figure 2-9: Case I: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made ..................................................... 41
Figure 2-10: Case II: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made .................................................................................... 42
Figure 2-11: Case III: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made .................................................................................... 43
Figure 2-12: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made . 45
Figure 2-13: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made .............................................................................................................................. 46
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Figure 2-14: Impact of design parameters α and β on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length .............. 48
Figure 2-15: Impact of design parameters α and β on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length ............................................ 50
Figure 2-16: Case I: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length............................................................................................................................. 53
Figure 2-17: Case II: Impact of sample size on the performance of RSPT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length............................................................................................................................. 54
Figure 2-18: Case I: Impact of sample size on the performance of RSPRT chart for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length ............................................ 57
Figure 2-19: Case II: Impact of sample size on the performance of RSPRT chart for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length ............................................ 58
Figure 2-20: Impact of shift in different rate parameter values on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10, α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length ...................................................................................................................... 61
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Figure 2-21: Impact of shift in different rate parameter values on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length .............. 63
Figure 2-22: Impact of delta on the performance of SPRT for J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1
1: p= (0.03, 0.15, 0.375, 0.6), H1
2: p = (0.035, 0.175, 0.4375, 0.7), H13: p = (0.04, 0.2,
0.5, 0.8), H14: p = (0.045, 0.225, 0.5625, 0.9)): ANI: average number of items, ANS:
average number of samples needed until a decision is made........................................ 66
Figure 3-1: Conceptual illustration of Kulldorff’s scan statistic ..................................... 71
Figure 3-2: Graphical illustration of spatial heterogeneity in and outside the scanning window R ...................................................................................................................... 76
Figure 3-3: Average p-values (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35)) (a) njR = 15, nj
S-R = 30, (b) nj
R = 30, njS-R = 60, (c) nj
R = 60, njS-R = 100 ............................................................ 80
Figure 3-4: Average p-values (J = 2, pS-R = (0.0183, 0.048)) (a) njR = 15, nj
S-R = 30, (b) nj
R = 30, njS-R = 60, (c) nj
R = 60, njS-R = 100 .................................................................. 82
Figure 3-5: Empirical distribution of p-values (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35), pR = (0.375, 0.075, 0.225, 0.525)) for sample sizes nj
R = 15, njS-R = 30; nj
R = 30, nj
S-R = 60; and njR = 60, nj
S-R = 100 ................................................................................ 84
Figure 3-6: Empirical distribution of LR values for pS-R = (0.2, 0.05, 0.15, 0.35) and pR = (0.2, 0.05, 0.15, 0.35) (null hypothesis is true) versus pR = (0.375, 0.075, 0.225, 0.525) (alternate hypothesis is true). ............................................................................. 85
Figure 4-1: Illustration of the need for normalization, (a) Exact probability distribution versus GCE of order 6, (b) Relative error of normalized versus conventional GCE of order 6, Relative Error = (Exact – Approximate) / Exact .......................................... 104
Figure 4-2: Impact of normalization and order on GCE accuracy: (a) Modified Kullback-Leibler statistic, (b) Total absolute deviation. NGCE: normalized GCE; JB1, JB2, JB3: parameter sets 1, 2 and 3 from Table 4-4, respectively ............................................. 104
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Figure 4-3: Impact of order on accuracy of the cumulant based approximations using (a) Hermite polynomials (NGCE) and (b) Laguerre polynomials. JB1, JB2, JB3: parameter sets 1, 2 and 3 from Table 4-4, respectively .............................................................. 106
Figure 4-4: Accuracy of cumulant based approximation via NGCE up to order 6 and Laguerre polynomials up to order 10 versus exact J-binomial probabilities and Monte Carlo estimates ............................................................................................................ 108
Figure 4-5: Comparison of accuracy of NGCE-o6: Normalized Gram-Charlier expansion of order 6, SPA-NR: Saddle point approximations using Newton-Raphson method, SPA-S: Saddle point approximations using secant method, TSPA-o3: Truncated saddle point approximation of order 3, and Cont.Corr.: Estimating the probabilities via numerical integration using continuation correction such that P(T = t) = P(t - 0.5 < T < t + 0.5) ......................................................................................................................... 110
Figure 4-6: Performance of NGCE of order 6 versus SPA in the body and tails of the distribution Relative Error = (Exact – Approximated) / Exact ................................... 111
Figure 4-7: Illustration of SPRT using the exact J-binomial probabilities, Normalized Gram-Charlier expansion of order 6 and saddle-point approximations: J = 4, nj = 10, j = 1, 2, 3, 4, H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), and p = (0.025, 0.125, 0.3125, 0.5) .................................................................................... 112
Figure 5-1: The input window for SPRT simulation performance analysis (a) when the Xj terms are known (b) when only T total counts are known (c) different choices of calculating J-binomial probabilities ............................................................................ 117
Figure 5-2: The output window for SPRT simulation performance analysis ................ 118
Figure 5-3: Format of the input data files for SPRT simulation performance analysis . 118
Figure 5-4: An example of the input window for constructing SPRTs ......................... 119
Figure 5-5: Format of the data file for constructing SPRTs .......................................... 119
Figure 5-6: Illustration of the risk- adjusted scan statistic procedure Monte Carlo simulation .................................................................................................................... 121
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Figure 5-7: The input window for calculation of J-binomial probabilities ................... 122
Figure 5-8: The input parameter file for calculation of J-binomial probabilities .......... 122
Figure 5-9: The logic of MCE of J-binomial probabilities ............................................ 124
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List of Tables Table 1-1: General format of data drawn from J heterogeneous binomial sub-populations
(nm,j ≥ 1, 0≤ xm,j ≤ nm,j ∀ m, j; m = 1, 2, ..., M; j = 1, 2, ..., J ) ........................................ 1
Table 1-2: General format of individual heterogeneous Bernoulli data with unique likelihoods (nm,j = 1, ∀ m, j; m = 1, 2, ..., M; j = 1, 2, ..., J ) ......................................... 2
Table 1-3: J-binomial probability distribution of when J = 3, nj = 5, ∀ j = 1, 2, 3, and p = (0.25, 0.15, 0.05) ....................................................................................................... 4
Table 1-4: Ventilator-associated Pneumonia (VAP) Bundle example .............................. 7
Table 1-5: Surgical site infection example: National Nosocomial Infections Surveillance (NNIS) risk categories (nm,j: the number of patients who have surgery, xm,j: number of patients who develop infection ....................................................................................... 8
Table 1-6: Risk-adjusted patient mortality example (nm,j: sample sizes, xm,j: number of surviving patients, i.e. xm,j = 1 indicates that patient did not survive, and pm,j: mortality rates) ................................................................................................................................ 8
Table 1-7: A power system with non-identical unit characteristics ................................... 9
Table 1-8: 2005-2006 NBA Season, Boston Celtics player Paul Pierce, (Nm: Total number of attempted shots, Tm: Total number of achieved shots in game m) .............. 11
Table 1-9: The probabilities computed using the correct weighted J-binomial distribution of total points scored per game by Boston Celtics player Paul Pierce and binomial distribution .................................................................................................................... 12
Table 1-10: CPU time (in seconds) to calculate the full J-binomial PDF (on a Pentium (R) 4, 2.80GHz CPU, 1GB RAM), MCE: Monte Carlo estimation, HW: half-width of the confidence interval .................................................................................................. 12
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Table 2-1: Tabular illustration of SPRT when Bernoulli data are sampled randomly, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of) ........... 22
Table 2-2: Tabular illustration of SPRT when binomial samples are observed with given probabilities, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of) ........................................................................................................................ 23
Table 2-3: Tabular illustration of SPRT when J Bernoulli data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of) ............ 26
Table 2-4: Tabular illustration of SPRT when all binomial data are sampled at one time H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of) ............ 28
Table 2-5: Impact of sampling order in cases for which Xm,j terms are known. PM: performance measure, ANS: average number of samples, ANI: average number of items needed until a decision is made, P(H0): probability of concluding that H0 is true....................................................................................................................................... 32
Table 2-6: Impact of type I and type II errors on the performance of SPRT for normal and binomial approximations versus exact J-binomial data ......................................... 34
Table 2-7: Impact of type I and type II errors on the performance of SPRT for J separate binomial events available at a time versus exact J-binomial data ................................ 36
Table 2-8: Case I: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), δ = 1.5) ....................................................................................................... 39
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Table 2-9: Case II: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5) ..... 40
Table 2-10: Case III: Impact of sample size on the performance of SPRT for normal and binomial approximations vs. exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9) ... 40
Table 2-11: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5)..................................................... 44
Table 2-12: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9)................................................... 44
Table 2-13: Impact of design parameters α and β on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length .............. 47
Table 2-14: Impact of design parameters α and β on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length .................................................................. 49
Table 2-15: Case I: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4,� α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length............................................................................................................................. 51
Table 2-16: Case II: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length............................................................................................................................. 52
xvii
Table 2-17: Case I: Impact of sample size on the performance of RSPRT for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length .................................................................. 55
Table 2-18: Case II: Impact of sample size on the performance of RSPRT for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length .................................................................. 56
Table 2-19: Impact of shift in different rate parameter values on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10, α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6) .......................................................................... 60
Table 2-20: Impact of shift in different rate parameter values on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length ............... 62
Table 2-21: Impact of delta on the performance of SPRT for J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1
1: p = (0.03, 0.15, 0.375, 0.6), H12: p = (0.035, 0.175, 0.4375, 0.7), H1
3: p = (0.04, 0.2, 0.5, 0.8), H1
4: p = (0.045, 0.225, 0.5625, 0.9)): P(H0): P (Accept H0), ANI: average number of individual items, ANS: average number of samples needed until a decision is made .......................................................................................................................... 65
Table 3-1: Spatial heterogeneity in and outside the scanning window R with radius r, where R
JrRr
Rr
Rr nnnn ,3,2,1, ≤⋅⋅⋅≤≤≤ , and RS
JrRS
rRS
rRS
r nnnn −−−− ≥⋅⋅⋅≥≥≥ ,3,2,1, ............................... 76
Table 3-2: Error analysis (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35)) with sample sizes njR = 15,
njS-R = 30; nj
R = 30, njS-R = 60; and nj
R = 60, njS-R = 100 ................................................ 81
Table 3-3: Error analysis (J = 2, pS-R = (0.0183, 0.048)) with sample sizes njR = 15,
njS-R = 30; nj
R = 30, njS-R = 60; and nj
R = 60, njS-R = 100 ................................................ 83
Table 4-1: First six Hermite polynomials and expansion coefficients in terms of moments µi of random variable z .................................................................................................. 92
xviii
Table 4-2: First six moments of random variable z and Hermite expansion coefficients in terms of cumulants iK of random variable T ............................................................... 93
Table 4-3: First six cumulants of J-binomial distribution ................................................ 95
Table 4-4: J-binomial distributions used for approximation analysis............................ 101
Table 4-5: Properties of orthogonal polynomials .......................................................... 102
Table 4-6: Sum of all estimated probabilities using GCE of order 6, without and with normalization .............................................................................................................. 103
Table 4-7: Impact of normalization and expansion order on the accuracy of cumulant based expansions using Hermite polynomials (GCE and NGCE) and Laguerre polynomials, for the parameter sets given in Table 4-4 ............................................. 105
Table 4-8: Comparison of accuracy of the approximation methods. NGCE-o6: Normalized Gram-Charlier expansion of order 6; SPA-S: Saddle point approximations using secant method; SPA-NR: Saddle point approximation using Newton-Raphson method; TSPA-o3: Truncated saddle point approximation of order 3; Cont.Corr.: Estimating the probabilities via numerical integration using continuation correction such that P(T = t) = P(t - 0.5 < T < t + 0.5) ................................................................. 111
Table 4-9: CPU times (in seconds) to calculate the entire J-binomial PDF .................. 113
Table 4-10: Comparison of accuracy of NGCE-o6: Normalized Gram-Charlier expansion of order 6, TSPA-o3: Truncated saddle point approximation of order 3, SPA-NR: Saddle point approximation using Newton-Raphson method, SPA-S: Saddle point approximations using secant method for the case J = 20, nj = 1, pj = 0.4.................. 114
1
Chapter 1 – Introduction
1.1. Motivation
This dissertation considers heterogeneous dichotomous data sampled from non-identical
binomial processes or sub-populations that can be encountered in many healthcare,
production, and service systems. Such data can be produced either as J binomial (nj, pj)
sub-populations, where trials within any sub-population are independent and identically
distributed (i.i.d.) but not i.i.d. between sub-populations, or individually unique, in the
sense that each Bernoulli trial has a different event probability. An event here can be
defined as the outcome of interest such as a process failure, disease occurrence, passenger
no-show or successful sports play. Table 1-1 and Table 1-2 respectively illustrate the
general format of such data where at time period m, nm,j and xm,j are the sample size and
count within category j and ∑ ==
J
j jmm nN1 , , ∑ =
=J
j jmm XT1 , , and mmm NTF = are the
total sample size, total number, and fraction of failures across all categories.
Table 1-1: General format of data drawn from J heterogeneous binomial sub-populations (nm,j ≥ 1, 0≤ xm,j ≤ nm,j ∀ m, j; m = 1, 2, ..., M; j = 1, 2, ..., J )
Time period
Sub-population 1 Sub-population 2 … Sub-population J Total nm,1 xm,1 nm,2 xm,2 nm,J xm,J Nm Tm
1 n1,1 x1,1 n1,2 x1,2 n1,J x1,J N1 T1
2 n2,1 x2,1 n2,2 x2,2 n2,J x2,J N2 T2
3 n3,1 x3,1 n3,2 x3,2 n3,J x3,J N3 T3
...
...
...
...
... ...
...
...
...
M - 1 nM-1,1 xM-1,1 nM-1,2 xM-1,2 nM-1,J xM-1,J NM-1 TM-1
M nM,1 xM,1 nM,2 xM,2 nM,J xM,J NM TM
2
Table 1-2: General format of individual heterogeneous Bernoulli data with unique likelihoods (n,m,j = 1, ∀ m, j; m = 1, 2, ..., M; j = 1, 2, ..., J )
Time period
Patient number
nm,j xm,j pm,j Nm Tm
1
1 1 x1,1 p1,1
N1 T1 2 1 x1,2 p1,2 3 1 x1,3 p1,3 4 1 x1,4 p1,4
...
...
...
...
...
...
...
M
1 1 xM,1 p1,1
NM TM
2 1 xM,2 p1,2
3 1 xM,3 p1,3
4 1 xM,4 p1,4
5 1 xM,5 p1,5
Examples of this type of non-homogeneity include the total number of defective items
produced by different manufacturing lines, automobile accidents combined across
different driver types, on-time shipments from different vendors, free throws scored by
different basketball players, and airline no-shows among different passenger types.
Important healthcare applications include patient mortality, hospital-acquired infections,
care protocol compliance, appointment no shows, and preventable hospital readmissions
across different disease groups, procedures, patient groups, providers, and medical
conditions. For example, readmission rates for congestive heart failure patients can range
from 0.09 to 0.455 (Philbin and DiSalvo, 1999), mortality rates from 0.01 to 0.95
(Higgins et al., 2005), and ventilator associated pneumonia rates from 0.06 to 0.52
(Jimenez et al., 2009). Because the rate of an event in the above examples is different
across different categories, the total number of events T and the fraction of events F
cannot be modeled by a binomial distribution, as would be the case if every individual or
category has the same occurrence probability of the event.
In some applications, furthermore, each Bernoulli event can carry a different weight, such
as when computing the total severity or cost of all adverse events, number of items in all
delayed shipments, power generated by all operating plants, cost of insurance claims, or
points scored by all types of basketball shots. Statistical and quality control methods
therefore should be adapted appropriately for the above examples to test for differences
3
or monitor longitudinally for process instability. Although not the main focus of this
study, all the methods proposed in this dissertation can be extended to weighted
J-binomial distribution case.
1.1.1. Distribution of Heterogeneity
Let X1, X2,…, XJ be independent binomial random variables each with parameters nj and
pj, where pk ≠ pj ∀ (k, j), T = X1 + X2 + … + XJ, and F = T / N. The random variables T
and F follow a J-binomial probability distribution function (PDF) with the 2J + 1
parameters J, n1,…, nJ, p1,…, pJ (Benneyan and Borgman, 2004). To motivate a general
form of this PDF, the simplest case with J = 2 categories can be written as the
convolution
∑−=
−====+==),min(
),0max(121121
1
21
))()(()()(nt
ntx
xtXPxXPtXXPtTP , (1-1)
where P(X1 = x1) and P(X2 = t - x1) are binomial probabilities and where P(T = t) is equal
to the sum product of all binomial probabilities for X1 and X2, such that X1 + X2 = T. To
reduce the number of terms computed in the convolution summation limits, the minimum
value that X1 can take so that x1 + x2 = t is t - n2 if t - n2 ≥ 0 and 0 if t - n2 ≤ 0, and the
maximum value is n1 if t ≥ n1 or t if t ≤ n1.
When J = 3 the right-most expression in Equation (1-1), P(X2 = t - x1), is replaced with
P(X2 + X3 = t - x1) which similarly can be calculated by summing across all possible
products of P(X2 = x2) and P(X3 = x3) such that x2 + x3 = t - x1. The PDF when J = 3 thus
extends to
)()( 321 tXXXPtTP =++==
.))()(()(),min(
),0max(21
),min(
),0max(32211
1
321
21
312
∑ ∑−−=
−
−−=
−−====
nt
nntx
nxt
nxtx
xxtXPxXPxXP (1-2)
4
To illustrate the calculation of J-binomial probabilities, consider the scenario when J = 3,
nj = 5, ∀ j = 1, 2, 3, and p = (0.25, 0.15, 0.05). Then the random variable T = X1 + X2 + X3
can take discrete values between 0 and 15. The probabilities P(T = 0), P(T = 1), and
P(T = 2) are calculated in the following way.
P(T = 0) = P(X1 = 0)*P(X2 = 0)*P(X3 = 0) = (0.237305)(0.443705)(0.773781)
= 0.081474 P(T = 1) = P(X1 = 1)*P(X2 = 0)*P(X3 = 0) + P(X1 = 0)*P(X2 = 1)*P(X3 = 0) + P(X1 = 0)*P(X2 = 0)*P(X3 = 1) = (0.395508)(0.443705)(0.773781) + (0.237305)(0.391505)(0.773781) + (0.237305)(0.443705)(0.203627) = 0.229119 P(T = 2) = P(X1 = 1)*P(X2 = 1)*P(X3 = 0) + P(X1 = 1)*P(X2 = 0)*P(X3 = 1) + P(X1 = 0)*P(X2 = 1)*P(X3 = 1)
= (0.395508)(0.391505)(0.773781) + (0.395508)( 0.443705)( 0.203627) + (0.237305)(0.391505)(0.203627) = 0.292623 Other probabilities are calculated similarly. Table 1-3 illustrates the whole PDF.
Table 1-3: J-binomial probability distribution of when J = 3, nj = 5, ∀ j = 1, 2, 3, and p = (0.25, 0.15, 0.05)
T P(T = t)
0 0.081474
1 0.229119
2 0.292623
3 0.224719
4 0.115807
5 0.04233
6 0.011311
7 0.002244
8 0.000333
9 0.000037
10 0.000003
11 0
12 0
13 0
14 0
15 0
5
For the general J category case, this logic extends to a sequence of J - 1 nested
summations,
∑ ∑∑
−=
−−
−−−=
−
−−=∑ ∑
⋅⋅⋅=
∑====
= ==
),min(
,0max
),min(
,0max
33
),min(
,0max
2211
1
21
321
4213
21
312
)()()()(nt
ntx
nxxt
nxxtx
nxt
nxtxJ
i iJ
i iJ
i i
xXPxXPxXPtTP
⋅⋅⋅
⋅⋅⋅
∑
∑−==
∑
∑∑⋅⋅⋅=⋅⋅⋅ ∑ ∑∑
−
−−=
−
=−−
−
−−=
−−
=
−
=−
−
=
+=
−
=
12
1
2
11
1
1
1
1
1
,min
,0max
1
111
,min
,0max
)()()(J
J
i i
JJ
i iJ
kk
i i
J
ki ik
i ik
nxt
nxtx
J
i iJJJ
nxt
nxtx
kk xtXPxXPxXP , (1-3)
where 0 ≤ T ≤ ∑ ==
J
j jnN1
. When binomial random variables in the convolution have
specific weights, jw , j = 1, 2, ..., J, then the resulting random variable
JJ XwXwXwT +++= ...2211 , is called a weighted J-binomial random variable, which
has the following probability mass function:
...)( ... )()()(0 0 0
2211
1
2
11
2
1
1
∑ ∑ ∑=
−
=
−
=
∑
=====
−
=
t
x
w
xwt
x
w
xwt
xkk
k
k
i ii
k
xXPxXPxXPtTP
......)()( ...
11
011
1
2
1
1
∑−
=
∑
=−
=
−
=−−∑
−
−
=
− J
Ji ii
J
w
xwt
xJJ w
xwtXPxXP
J
J
i ii
J
, (1-4)
where 0 ≤ T ≤ ∑ =
J
j jmj nw1 , . Previous work by Benneyan and Borgman (Benneyan and
Borgman, 2004) has shown that the expected values of J-binomial distribution and its
binomial counterpart are equal whereas J-binomial has a smaller variance, i.e. is under-
dispersed relative to the binomial. This under-dispersion might result in significantly
different and misleading conclusions in many statistical process control methods such as
statistical control charts (Shewhart, EWMA, and so on) and sequential probability ratio
tests (SPRTs), as discussed in the following chapters.
6
1.1.2. Examples of J-binomial Data
This subsection presents examples of heterogeneity and J-binomial random variables
from health care, electric power systems, reliability and sports. The examples discussed
herein also demonstrate how J-binomial and weighted J-binomial distributions can be
different from the corresponding binomial distributions. These examples further
emphasize the need to properly adapt statistical techniques in case of heterogeneity.
1.1.2.1. Healthcare The unique characteristics of patients and procedures in health care produce many
examples of J-binomial data. Often of interest in such applications is the development of
one-time and longitudinal statistical methods to test and detect changes in processes
producing these types of data, where it is important to account for different risk
probabilities.
The data in Table 1-4 represent care provider adherence to the elements of a ventilator-
associated pneumonia (VAP) bundle (adapted from (Jimenez et al., 2009)) that is widely
used and disseminated by The Joint Commission, the Premier group and the Institute for
Healthcare Improvement. The VAP bundle consists of four care process elements
(Crocker and Kinnear, 2008, Institute for Healthcare Improvement, 2010):
(i) elevating the head of the patient’s bed to between 30o and 45o to reduce
gastroesophageal reflux and aspiration that can lead to VAP,
(ii) daily sedation vacationing to test whether the patient is ready to stop using
a ventilator,
(iii) peptic ulcer disease prophylaxis (PUD) to prevent oral ulcers, and
(iv) deep-vein thrombosis (DVT) prophylaxis to prevent blood clotting.
At the end of each row, the total number of patients times the number of bundle elements,
Nm, and the total number of bundle elements properly observed, Tm, are given for each
7
week. Here the Tm terms do not have a binomial distribution since the compliance
probability for each bundle element is different.
Table 1-4: Ventilator-associated Pneumonia (VAP) Bundle example
Bundle Element
Head of the Bed > 30o
Sedation Vacation
PUD Prophylaxis
DVT Prophylaxis
Total
Day nm,1 xm,1 nm,2 xm,2 nm,3 xm,3 nm,4 xm,4 Nm Tm 1 3 1 3 2 3 2 3 2 12 7 2 2 0 2 1 2 2 2 1 8 4 3 3 2 3 1 3 2 3 1 12 6 4 2 0 2 0 2 1 2 1 8 2 5 3 0 3 2 3 2 3 2 12 6
...
...
...
...
...
...
...
...
...
...
...
12 2 1 2 1 2 1 2 1 8 4 13 2 0 2 1 2 2 2 1 8 4 14 2 0 2 1 2 2 2 2 8 5
:ˆ jp 0.14 0.67 0.93 0.87 P̂ = 0.65
Table 1-5 illustrates a common infection control concern, where patients who develop a
surgical site infection (SSI) are stratified into one of four risk categories defined by the
National Nosocomial Surveillance Index (NNSI), with nm,j and xm,j being the number of
patients who have surgery and develop infections, respectively. The right-most column
contains the total number of patients sampled (Nm) and that had a SSI (Tm) across all
categories in week m, with P denoting the overall pooled probability of an infection and
the symbol ^ denoting parameter estimates. Again note that the Tm terms do not have
binomial distributions since the probability each patient develops an SSI differs by risk
category; i.e. pj ≠ P for one or more j. Table 1-6 illustrates a related scenario where each
individual has a unique mortality rate, such that the sample size for each category is
essentially 1.
8
Table 1-5: Surgical site infection example: National Nosocomial Infections Surveillance (NNIS) risk categories (nm,j: the number of patients who have surgery, xm,j: number of patients who develop infection
Week
Category 1 Category 2 Category 3 Category 4 Total nm,1 xm,1 nm,2 xm,2 nm,3 xm,3 nm,4 xm,4 Nm Tm
1 41 2 38 2 17 5 1 0 97 9 2 60 5 34 0 17 6 1 1 112 12 3 50 1 46 3 15 5 2 2 113 11 4 47 1 32 4 20 4 4 2 103 11 5 48 0 36 5 10 2 1 1 95 8 6 36 0 41 4 12 2 2 1 91 7 7 64 0 25 0 10 4 3 0 102 4 8 44 1 33 2 20 3 3 1 100 7 9 45 4 32 2 11 3 4 0 92 9 10 57 2 24 1 17 3 3 2 101 8 11 52 1 28 4 15 4 4 3 99 12 12 54 0 32 1 16 5 0 0 102 6 13 38 2 32 2 8 1 0 0 78 5 14 25 3 16 3 9 3 0 0 50 9 15 20 1 19 6 5 2 0 0 44 9 16 19 0 18 1 7 3 0 0 44 4 17 2 0 4 0 4 2 0 0 10 2
Total 702 23 490 40 213 57 28 13 1433 133
:ˆ jp 0.0328 0.0816 0.2676 0.4643 =P̂ 0.09281
Table 1-6: Risk-adjusted patient mortality example (nm,j: sample sizes, xm,j: number of surviving patients, i.e. xm,j = 1 indicates that patient did not survive, and pm,j: mortality rates)
Time Period
Patient Number
nm,j
xm,j pm,j
Nm
Tm
1 1 1 0 0.0307468
3 0 2 1 0 0.0998745 3 1 0 0.0734482
2
1 1 0 0.0822161
5 1 2 1 1 0.0506398 3 1 0 0.0202584 4 1 0 0.0450202 5 1 0 0.0159333
3
1 1 1 0.0389338
4 2 2 1 1 0.4456432 3 1 0 0.0734482 4 1 0 0.091927
4
1 1 0 0.0845525
6 1
2 1 0 0.4004309 3 1 0 0.0568521 4 1 1 0.1353115 5 1 0 0.0584448 6 1 0 0.1556523
9
1.1.2.2. Electric Power Systems In electric power systems, it is important to accurately determine the distribution of
system capacity to calculate the loss of load probabilities and the probability of a possible
capacity shortage in the process. In an environment with two or more independent units,
the overall capacity or load flow of the system is represented as linear combinations of
independent random variables (convolutions). With identical units, the capacity can be
described with a binomial random variable. However, when there are components with
diverse capacities and availabilities, it is no longer appropriate to simply model the
probability distribution of overall system capacity with a binomial random variable.
Table 1-7 lists the characteristics of non-identical components where the overall capacity
of the system is a weighted J-binomial random variable with J = 17 and the weights are
the capacity of each type of unit (example taken from Stremel and Rau (Stremel and Rau,
1979)).
Table 1-7: A power system with non-identical unit characteristics
Type of Units Number of Units Capacity (Cj) Availability ( Pj) 1 1 4621 1 2 1 200 0.9115 3 1 400 0.935 4 1 600 0.88 5 4 100 0.92 6 4 200 0.91 7 4 64 0.94 8 8 287 0.87 9 4 525 0.94 10 4 547 0.92 11 8 530 0.9 12 1 206 0.8 13 4 514 0.88 14 1 685 0.85 15 1 685 0.84 16 1 685 0.82 17 1 644 0.79
10
1.1.2.3. Reliability A ‘K of N system’ in reliability refers to a system which operates when at least any K of
the N units function. To be able to calculate the reliability of such a system, first the
reliability of each unit is needed. Consider a K of N system, with n components each with
a different failure rate λj, as illustrated in Figure 1-1. If the lifetime of each component is
exponentially distributed, Xj ~ expo(λj), one can easily calculate the reliability of each
component, Rj = P(Xj > xj). If the number components that survive in the system is
denoted by T, then T has a J-binomial distribution with J = N and the risk parameter of
each component being equal to 1-Rj. The survival probability of the system, P(T ≥ K)
then is a J-binomial probability.
Figure 1-1: K of N system with N units, each having reliability Rj, j = 1, ..., N
Aggarwal (Aggarwal, 1993) describes such an electrical system where the overall system
works when at least 3 of the 4 components work. If each of the components has a failure
rate of λ = 0.0003, and L denotes the lifetime of a component, then the reliability of each
component is computed as pj = P(L > 2500) = e-2500*0.0003 = 0.4724. The survival
probability of the system is computed as P(T ≥ 3) = 0.2721 using a binomial distribution
with parameters n = 4, p = 0.4724. If the system has components with different failure
rates such that λ1 = λ2 = 0.0001, and λ3 = λ4 = 0.0005, the reliability of components 1 and
2 then will be 0.7788, 3 and 4 will be 0.2865. Consequently, the survival probability of
the overall system can be computed as P(T ≥ 3) = 0.3260, using J-binomial distribution
with parameters J = 4, nj = 1, j = 1, 2, 3, 4, and p1 = p2 = 0.7788, p3 = p4 = 0.2865. So, if
λ1, R1
λ2, R2
λ3, R3
λN, RN
11
all components are assumed to be homogeneous with a failure rate of 0.0003 in this
scenario, the mean time to failure of the system will be computed as 1 / 0.2721 = 3.6751
although it really is 1 / 0.326 = 3.0675.
1.1.2.4. Sports The data in Table 1-8 show the shots attempted and achieved by the Boston Celtics
player Paul Pierce in the 2005-2006 NBA season. If the total number of shots achieved is
Tm, then the random variable Tm = Xm,1 + Xm,2 + Xm,3, is a convolution of non-identical
binomial random variables, namely a J-binomial random variable, since the probability of
success for a field goal (FG), three point shots (3P), and free throw (FT) will not be the
same for a player. Similarly, if the random variable Tm represents the total number of
points achieved, then Tm= 1*Xm,1 + 2*Xm,2 + 3*Xm,3 is a weighted J-binomial with the
points of each shot being the weights. Figure 1-2 and Table 1-9 illustrate how the
weighted J-binomial distribution probabilities are different from their counterparts under
the assumption of equal successful shot probabilities.
Table 1-8: 2005-2006 NBA Season, Boston Celtics player Paul Pierce, (Nm: Total number of attempted shots, Tm: Total number of achieved shots in game m)
FG FGA 3P 3PA FT FTA Total Date xm,1 nm,1 xm,2 nm,2 xm,3 nm,3 Nm Tm
11/2/2005 5 18 0 2 18 24 44 23 11/4/2005 9 18 0 1 8 12 31 17 11/5/2005 10 20 1 3 9 10 33 20 11/9/2005 8 13 3 3 8 10 26 19 11/11/2005 7 19 0 1 4 5 25 11 11/13/2005 4 13 3 8 6 6 27 13 11/15/2005 8 18 3 5 2 2 25 13 11/16/2005 8 16 1 2 3 5 23 12 11/18/2005 6 16 2 4 10 12 32 18 11/22/2005 9 24 2 4 2 2 30 13
12
Table 1-9: The probabilities computed using the correct weighted J-binomial distribution of total points scored per game by Boston Celtics player Paul Pierce and binomial distribution
P(T > 25) P(T > 35) P(T > 40) J-binomial 0.620193 0.058368 0.006153 Binomial 0.819469 0.179957 0.032067
Figure 1-2: The correct weighted J-binomial distribution of total points scored per game by Boston Celtics
player Paul Pierce, and its counterpart under homogeneity assumption
1.1.3. Computation Issues
Although J-binomial probabilities can be computed easily for small number of
heterogeneous sub-populations (e.g. J ≤ 4) in the convolution, this may not always be
possible due to the lack of a closed form for the PDF given in Equation (1-3). The
amount of summation effort in the convolution significantly increases as the number of
random variables J or their sample sizes nj increases, in turn prohibitively increasing the
time to calculate exact probabilities (see Table 1-10).
Table 1-10: CPU time (in seconds) to calculate the full J-binomial PDF (on a Pentium (R) 4, 2.80GHz CPU, 1GB RAM), MCE: Monte Carlo estimation, HW: half-width of the confidence interval
Case Parameters MCE
(HW=0.001) Exact
1 J = 4, N = 18 9.844 0
2 J = 4, N = 100 30.563 0.406
3 J = 10, N = 100 31.531 3829.43
4 J = 25, N = 1278 82.469 > one week
Total Points Scored per Game
WeightedJ-Binomial(correct)
Binomial(incorrect)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56
T
P(T
=t)
13
Statistical procedures that involve likelihood ratio type test statistics, such as sequential
probability ratio tests (SPRTs) (Wald, 1945) and scan cluster detection statistics
(Kulldorff, 1997), moreover, necessitate repeated calculation of numerous probabilities,
exacerbating this problem. An SPRT with m = 100 data, for example, would require
calculation of 2m = 200 J-binomial probabilities, while scan statistics can require
thousands of probability calculations as they iteratively search for the location and size of
a cluster. Highly accurate and fast approximations of J-binomial probabilities, therefore,
are especially important for such methods. Although binomial and normal distributions
might seem reasonable in many general cases, the inaccuracies of both, often can
compromise the performance of many statistical methods (Benneyan et al., 2007,
Benneyan and Borgman, 2004). Although the normal distribution often is considered as
the probability distribution of sums of independent random variables based on the central
limit theorem, if the number of random variables in the sum or the value of the pj terms
are small, it may not estimate the tail probabilities with accuracy (Schlenker, 1986).
Monte Carlo simulation, on the other hand, provides results that are highly close to the
exact values, but can be too time consuming in many practical applications, and is used
for comparison purposes in most studies (Zhang and Lee, 2004).
1.2. Focus of Dissertation
The first focus area of this dissertation is sequential testing and monitoring of non-
homogeneous dichotomous events over time. Sequential probability ratio tests (SPRTs)
and resetting SPRT (RSPRT) charts are developed and investigated for processes that
produce non-identical binomial outcomes, where samples can be available in two
different ways based on whether the outcome of each binomial event is available or not.
In the former case, data representing the number of observed events can be obtained one
by one from each process, while in the latter case only the total number of outcomes of
all processes is known, but not the number of outcomes for individual processes,
corresponding to what is referred to as a “pure J-binomial” random variable. The
performance of each method is investigated under various scenarios via Monte Carlo
simulation. The impacts of sample size, the deviation between null and alternate
14
hypotheses, and design parameters on the average run length (ARL), average number of
samples and items until decision (ANS and ANI), and operating characteristic (OC)
curves are shown to have significant effects on the performance of both SPRTs and
RSPRT charts. The results also illustrate and enhance the significance of properly
adopting the techniques for heterogeneity.
Although examples of clusters can be found in many areas such as genetics, sociology,
reliability, quality control, and so on (Glaz and Balakrishnan, 1999), detecting existence
of clusters is especially important in controlling health events such as exposure and
spread of diseases or high mortality rates. A geographical cluster occurs when the number
of events in a certain part of a study region is significantly different than in the rest of the
geographical area (Jacquez et al., 1996b). The second part of this study focuses on
detection of spatial clusters where the type of heterogeneity described above exists. One
of the most popular methods of cluster detection, Kulldorff’s scan statistic (Kulldorff,
1997, Kulldorff, 2010) detects spatial clusters by scanning over a geographical area with
a circular window and determining (via Monte Carlo simulation) if the number of cases
inside the window is significantly higher than the rest of the area. While computing the
scan test statistic, Kulldorff’s Bernoulli model, however, assumes homogeneity within
the scanning window as well as homogeneity over the rest of the study space. Two
methods of risk-adjustment to Kulldorff’s Bernoulli model are proposed in Chapter 3 of
this dissertation and their performance compared to the conventional model. The risk-
adjusted scan statistic based on the J-binomial distribution is shown to have larger power.
Investigation of fast ways of approximating J-binomial distribution is a fundamental part
of this dissertation. Among the many methods that exist in the literature, the two
methods, cumulant based approximations using orthogonal polynomials and saddle point
approximations (SPA) both are found to be practical and suitable for this application.
Cumulant based orthogonal polynomial expansions have been used in variety of
applications such as to estimate the loss of load or capacity shortage probabilities (Gupta
and Manohar, 2005, Schellenberg et al., 2005, Tian et al., 1989, Zhang and Lee, 2004,
Singh and Kim, 1991, Stremel and Rau, 1979). Saddle point approximations, developed
15
by Daniels (Daniels, 1954), are shown to be fast and to produce highly close estimates of
tail probabilities by many researchers (Schlenker, 1986, Giles, 2001, Guotis and Casella,
1999, Matis and Guardiola, 2006). While orthogonal polynomial expansions usually
produce fast results that are quite close to the exact values, different types of polynomials
tend to be effective in different types of applications. Of six types of orthogonal
polynomials investigated in this study, Hermite and Laguerre polynomials produce
reasonable estimates to a range of J-binomial distributions. The accuracy of a normalized
version of cumulant based Hermite polynomial expansion and SPA is assessed by
comparing to the Monte Carlo estimations and exact probability calculations. An easily
adaptable computer program of normalized Hermite polynomial expansions is developed
and integrated in the risk-adjusted SPRT, RSPRT, and scan statistic Monte Carlo
simulations.
The remainder of this dissertation is organized as follows. Chapter 2 develops SPRTs and
RSPRT charts for heterogeneous dichotomous events and evaluates the performance of
methods over a range of parameters, also providing a detailed comparison to
conventional methods based on the homogeneity assumption. Two methods of risk-
adjustment for the Bernoulli model of Kulldorff’s scan statistic are proposed in Chapter
3. The comparative performance of the developed and existing models is investigated and
results are presented. Chapter 4 assesses the accuracy and computational efficiency of
cumulant based orthogonal polynomial expansions and saddle point approximations for
estimating J-binomial probabilities through comparison of exact and MCE values. A
normalized cumulant based Hermite polynomial expansion is developed and its
advantages and disadvantages versus SPA are discussed. Finally, Chapter 5 summarizes
the dissertation, discusses the implications of results, describes the computer programs
developed, and provides recommendations for possible future work.
16
Chapter 2 – Risk-adjusted Non-resetting and Resetting Sequential Probability Ratio Tests
2.1. Background
Wald’s SPRT (Wald, 1945) is an extension of the Neyman and Pearson hypotheses test to
a sequential context. Neyman and Pearson (Neyman and Pearson, 1928) showed that
given a set of null and alternate hypotheses H0: θθθθ = θθθθ0 versus H1: θθθθ = θθθθ1 for a random
variable with some parameter vector θθθθ, the most powerful test for a fixed sample size is
based on the likelihood ratio
)|()|()|(
)|()|()|(
)|(
)|(
00201
11211
10
11
θθθ
θθθ
θ
θ
M
MM
mm
M
mm
xfxfxf
xfxfxf
xf
xf
LR⋅⋅⋅⋅
⋅⋅⋅⋅==
∏
∏
=
=, (2-1)
where f(xm|θθθθ0) and f(xm|θθθθ1) are the probability density functions for Xm given the null and
alternate parameter vectors θθθθ0 and θθθθ1, respectively. This test rejects H0 if LR > k or
decides in favor of H0 if LR < k, where k is a constant that is chosen so that the
probability of a type I error does not exceed α and the sample size M is the smallest
integer for which the probability of type II error does not exceed a predetermined value β.
In Wald’s SPRT, on the other hand, one continues sampling as long as the inequality
B < LR < A is met, where the constants B and A are chosen such that the probability of
type I error does not exceed α and the probability of type II error does not exceed β. A
decision in favor of H0 or H1 is made if LR ≤ B or LR ≥ A, respectively. Wald shows that
these limits can be approximated by the lower bound B ≅ β / (1- α) and upper bound
A ≅ (1- β) / α. These approximate limits usually lead to smaller actual probabilities of
17
type I and type II errors although they may increase the average number of samples
(ANS) needed for a decision (Wald, 1945).
Cumulative sum (CUSUM) (Steiner et al., 2000, Novick et al., 2006, Steiner et al., 2001,
Beiles and Morton, 2004), sequential probability ratio test (SPRT) (Matheny et al., 2008,
Spiegelhalter et al., 2003) and resetting SPRT (RSPRT) methods (Grigg et al., 2003,
Cook et al., 2008, Grigg and Farewell, 2004, Sibanda et al., 2007) are discussed within
healthcare risk-adjusted contexts only for the simplest case where the risks and outcomes
for each patient are known and the same percent change is considered for all patients (a
constant increase in the odds ratio), either of which may not be true in some applications
as illustrated in Section 1.1.2.
Three types of risk-adjusted non-resetting and resetting SPRTs constructed below for the
cases when patients either have unique failure rates or are categorized into different risk
groups or sub-populations, where data become available one Bernoulli or binomial sub-
sample Xm,j at a time, from all sub-samples simultaneously (i.e. all Xm,j terms at once), or
only as the total Tm or fraction Fm (i.e., with the number of failures in individual sub-
samples unknown). The performance of each of these SPRTs and RSPRT charts then is
investigated over a range of sample sizes, design parameters (type I and type II error
probabilities), differences between the null and alternate hypotheses (δ), manners by
which data are drawn from each category j and over several scenarios including cases for
which only the smallest, the largest, or random 50% of the rate parameters change. The
parameter delta is the magnitude δ = p1/ p0 by which the alternate hypothesis differs from
the null. If H0: p = p0 and H1: p = 1.5*p0, for example, then δ = 1.5. The ARL until each
test terminates, the probability of making the correct decision for SPRTs, and the average
time to detection of shifts in the process parameters for RSPRT charts are compared to
those of their conventional homogeneous counterparts for which instead a common risk
parameter is assumed, (i.e. assuming a homogeneous parameter,∑
∑=
==J
j j
J
j jj
n
pnP
1
1 , applied to
all trials), or to binomial and normal approximations.
18
2.2. Methodology
2.2.1. SPRTs and Resetting SPRT Charts for Homogeneous Events
It is convenient to first review how a SPRT is conducted for conventional i.i.d. Bernoulli
and binomial random variables before extending this to the heterogeneous case. Since the
likelihood function for identical Bernoulli random variables is L = P(X1, X2, …, XM| p) =
∏ =
−−M
m
xx mm pp1
1 ,)1( where xm = 1 if the mth Bernoulli trial is a failure and xm = 0 if it is a
success, the likelihood ratio given the null and alternative hypotheses H0: p = p0 and
H1: p = p1 is .)1()1(1
1001
111 ∏∏ =
−
=
− −−=M
m
xxM
m
xx mmmm ppppLR The SPRT inequality then is
Ap
p
p
pB
mmx
M
m
x
<
−
−
<
−
=∏
1
1 0
1
0
1
1
1, (2-2)
which by taking natural logarithms becomes
)ln()1(1
1lnln)ln(
10
1
10
1 Axp
px
p
pB
M
mm
M
mm <−
−
−+
< ∑∑
==
. (2-3)
In practice, the center term is updated after each Bernoulli trial is observed. If there are a
total of sxM
m m =∑ =1failures after M data, the log likelihood ratio ln(LR)M becomes
−
−−+
=
0
1
0
1
1
1ln)(ln)ln(
p
psM
p
psLR M , (2-4)
which also can be computed recursively via the updating equation
,)1(
)1(ln
1
1ln)ln()ln(
10
01
0
11
−
−+
−
−+= − pp
ppx
p
pLRLR MMM (2-5)
where ln(LR)0 = 0. More generally, when the data are drawn instead in binomial samples
of size nm, the general form of the SPRT follows very similarly as
19
A
ppx
n
ppx
n
BM
m
xnx
m
m
M
m
xnx
m
m
mmm
mmm
<
−
−
<
∏
∏
=
−
=
−
100
111
)1(
)1(
. (2-6)
Cancelling the combination terms, taking natural logarithms, and rearranging produces
∑∑==
<
−−
+
−−
<M
mm
M
mm A
p
pn
pp
ppxB
0 0
1
0 10
01 )ln(1
1ln
)1(
)1(ln)ln( , (2-7)
with the updating equation
−
−+
−
−+= − )1(
)1(ln
1
1ln)ln()ln(
10
01
0
11 pp
ppx
p
pnLRLR MMMM , (2-8)
where nM denotes the size of the Mth sample (Kenett and Zacks, 1998, Ghosh, 1970).
With a bit of algebra, Equation (2-7) can be rearranged to isolate a cumulative sum term
in the middle
−−
−−
−
<<
−−
−−
− ∑∑
∑=
=
=
)1(
)1(ln
1
1ln)ln(
)1(
)1(ln
1
1ln)ln(
10
01
0
1
1
1
10
01
0
1
1
pp
pp
p
pnA
x
pp
pp
p
pnB
M
mmM
mm
M
mm
, (2-9)
where these expressions can be simplified further to the Bernoulli case when nm = n = 1.
While, typically, one is interested in alternate hypotheses where p1 > p0, if instead the
alternate hypothesis is p0 > p1, then all above inequalities simply change direction.
Resetting SPRT charts can be considered as a sequence of SPRTs, where Equation (2-7)
or (2-9) is updated to increment the middle, left hand side, and right hand side after each
Xm data is observed and instead of stopping and deciding in favor of H0 when the lower
bound is crossed as in a conventional SPRT, the likelihood ratio now is reset to zero and
sampling continues until upper bound is crossed indicating an out of control signal.
Again, to monitor for rate decreases (with H1: p1 < p0), the direction of the above
20
inequalities, and the resetting logic reverse (likelihood ratio reset to zero when a result
falls above the right hand side), with two-sided RSPRTs working in the obvious way (i.e.
dual one-sided RSPRTs) to monitor for rate increases or decreases simultaneously.
As discussed by others (Woodall, 2006, Grigg and Farewell, 2004, Steiner et al., 2001,
Grigg et al., 2003), it is not possible to measure the performance of a RSPRT chart by
type I and type II error probabilities because the plotted values are not independent
random variates and because sampling continues until the upper bound eventually is
exceeded. The fundamental performance measures instead now are ANS or average run
length, ARL, and ANI until an out of control signal is obtained. Thus in the resetting
case, the design parameters α and β, that are used to compute the limits, no longer strictly
refer to type I and II error rates.
2.2.2. SPRTs and Resetting SPRT Charts for Non-Homogeneous Dichotomous Events
SPRTs and RSPRTs can be developed in much the same manner as above for the case
when each trial now has a unique failure probability pm,j or the trials are partitionable into
J intra-homogeneous sub-populations of size nm,j with inter-heterogeneous parameters
pm,j, j = 1, 2, …, J, m = 1, 2, …, M. In either case, the analyst may know the outcomes of
each individual Bernoulli trial, only the sub-sample counts, or only the total count. When
the number of failures in each category (the Xm,j terms) is knowable, the data may be
available as either one Xm,j value from each category at a time or all together
simultaneously. When only the total number of failures in J categories, ∑ ==
J
j jmm xT1 , , is
known Tm has J-binomial distribution as given by Equation (1-3) in Section 1.1.1. If the
data are available one category at a time, this might occur in three possible ways:
(i) In sequential order, where each category is available in a repeating order (e.g.
patient or passenger no-show rates unique for each day of the week),
21
(ii) In random order, where every category is drawn randomly (without replacement)
such that each category is sampled once before repeating (e.g. defect rates from four
manufacturing lines, sampled once per day in no defined order), or
(iii) Randomly (with replacement) where each category has a constant probability of
being selected (e.g. post-surgery infections where each patient falls into one of five risk-
categories or has a unique complication probability).
Sections 2.2.2.1 and 2.2.2.2 illustrate SPRTs for each of the above three cases for an
example with J = 4, sample sizes of 1 or 10, α = β = 0.05, A = 19, B = 0.05263, and
p0 = (0.02, 0.1, 0.25, 0.4) and p1 = (0.03, 0.15, 0.375, 0.6), where the alternate failure
rate vector p1 corresponds to a 50% increase in all of the null pj parameters.
2.2.2.1. Individual Xm,j Terms Known
a. One Bernoulli event at a time
When non-identical Bernoulli trials are observed one at a time, the SPRT can be
developed similar to Equation (2-2). Because each trial now has a unique failure
probability, the general form of the SPRT is
A
pp
pp
BM
m
xm
xm
M
m
xm
xm
mm
mm
<−
−<
∏
∏
=
−
=
−
1
100
1
111
)1()(
)1()(, (2-10)
where xm is the outcome at time m, such that xm = 1 in case of a failure and 0, otherwise,
and 0mp and 1
mp represent the unique failure probabilities of the mth Bernoulli trial under
the null and alternate hypotheses, respectively.
After taking the natural logarithm and rearranging terms, the ln(LR) after M data becomes
∑∑==
−
−+
−
−=
M
m m
mM
m mm
mmmM p
p
pp
ppxLR
10
1
110
01
1
1ln
)1(
)1(ln)ln( , (2-11)
22
where the log likelihood term is updated and tested after each Xm individual term is
observed. Table 2-1 and Figure 2-1 illustrate the application of this SPRT in the case for
which the sub-samples are drawn in random order without replacement.
Table 2-1: Tabular illustration of SPRT when Bernoulli data are sampled randomly, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null
hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of)
(a) H0 is true (b) H1 is true Category
drawn Sample number
True p
NoF ln(LR) Decision Category drawn
Sample number
True p
NoF ln(LR) Decision
2 1 0.1 0 -0.05716 c.s. 2 1 0.15 0 -0.05716 c.s. 4 2 0.4 0 -0.46262 c.s. 4 2 0.6 0 -0.46262 c.s. 3 3 0.25 0 -0.64495 c.s. 3 3 0.375 0 -0.64495 c.s. 1 4 0.02 0 -0.65520 c.s. 1 4 0.03 0 -0.65520 c.s. 4 5 0.4 1 -0.24974 c.s. 4 5 0.6 1 -0.24974 c.s. 3 6 0.25 0 -0.43206 c.s. 3 6 0.375 0 -0.43206 c.s. 1 7 0.02 0 -0.44232 c.s. 1 7 0.03 0 -0.44232 c.s. 2 8 0.1 0 -0.49947 c.s. 2 8 0.15 0 -0.49947 c.s. 2 9 0.1 0 -0.55663 c.s. 2 9 0.15 0 -0.55663 c.s. 1 10 0.02 0 -0.56689 c.s. 1 10 0.03 0 -0.56689 c.s. 4 11 0.4 1 -0.16142 c.s. 4 11 0.6 1 -0.16142 c.s. 3 12 0.25 1 0.24404 c.s. 3 12 0.375 1 0.24404 c.s. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 4 152 0.4 0 -2.90453 c.s. 3 30 0.375 1 2.77519 c.s. 1 153 0.1 0 -2.96169 DIFO H0
4 31 0.6 1 3.18066 DIFO H1
(a) (b)
Figure 2-1: Graphical illustration of SPRT when Bernoulli data are sampled randomly, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439,
ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true
-4-3-2-101234
1 11 21 31 41 51 61 71 81 91 101
111
121
131
141
151
Ln
like
liho
od
rat
io
Sample number
H0 is true
-4-3-2-101234
1 3 5 7 9 1113 15 17 19 21 2325 27 29 31
Ln
like
liho
od
rat
io
Sample number
H1 is true
23
b. One binomial event at a time
For the binomial extension with the Xm,j counts within each sample are known, the
general form of the non-homogeneous SPRT becomes
A
ppx
n
ppx
n
BM
m
xnm
xm
m
m
M
m
xnm
xm
m
m
mmm
mmm
<
−
−
<
∏
∏
=
−
=
−
1
00
1
11
)1()(
)1()(
, (2-12)
where nm and xm are the sample size and outcome at time m, and 0mp and 1
mp are the failure
probabilities of the mth binomial sample under null and alternate hypotheses, respectively.
The combination terms again cancel out, and taking the natural logarithm of LR produces
∑∑==
−
−+
−
−=
M
m m
mm
M
m mm
mmmM p
pn
pp
ppxLR
10
1
110
01
1
1ln
)1(
)1(ln)ln( . (2-13)
Table 2-2 and Figure 2-2 illustrate the application of this test for the case when sub-
samples are drawn with given probabilities and with replacement.
Table 2-2: Tabular illustration of SPRT when binomial samples are observed with given probabilities, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439,
ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of)
(a) H0 is true (b) H1 is true Category
drawn Sample number
True p NoF ln(LR) Decision Category
drawn Sample number
True p NoF ln(LR) Decision
3 1 0.25 3 -0.05986 c.s. 3 1 0.375 4 0.527931 c.s. 2 2 0.1 0 -0.63144 c.s. 2 2 0.15 0 -0.04365 c.s. 1 3 0.02 0 -0.73401 c.s. 1 3 0.03 0 -0.14622 c.s. 1 4 0.02 0 -0.83657 c.s. 1 4 0.03 0 -0.24878 c.s. 2 5 0.1 0 -1.40816 c.s. 2 5 0.15 1 -0.35774 c.s. 2 6 0.1 1 -1.51712 c.s. 2 6 0.15 1 -0.46671 c.s. 1 7 0.02 0 -1.61968 c.s. 1 7 0.03 0 -0.56927 c.s. 1 8 0.02 0 -1.72225 c.s. 1 8 0.03 0 -0.67184 c.s. 3 9 0.25 1 -2.95768 DIFO H0 3 9 0.375 4 -0.14391 c.s. 2 10 0.15 3 0.67238 c.s. 4 11 0.6 6 1.48331 c.s. 2 12 0.15 3 2.29959 c.s. 3 13 0.375 3 2.23974 c.s. 1 14 0.03 2 2.96862 DIFO H1
24
(a) (b)
Figure 2-2: Graphical illustration of SPRT when binomial samples are gathered with predefined probabilities, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05,
ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true
In some applications, data from all J categories instead may become available
simultaneously, and the test statistic and limits are updated with all J data at once before
comparison to upper and lower limits. While dealing with binomial rather than Bernoulli
samples (i.e. nm,j > 1 for at least one category or time period), the LR and ln(LR) terms
become
∏∏
∏∏
∏∏
∏∏
= =
−
= =
−
= =
= =
−
−
==
=
=M
m
J
j
xnjm
xjm
M
m
J
j
xn
jm
x
jm
M
m
J
jjmjmjmjm
M
m
J
jjmjmjmjm
Mjmjmjm
jmjmjm
pp
pp
pnxXP
pnxXP
LR
1 1
0,
0,
1 1
1,
1,
1 1
0,
0,,,
1 1
1,
1,,,
,,,
,,,
)1()(
)1()(
),|(
),|(
(2-14)
and
∑∑= =
−
−+
−
−=
M
m
J
j jm
jmjm
jmjm
jmjmjmM p
pn
pp
ppxLR
1 10
,
1,
,1,
0,
0,
1,
, 1
1ln
)1(
)1(ln)ln( , (2-15)
where xm,j and nm,j are the number of failures and the sample size for the j th binomial
random variable at time m, respectively. Again, the formulae for Bernoulli data can be
obtained by setting the sample sizes equal to 1.
-4-3-2-101234
1 2 3 4 5 6 7 8 9 10 11
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Sample number
H0 is true
-4-3-2-101234
1 2 3 4 5 6 7 8 9 101112131415
Ln
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Sample number
H1 is true
25
c. J Bernoulli events at a time
When results for all J categories are available simultaneously rather than one at a time as
above, the likelihood and the log likelihood ratios are
∏∏
∏∏
∏∏
∏∏
= =
−
= =
−
= =
= =
−
−
==
=
=M
m
J
j
xjm
xjm
M
m
J
j
xjm
xjm
M
m
J
jjmjmjm
M
m
J
jjmjmjm
Mjmjm
jmjm
pp
pp
pxXP
pxXP
LR
1 1
10,
0,
1 1
11,
1,
1 1
0,,,
1 1
1,,,
,,
,,
)1()(
)1()(
)|(
)|(
(2-16)
and
∑∑= =
−
−+
−
−=
M
m
J
j jm
jm
jmjm
jmjmjmM p
p
pp
ppxLR
1 10
,
1,
1,
0,
0,
1,
, 1
1ln
)1(
)1(ln)ln( , (2-17)
where xm,j is the j th Bernoulli outcome at time m, such that xm,j = 1 in case of a failure and
0, otherwise. The 0, jmp
and 1
, jmp terms are the failure probabilities of the item belonging
to j th category at time m, under null and alternate hypotheses, respectively. The test
statistic now is updated with all J results before comparing it to the thresholds A and B
(or their logarithms). Table 2-3 and Figure 2-3 illustrate the application of SPRT for this
case.
26
Table 2-3: Tabular illustration of SPRT when J Bernoulli data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439,
ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of)
(a) H0 is true (b) H1 is true Category
drawn Sample number
True p
NoF ln(LR) Decision Category drawn
Sample number
True p
NoF ln(LR) Decision
1 1 0.02 0 -0.010256 1 1 0.03 0 -0.010256 2 1 0.1 0 -0.067415 2 1 0.15 0 -0.067415 3 1 0.25 0 -0.249737 3 1 0.375 0 -0.249737 4 1 0.4 1 0.155729 4 1 0.6 1 0.155729
Perform the test c.s. Perform the test c.s. 1 2 0.02 0 0.145472 1 2 0.03 0 0.145472 2 2 0.1 0 0.088314 2 2 0.15 0 0.088314 3 2 0.25 0 -0.094008 3 2 0.375 0 -0.094008 4 2 0.4 0 -0.499473 4 2 0.6 0 -0.499473
Perform the test c.s. Perform the test c.s. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 1 71 0.02 0 -2.310629 1 9 0.03 0 2.427482 2 71 0.1 0 -2.367787 2 9 0.15 1 2.832947 3 71 0.25 0 -2.550109 3 9 0.375 0 2.650625 4 71 0.4 0 -2.955574 4 9 0.6 1 3.056091
Perform the test DIFO H0 Perform the test DIFO H1
(a) (b)
Figure 2-3: Graphical illustration of SPRT when all Bernoulli data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439,
ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true
-4-3-2-101234
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
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Sample number
H0 is true
-4-3-2-101234
1 2 3 4 5 6 7 8 9 10
Ln
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Sample number
H1 is true
27
d. J binomial events at a time
Similarly, if dealing with binomial rather than Bernoulli samples (i.e. nm,j > 1 for at least
one category or time period), then the LR and ln(LR) become
∏∏
∏∏
∏∏
∏∏
= =
−
= =
−
= =
= =
−
−
==
=
=M
m
J
j
xnjm
xjm
M
m
J
j
xn
jm
x
jm
M
m
J
jjmjmjmjm
M
m
J
jjmjmjmjm
Mjmjmjm
jmjmjm
pp
pp
pnxXP
pnxXP
LR
1 1
0,
0,
1 1
1,
1,
1 1
0,
0,,,
1 1
1,
1,,,
,,,
,,,
)1()(
)1()(
),|(
),|( (2-18)
and
∑∑= =
−
−+
−
−=
M
m
J
j jm
jmjm
jmjm
jmjmjmM
p
pn
pp
ppxLR
1 10
,
1,
,1,
0,
0,
1,
,1
1ln
)1(
)1(ln)ln( , (2-19)
where xm,j and nm,j are the number of failures and the size of the j th sample at time m, and
0, jmp and 1
, jmp are the failure probabilities for the j th binomial random variable under the
null and alternate hypotheses, respectively. As above, the test statistic is updated with all
J xm,j values before comparison to the appropriate thresholds. Table 2-4 and Figure 2-4
illustrate how the SPRT is conducted in this case.
Note that in all of the above scenarios, unlike the homogeneous cases, it is not
mathematically possible to isolate a cumulative sum term for simple comparison to left
and right hand thresholds. Thus technically no heterogeneous or risk-adjusted cumulative
sum exists and Equations (2-10) to (2-19) must be left in their SPRT forms.
28
Table 2-4: Tabular illustration of SPRT when all binomial data are sampled at one time H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439:
(a) null hypothesis is true, (b) alternate hypothesis is true (NoF: number of failures, c.s.: continue sampling, DIFO: decide in favor of)
(a) H0 is true (b) H1 is true Category
drawn Sample number
True p
NoF ln(LR) Decision Category drawn
Sample number
True p
NoF ln(LR) Decision
1 1 0.02 1 0.313157 1 1 0.03 1 0.313157 2 1 0.1 1 0.204196 2 1 0.15 1 0.204196 3 1 0.25 3 0.14434 3 1 0.375 3 0.14434 4 1 0.4 5 0.14434 4 1 0.6 8 2.577131
Perform the test c.s. Perform the test c.s. 1 2 0.02 0 0.041775 1 2 0.03 0 2.474566 2 2 0.1 0 -0.52981 2 2 0.15 0 1.902981 3 2 0.25 3 -0.58967 3 2 0.375 5 3.018699 4 2 0.4 5 -0.58967 4 2 0.6 7 4.640559
Perform the test c.s. Perform the test DIFO H1 . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 1 5 0.02 0 -2.60653 2 5 0.1 2 -2.25287 3 5 0.25 1 -3.4883 4 5 0.4 4 -4.29923
Perform the test DIFO H0
(a) (b)
Figure 2-4: Graphical illustration of SPRT when all binomial data are sampled at one time, H0: p = (0.02, 0.1, 0.25, 0.4), and H1: p = (0.03, 0.15, 0.375, 0.6), α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439:
(a) null hypothesis is true, (b) alternate hypothesis is true
-6
-4
-2
0
2
4
1 2 3 4 5
Ln
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Sample number
H0 is true
-4
-2
0
2
4
6
1 2
Ln
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Sample number
H1 is true
29
2.2.2.2. Only Count Statistic Tm is Known In some cases even if the Xm,j terms are knowable, the total count Tm itself may be
reported and monitored due to the setting or analyst preference (such as an overall
compliance or reliability rate). For the case when only the total count, ,1 ,∑ =
=J
j jmm XT is
known and not the individual Xm,j terms (herein termed the “pure J-binomial” case), the
likelihood ratio becomes
),,|(),,|(
),,|(),,|(
),,|(
),,|(
11
11
1
100
11
0
1
pnpn
pnpn
pn
pn
mm
mm
m
m
JtTPJtTP
JtTPJtTP
JtTP
JtTPLR
MMJBJB
MMJBJB
M
mmmJB
M
mmmJB
M =⋅⋅⋅=
=⋅⋅⋅==
=
==
∏
∏
=
= , (2-20)
where PJB(Tm = tm|nm, p0) and PJB(Tm = tm|nm, p1) are J-binomial probabilities, nm is the
sample size vector at time m, and p0 and p1 are the rate parameter vectors under the null
and alternate hypotheses, respectively. Every time a value of Tm is observed, the
corresponding probabilities under the null and alternate hypotheses are computed and
multiplied into the numerator and denominator of Equation (2-20). Note that a
cumulative sum term again cannot be isolated, so again no exact risk-adjusted cumulative
sum exists. It may be approximated, however, under some conditions by binomial or
normal based RSPRTs with ,µ σ , and P set to equate means and variances to those of
the J-binomial random variables as described below.
2.2.2.3. Approximations and Model Mis-specification
The J-binomial probability distribution given by Equation (1-3) becomes computationally
intensive as the number of categories, J, increases. This also affects the time to calculate
the likelihood ratio in Equation (2-20) to a degree that performing the SPRT or
constructing the RSPRT charts might become impractical. It is very common to model
the sum of independent random variables by normal distribution, based on the central
limit theorem, and because J-binomial distribution is a convolution of independent non-
identical binomial random variables, it might reasonably be approximated by its normal
30
counterpart. The assumption of homogeneity with a common rate parameter for all
individuals or categories, conversely, might lead to using a binomial approximation
instead of the true J-binomial distribution in the SPRT likelihood ratio. Both these
approximations, however, impact the performance of SPRTs and RSPRT charts and in
general should be avoided. To illustrate this impact, this sub-section first explains how
RSPRTs might be constructed based on normal and binomial distributions. The effects of
these model mis-specifications on the performance then are explored in detail in the
results section.
a. Normal approximation to J-binomial distribution
The J-binomial distribution might be approximated by a normal distribution with mean µ
and variance σ2 by setting
∑ =
=≡J
j jmjmmm pnTE1 ,,)(µ (2-21)
and
)1()( ,1 ,,2
jm
J
j jmjmmm ppnTV −=≡ ∑ =σ . (2-22)
The corresponding LR then can be approximated as
∏∏
∏
=
−−
−
=
−−
=
−−
=≈
M
m
TTM
m
m
M
m
T
m
M
m
T
mm
m
mm
m
mm
m
mm
m
mm
e
e
e
LR1
2
1
1
0
1
2
1
0
1
2
1
1
2
1
12
0
0
2
0
0
2
1
1
1
1σ
µ
σ
µ
σ
µ
σ
µ
σσ
σ
σ. (2-23)
Each time a sample is drawn, the ln(LR) can be updated by using the recursive equation
σ
µ−−
σ
µ−+σ−σ+= −
2
1
12
0
010
1 2
1)ln()ln()ln()ln(
M
MM
M
MMMMMM
TTLRLR , (2-24)
where µ0, µ1, σ0, and σ1 are the null and alternate Gaussian parameters obtained from the
J-binomial parameters as above.
31
b. Binomial approximation to J-binomial distribution
Similar to the normal approximation, the J-binomial distribution might be approximated
by a binomial distribution with parameters Nm and Pm, such that their expected values are
equal, i.e.
NmPm = ∑ =
J
j jmjm pn1 ,, , (2-25)
where ∑ ==
J
j jmm nN1 , and m
J
j jmjmm NpnP ∑ ==
1 ,, . The updating formula for the ln(LR)
now becomes
∑=
−
−⋅+
−
−⋅=
M
m m
mm
mm
mmmM P
PN
PP
PPTLR
10
1
10
01
1
1ln
)1(
)1(ln)ln( , (2-26)
where m
J
j jmjmm NpnP ∑ ==
1
1,,
1 and m
J
j jmjmm NpnP ∑ ==
1
0,,
0 .
2.3. Results This section investigates the performance of the risk-adjusted SPRTs and RSPRT charts.
2.3.1. Risk-adjusted SPRTs
2.3.1.1. Impact of Sampling Order For the cases where the Xm,j values are known, Table 2-5 summarizes the SPRT
performance under each sampling order described in Section 2.2.2 given a null
hypothesis of H0: p = (0.02, 0.1, 0.25, 0.4), two constant sample sizes nm,j = 1 and
nm,j = 10, error probabilities α = β = 0.05, and three alternate hypotheses varying in the
magnitude δ = p1 / p0 by which they differ from the null hypothesis; p1 = (1.1)p0,
p1 = (1.5) p0, and p1 = (1.9) p0. The notation in the third column indicates whether H0 is
true, H1 is true, or all failure rates are halfway between their null and alternate values,
i.e. p = (p1 + p0) / 2. The tabulated values are the probability of making a decision in
32
favor of the null hypothesis (P(H0)), the average number of samples (ANS), and total
number of items (ANI) until the test concludes. As shown, the manner by which the Xm,j
values are drawn does not affect any of these performance measures significantly, with
the random order (without replacement) usually resulting in only slightly smaller ANI
and ANS values.
Table 2-5: Impact of sampling order in cases for which Xm,j terms are known. PM: performance measure, ANS: average number of samples, ANI: average number of items needed until a decision is made,
P(H0): probability of concluding that H0 is true Bernoulli samples (nm,j = 1) Binomial samples (nm,j = 10)
H1 PM
True parameter
values
Sequential order
Random order (w/o
replacement)
With given prob. (with
replacement)
Sequential order
Random order (w/o
replacement)
With given prob. (with
replacement)
Cas
e I:
P1 =
(0.
022,
0.1
1,
0.27
5, 0
.44)
P(H0)
H0 is true 0.95112 0.95187 0.95161 0.95610 0.95607 0.95502
H1 is true 0.04922 0.04991 0.04912 0.04481 0.04499 0.04555
Halfway 0.49534 0.49722 0.49674 0.49663 0.49583 0.49683
ANS
H0 is true 1952.967 1958.085 1948.882 203.525 202.384 202.474
H1 is true 1930.067 1932.707 1936.302 201.651 200.868 201.230
Halfway 3204.441 3205.495 3202.544 338.143 337.476 340.215
ANI
H0 is true 1952.967 1958.085 1948.882 2035.254 2023.838 2024.743
H1 is true 1930.067 1932.707 1936.302 2016.507 2008.685 2012.301
Halfway 3204.441 3205.495 3202.544 3381.428 3374.757 3402.151
Cas
e II:
P1 =
(0.
022,
0.1
1,
0.27
5, 0
.44)
P(H0)
H0 is true 0.95733 0.95808 0.95678 0.97141 0.97086 0.97073
H1 is true 0.04294 0.04328 0.04292 0.02879 0.03011 0.02869
Halfway 0.48976 0.49116 0.48981 0.48504 0.48788 0.48613
ANS
H0 is true 88.173 86.771 87.409 11.309 10.231 10.716
H1 is true 85.148 84.474 84.903 11.041 9.977 10.397
Halfway 143.902 143.160 143.929 19.684 18.576 19.057
ANI
H0 is true 88.173 86.771 87.409 113.095 102.306 107.156
H1 is true 85.148 84.474 84.903 110.407 99.775 103.974
Halfway 143.902 143.160 143.929 143.902 143.160 143.929
Cas
e III
: P
1 = (
0.02
2, 0
.11,
0.
275,
0.4
4)
P(H0)
H0 is true 0.96054 0.96105 0.96009 0.98257 0.98208 0.98074
H1 is true 0.03761 0.03789 0.03832 0.01733 0.01693 0.01710
Halfway 0.50073 0.50243 0.50268 0.49720 0.50378 0.49746
ANS
H0 is true 28.233 27.187 27.582 4.945 3.793 4.259
H1 is true 28.289 27.280 27.661 4.841 3.639 4.068
Halfway 46.402 45.517 46.023 8.483 7.272 7.722
ANI
H0 is true 28.233 27.187 27.582 49.445 37.929 42.595
H1 is true 28.289 27.280 27.661 48.408 36.389 40.679
Halfway 46.402 45.517 46.023 84.829 72.721 77.218
Figure 2-5 compares performance under each sampling approach and alternate
hypothesis, including a fourth case for which all J of the Xm,j terms are known and
evaluated at the same time. Again note that the manner by which the data are sampled has
negligible impact. Since purely random sampling is a likely scenario in many
33
applications, the remainder of this study focuses only on this case when the Xm,j values
are observable individually. Conducting the test with all J Xm,j data simultaneously results
in higher probabilities of drawing the correct conclusion, presumably due to the larger
size of the aggregate sample, but has minor impact on ANI (in some cases slightly
increasing it). Generally, both performance measures, P(Accept H0) and ANI, improve
for greater differences between the null and alternate hypotheses, as would be expected.
(a) H0 is true (b) H1 is true
Figure 2-5: Comparison of performance under four sampling scenarios (S: sequentially, R: randomly, WGP: with given probability, JAT: J at a time) on the probability of accepting H0 and the average number
of items until a decision is made (ANI). H0: p = (0.02, 0.1, 0.25, 0.4); (i) H1: p = (1.1)p0, (ii) H1: p = (1.5)p0, (iii) H 1 : p = (1.9)p0: (a) null hypothesis is true, (b) alternate hypothesis is true
0.94
0.95
0.96
0.97
0.98
0.99
1
S R
WG
P
JAT S R
WG
P
JAT S R
WG
P
JAT
P(A
ccep
t H
0)
Sampling scenario for Xm,j terms
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
intended α = 0.050
0.01
0.02
0.03
0.04
0.05
0.06
S R
WG
P
JAT S R
WG
P
JAT S R
WG
P
JAT
P(A
ccep
t H
0)
Sampling scenario for Xm,j terms
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
intended β = 0.05
0
500
1000
1500
2000
S R
WG
P
JAT S R
WG
P
JAT S R
WG
P
JAT
AN
I
Sampling scenario for Xm,j terms
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
0
500
1000
1500
2000
S R
WG
P
JAT S R
WG
P
JAT S R
WG
P
JAT
AN
I
Sampling scenario for Xm,j terms
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
34
2.3.1.2. Impact of Design Parameters α and β Table 2-6 and Figure 2-6 summarize the impact of design parameters α and β on the
performance of the SPRT when only the total count statistic Tm is known and for the
scenario where J = 4, p0 = (0.02, 0.1, 0.25, 0.4), p1 = (0.03, 0.15, 0.375, 0.6), and nm,j = 1.
As previously, the column denoted “halfway” contains results for the case where all pm,j
terms are halfway between their null and alternate values. As shown, the binomial
approximation often leads to smaller type I and type II error rates than intended, with the
consequence of requiring more samples until the test concludes.
Table 2-6: Impact of type I and type II errors on the performance of SPRT for normal and binomial approximations versus exact J-binomial data
Design Parameters Method P(deciding in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
01.0,1.0 == βα J-binomial 0.92689 0.33349 0.00754 137.163 193.756 80.605
Normal 0.90084 0.32471 0.00996 130.119 168.627 75.001
Binomial 0.95130 0.30606 0.00213 179.318 179.372 95.236
05.0== βα J-binomial 0.96263 0.48331 0.03839 96.206 164.712 94.847
Normal 0.94276 0.47496 0.04644 91.149 142.491 86.314
Binomial 0.97772 0.45867 0.01735 124.676 240.347 116.424
1.0,01.0 == βα J-binomial 0.99262 0.63329 0.07795 80.822 194.110 133.282
Normal 0.98535 0.61972 0.08974 78.572 167.751 119.319
Binomial 0.99652 0.60844 0.04164 101.987 289.078 167.874
35
(a) (b)
Figure 2-6: Impact of type I and type II errors on the performance of SPRT (Xm,j terms are unknown), J = 4, nm,j = 1, H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6): (a) P(Accept H0),
(b) ANI: Average number of items, ANS: Average number of samples needed until a decision is made
00.10.20.30.40.50.6
0.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
αααα = 0.01, ββββ = 0.1
J-binomial exactNormal approximationBinomial approximation
intended α = 0.01
intended β = 0.1
0
10
20
30
40
50
60
70
80
0
40
80
120
160
200
240
280
320
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
AN
S
AN
I
Relative distance between H0 and H1
αααα = 0.01, ββββ = 0.1
J-binomial exactNormal approximationBinomial approximation
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
αααα = ββββ = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.050
10
20
30
40
50
60
70
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280
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
AN
S
AN
I
Relative distance between H0 and H1
αααα = ββββ = 0.05
J-binomial exactNormal approximationBinomial approximation
0
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P(A
ccep
t H
0)
Relative distance between H0 and H1
αααα = 0.1, ββββ = 0.01
J-binomial exactNormal approximationBinomial approximation
intended α = 0.1
intended β = 0.010
10
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30
40
50
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AN
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I
Relative distance between H0 and H1
αααα = 0.1, ββββ = 0.01
J-binomial exactNormal approximationBinomial approximation
36
2.3.1.3. Impact of Aggregated Data Table 2-7 and Figure 2-7 compare the performance of the SPRTs when the information
from all categories (the Xm,j terms) is available all at once versus when only the count
statistic Tm is known, for several values of α and β and the same scenario as above. Both
Table 2-7 and Figure 2-7 indicate that the design parameters α and β can significantly
impact performance. While the operating characteristic (OC) curves for J-binomial data
versus J separate simultaneous binomial events do not differ much, the time until a
decision is always slightly longer for pure J-binomial data than when the number of
failures in each category is known. This slight improvement is not unexpected, given the
analyst has more information, although it is interesting that the improvement is not more
dramatic and that the OC curves (i.e. the accuracy of the test) are essentially unaffected,
having implications on the level of data availability needed.
Table 2-7: Impact of type I and type II errors on the performance of SPRT for J separate binomial events available at a time versus exact J-binomial data
Design Parameters Method P(deciding in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
01.0,1.0 == βα Pure J-binomial 0.92689 0.33490 0.00754 137.163 193.756 80.605
J binomial events at a time 0.92580 0.33772 0.00814 131.239 184.605 76.335
05.0== βα Pure J-binomial 0.96263 0.48331 0.03839 96.206 164.712 94.847
J binomial events at a time 0.96195 0.48757 0.03853 92.516 157.185 90.143
1.0,01.0 == βα Pure J-binomial 0.99262 0.63329 0.07795 80.822 194.110 133.282
J binomial events at a time 0.99263 0.63761 0.07667 78.192 185.708 127.353
37
(a) (b)
Figure 2-7: Impact of type I and type II errors on the performance of SPRT for J separate binomial events available at a time vs. exact J-binomial data, J = 4, nm,j = 1, H0: p0 = (0.02, 0.1, 0.25, 0.4),
H1: p1 = (0.03, 0.15, 0.375, 0.6): (a) P(Accept H0), (b) ANI: Average number of items, ANS: Average number of samples needed until a decision is made
0
0.1
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P(A
ccep
t H
0)
Relative distance between H0 and H1
αααα = 0.01, ββββ = 0.1
J-binomial T knownJ Xij terms known simultaneously
intended α = 0.01
intended β = 0.1
0
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I
Relative distance between H0 and H1
α α α α = 0.01, ββββ = 0.1
J-binomial T knownJ Xij terms known simultaneously
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
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Relative distance between H0 and H1
αααα = ββββ = 0.05
J-binomial T knownJ Xij terms known simultaneously
intended α = 0.05
intended β = 0.050
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αααα = ββββ = 0.05
J-binomial T known
J Xij terms known simultaneously
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
αααα = 0.1, ββββ = 0.01
J-binomial T knownJ Xij terms known simultaneously
intended α = 0.1
intended β = 0.010
10
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I
Relative distance between H0 and H1
αααα =0.1, ββββ =0.01
J-binomial T knownJ Xij terms known simultaneously
38
2.3.1.4. Impact of Sample Size and δ
Figure 2-8 illustrates the impact of sample size and δ for the same scenarios as above and
the cases when nm,j = 1, 10, and 100. For all three alternate hypotheses, increasing the
sample size decreases both error rates but increases the ANI. As delta increases, the
observed error probabilities decrease significantly below those intended (even to zero
when H1: p = (1.9)p0, and nm,j = 100). For a fixed sample size, the SPRT is likely to
conclude earlier as the null and alternate hypotheses diverge. Furthermore, evaluating
sub-samples one at a time produces larger error rates but smaller ANI values than when
all J data are evaluated at the same time.
(a) H0 is true (b) H1 is true
Figure 2-8: Impact of sample size and δ (Xm,j terms are known,1 at a time: 1 Bernoulli or binomial event known at a time, J at a time: J Xm,j terms known simultaneously) on P(Accept H0) and the average number
of items needed until a decision is made (ANI) under null hypothesis H0: p = (0.02, 0.1, 0.25, 0.4), and alternate hypotheses (i) H1: p = (1.1)p0, (ii) H1: p = (1.5)p0 , (iii) H1: p = (1.9)p0, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439: (a) null hypothesis is true, (b) alternate hypothesis is true
1 at a time
J at a time
0.94
0.95
0.96
0.97
0.98
0.99
1
1 10 50 100 1 10 50 100 1 10 50 100
P(A
ccep
t H
o)
Sample sizes
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
intended α = 0.05
1 at a time
J at a time
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0.02
0.03
0.04
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1 10 50 100 1 10 50 100 1 10 50 100
P(A
ccep
t H
o)
Sample sizes
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
intended β = 0.05
1 at a time
J at time
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1 10 50 100 1 10 50 100 1 10 50 100
AN
I
Sample sizes
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9* p0
1 at a time
J at a time
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1 10 50 100 1 10 50 100 1 10 50 100
AN
I
Sample sizes
H1:p1 = 1.1 * p0 H1:p1 = 1.5* p0 H1:p1 = 1.9 * p0
39
Tables 2-8 to 2-10 summarize the performance when only the Tm values are known for
different sample sizes and values of δ:
(i) Case I: J = 4, p0 = (0.02, 0.1, 0.25, 0.4), p1 = (0.03, 0.15, 0.375, 0.6) (δ = 1.5),
(ii) Case II: J = 2, p0 = (0.02, 0.25), p1 = (0.03, 0.375) (δ = 1.5),
(iii) Case III: J = 2, p0 = (0.01, 0.49), p1 = (0.019, 0.931) (δ = 1.9),
In general, the normal approximation results in smaller ANIs and closer error
probabilities to the J-binomial than the binomial approximation results. Figures 2-9 to
2-11 indicate that the difference between J-binomial, normal, and binomial
approximations become more noticeable as sample sizes decrease, J decreases, and the
separation of null and alternate parameter values increases.
Table 2-8: Case I: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439,
H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), δ = 1.5)
Sample sizes Method
P(decide in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
nj = 1 J-binomial 0.96263 0.48331 0.03839 96.206 164.712 94.847
Normal 0.94276 0.47496 0.04644 91.149 142.491 86.314
Binomial 0.97772 0.45867 0.01735 124.676 240.347 116.424
nj = 10 J-binomial 0.98042 0.48001 0.02247 128.158 243.874 128.149
Normal 0.97541 0.47343 0.01957 131.876 236.990 122.406
Binomial 0.98837 0.45501 0.01101 153.541 330.144 149.434
nj = 100 J-binomial 0.99917 0.47439 0.00099 410.248 704.412 410.996
Normal 0.99915 0.45958 0.00097 410.270 703.800 410.964
Binomial 0.99915 0.42010 0.00031 418.296 808.392 411.296
40
Table 2-9: Case II: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439,
H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5)
Sample sizes Method
P(decide in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
nj = 1 J-binomial 0.95779 0.47559 0.04415 150.896 244.620 142.275
Normal 0.88400 0.46095 0.09279 97.766 127.756 90.725
Binomial 0.97282 0.46064 0.02547 185.807 332.519 171.181
nj = 10 J-binomial 0.95809 0.40538 0.02663 183.172 293.758 154.483
Normal 0.95810 0.46910 0.03444 173.552 283.878 157.305
Binomial 0.98182 0.45341 0.01781 212.303 409.488 197.432
nj = 100 J-binomial 0.98252 0.34543 0.00590 322.196 555.310 270.324
Normal 0.99013 0.45620 0.01520 296.840 582.458 292.186
Binomial 0.99344 0.43717 0.00648 324.941 702.462 313.226
Table 2-10: Case III: Impact of sample size on the performance of SPRT for normal and binomial approximations vs. exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439,
H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9)
Sample sizes Method
P(deciding in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
nj = 1 J-binomial 0.95677 0.60954 0.02196 10.258 18.060 13.052
Normal 0.87947 0.62505 0.15884 3.981 5.512 6.514
Binomial 0.99583 0.41341 0.00005 30.039 89.057 26.529
nj = 10
J-binomial 0.98518 0.63162 0.00366 23.625 38.861 22.955
Normal 0.98577 0.81217 0.00389 23.601 37.960 23.203
Binomial 0.99929 0.40544 0.00002 40.0784 134.594 40.751
nj = 100
J-binomial 1.00000 0.82186 0.00000 200.002 247.654 200.000
Normal 1.00000 0.92965 0.00000 200.000 216.868 200.000
Binomial 0.99999 0.35205 0.00000 200.018 363.620 200.000
41
(a) (b)
Figure 2-9: Case I: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439,
H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made
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P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 1, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.05
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J-binomial exactNormal approximationBinomial approximation
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
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nm,j = 10, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.050
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I
Relative distance between H0 and H1
nm,j = 10, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
00.10.20.3
0.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 100, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.050
0.5
1
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AN
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I
Relative distance between H0 and H1
nm,j = 100, α = β α = β α = β α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
42
(a) (b)
Figure 2-10: Case II: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25), H1: p = (0.03, 0.375), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items,
ANS: average number of samples needed until a decision is made
0
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P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 1, α = β α = β α = β α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended β = 0.05
intended α = 0.05
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J-binomial exactNormal approximationBinomial approximation
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t H
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Relative distance between H0 and H1
nm,j = 10, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.050246810121416182022
04080
120160200240280320360400440
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
AN
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I
Relative distance between H0 and H1
nm,j = 10, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
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P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 100, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended β = 0.05
intended α = 0.05
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Relative distance between H0 and H1
nm,j = 100, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
43
(a) (b)
Figure 2-11: Case III: Impact of sample size on the performance of SPRT for normal and binomial approximations versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439,
H0: p = (0.01, 0.49), H1: p = (0.019, 0.931), δ = 1.9): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 1, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.0505101520253035404550
0102030405060708090
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
AN
SAN
I
Relative distance between H0 and H1
nm,j = 1, αααα = ββββ = 0.05
J-binomial exactNormal approximationBinomial approximation
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 10, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
intended α = 0.05
intended β = 0.050
1
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AS
N
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I
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nm,j = 10, αααα = ββββ = 0.05
J-binomial exactBinomial approximationNormal approximation
00.10.20.30.40.50.60.70.80.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P(A
ccep
t H
0)
Relative distance between H0 and H1
nm,j = 100, αααα = ββββ = 0.05
J-binomial exactNormal approximationBinomial approximation
intend
intend
intended β = 0.05
intended α = 0.05
0
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1
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nm,j = 100, α = β = α = β = α = β = α = β = 0.05
J-binomial exactNormal approximationBinomial approximation
44
Tables 2-11 and 2-12, Figures 2-12 and 2-13 similarly compare the impact of sample size
and δ when all Xm,j terms are available simultaneously versus when only the count
statistic Tm is known. Figure 2-12 considers the scenario where J = 2, p0 = (0.02, 0.25),
and p1 = (0.03, 0.375) (δ = 1.5). Figure 2-13 is for the case where J = 2, p0 = (0.01, 0.49),
and p1 = (0.019, 0.931) (δ = 1.9). While both sampling scenarios produce nearly identical
error rates when one of the hypotheses is true, performance when the Xm,j terms are
known tends to be a bit more conservative at the “in between” state in the sense of
concluding that H0 is true with higher probability. As previously, larger separation
between the null and alternate parameter values produces smaller error rates and shorter
ANIs.
Table 2-11: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25),
H1: p = (0.03, 0.375), δ = 1.5)
Sample Sizes Method
P(deciding in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
nj = 1 Pure J-binomial 0.95779 0.47559 0.04415 150.896 244.620 142.275
J binomial events at a time 0.95782 0.47637 0.04590 149.245 242.570 140.175
nj = 10 Pure J-binomial 0.95809 0.40538 0.02663 183.172 293.758 154.483
J binomial events at a time 0.96966 0.47134 0.03201 175.887 309.309 166.790
nj = 100 Pure J-binomial 0.98252 0.34543 0.00590 322.196 555.310 270.324
J binomial events at a time 0.99044 0.46090 0.01048 296.174 581.406 292.744
Table 2-12: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49),
H1: p = (0.019, 0.931), δ = 1.9)
Sample Sizes Method
P(deciding in favor of H0) ANI
H0 true Halfway H1 true H0 true Halfway H1 true
nj = 1 Pure J-binomial 0.95677 0.60954 0.02196 10.258 18.060 13.052
J binomial events at a time 0.95895 0.61717 0.02278 10.153 17.958 13.077
nj = 10 Pure J-binomial 0.98518 0.63162 0.00366 23.625 38.861 22.955
J binomial events at a time 0.98997 0.68687 0.00422 23.323 40.111 23.517
nj = 100 Pure J-binomial 1.00000 0.82186 0.00000 200.002 247.654 200.000
J binomial events at a time 1.00000 0.87900 0.00000 200.000 228.160 200.000
45
(a) (b)
Figure 2-12: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.25),
H1: p = (0.03, 0.375), δ = 1.5): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made
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(a) (b)
Figure 2-13: Performance of SPRT for J separate binomial events available at a time versus exact J-binomial data (J = 2, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.01, 0.49),
H1: p = (0.019, 0.931), δ = 1.9): (a) P(Accept H0), (b) ANI: average number of items, ANS: average number of samples needed until a decision is made
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2.3.2. Risk-adjusted RSPRT Charts
2.3.2.1. Impact of Design Parameters α and β Table 2-13 and Figure 2-14 summarize the ARL and ANI until detecting a process
change for the case when Xm,j terms known and where H0: p = (0.02, 0.1, 0.25, 0.4) versus
H1: p = (0.03, 0.15, 0.375, 0.6) – i.e., a 50% increase in all defect rates – with J = 4,
nm,j = 10 for all sub-populations over all time periods, and assuming 3 different
combinations of α and β design parameters.
Note that in some cases, constructing the chart based on the homogeneity assumption can
significantly alter in-control and out-of-control performance. In general, assuming a
common rate parameter P for each trial or category results in larger detection times when
all Xm,j terms are known together but shorter detection times when plotting 1 sub-
population at a time, which also causes faster false alarm rates. Also note that for smaller
α it takes longer to detect an out of control signal when the process remains in statistical
control, and that for larger β it takes longer to detect a true out-of-control state, as would
be expected.
Table 2-13: Impact of design parameters α and β on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10,
H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
Design parameters Sampling Method ARL (ANI)
H0 true Halfway H1 true
01.0,1.0 == βα 1 sub-sample at a time
J-binomial 261.411 (2614.1) 33.648 (336.5) 8.558 (85.6)
Binomial 93.405 (934.0) 23.967 (239.7) 7.581 (75.8)
J sub-samples together J-binomial 107.445 (4297.8) 10.652 (426.1) 2.688 (107.5)
Binomial 183.666 (7346.6) 13.773 (550.9) 3.058 (122.3)
05.0== βα 1 sub-sample at a time
J-binomial 371.848 (3718.5) 36.576 (365.8) 10.299 (102.9)
Binomial 127.512 (1275.0) 26.096 (261.0) 9.495 (94.9)
J sub-samples together J-binomial 154.733 (6189.3) 11.593 (463.7) 3.142 (125.7)
Binomial 338.359 (13534.4) 15.243 (609.7) 3.779 (151.2)
1.0,01.0 == βα 1 sub-sample at a time
J-binomial 1545.492 (15454.9) 59.154 (591.5) 14.806 (148.0)
Binomial 572.335 (5723.3) 46.957 (469.6) 14.760 (147.6)
J sub-samples together J-binomial 649.507 (25980.3) 17.893 (715.8) 4.282 (171.3)
Binomial 1877.856 (75114.3) 23.668 (946.7) 5.200 (208.0)
48
Figure 2-14: Impact of design parameters α and β on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10,
H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
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49
Table 2-14 and Figure 2-15 summarize the impact of design parameters α and β on the
performance of RSPRT charts when only the ∑ ==
J
j jmm XT1 , terms are known for the
scenario where H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6) with
J = 4 and nm,j = 10. The normal approximation produces slightly shorter ARLs when
there is a large percent shift but much faster false alarms, whereas the binomial
approximation causes delays to detect of changes in process parameters, as also observed
in the case when the Xm,j terms are known.
Table 2-14: Impact of design parameters α and β on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4)
versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
Design parameters Method ARL (ANI)
H0 true Halfway H1 true
1.0,01.0 == βα J-binomial 658.914 (26356.6) 18.188 (727.5) 4.482 (179.3)
Binomial 1877.109 (75084.4) 23.668 (946.7) 5.151 (206.1)
Normal 493.247 (19729.9) 17.259 (690.4) 4.244 (169.8)
05.0== βα J-binomial 158.313 (6332.5) 11.833 (473.3) 3.286 (131.4)
Binomial 341.700 (13668.0) 14.823 (592.9) 3.748 (149.9)
Normal 130.978 (5239.1) 11.338 (453.5) 3. 124 (124.9)
01.0,1.0 == βα J-binomial 116.551 (4662.1) 10.979 (439.2) 2.841 (113.6)
Binomial 187.255 (7490.2) 13.454 (538.2) 3.022 (120.8)
Normal 90.053 (3602.1) 10.412 (416.5) 2.649 (105.9)
50
Figure 2-15: Impact of design parameters α and β on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, H0: p = (0.02, 0.1, 0.25, 0.4)
versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
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51
2.3.2.2. Impact of Sample Size and δ Tables 2-15 and 2-16 and Figures 2-16 and 2-17 illustrate the impact of different sample
sizes and larger separations between H0 and H1 for two different sets of hypotheses;
namely H0: p = (0.02, 0.1, 0.25, 0.4) versus the alternate hypotheses (case I) p1 = (0.03,
0.15, 0.375, 0.6) (δ = 1.5), and (case II) p1 = (0.038, 0.19, 0.475, 0.76) (δ = 1.9) both with
α = β = 0.05 and nm,j = 1, 10, and 25 when the Xm,j terms are known. Smaller ARLs and
ANIs occur for Bernoulli data (nm,j = 1) using the correct risk-adjusted approach, whereas
they are larger for nm,j = 10 or 25 when sampling from 1 sub-population at a time.
Table 2-15: Case I: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4,� α = β = 0.05,
δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
Sample size Sampling Method
ARL (ANI)
H0 true Halfway H1 true
nm,j = 1
1 sub-sample at a time J-binomial 2032.140 (2032.1) 282.650 (282.6) 88.499 (88.5)
Binomial 4281.878 (4281.9) 397.608 (397.6) 110.778 (110.8)
J sub-samples together J-binomial 613.106 (2452.0) 77.024 (308.0) 23.459 (94.0)
Binomial 1424.591 (5698.4) 111.517 (446.1) 29.622 (118.5)
nm,j = 10
1 sub-sample at a time J-binomial 371.848 (3718.5) 36.575 (365.8) 10.299 (102.9)
Binomial 127.512 (1275.0) 26.096 (261.0) 9.495 (94.9)
J sub-samples together J-binomial 154.734 (6189.3) 11.593 (463.7) 3.142 (125.7)
Binomial 338.359 (13534.4) 15.243 (609.7) 3.779 (151.2)
nm,j = 25
1 sub-sample at a time J-binomial 247.168 (6179.2) 17.998 (449.9) 4.660 (116.5)
Binomial 13.609 (340.2) 5.058 (126.4) 3.020 (75.5)
J sub-samples together J-binomial 143.477 (14347.8) 6.382 (638.2) 1.638 (163.8)
Binomial 291.827 (29182.7) 7.935 (793.5) 1.877 (187.7)
52
Table 2-16: Case II: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, α = β = 0.05, δ = 1.9,
H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length
Sample size Sampling Method
ARL (ANI)
H0 true Halfway H1 true
nm,j = 1
1 sub-sample at a time J-binomial 700.769 (700.7) 92.354 (92.4) 28.336 (28.3)
Binomial 1619.575 (1619.6) 135.346 (135.5) 36.864 (36.8)
J sub-samples together J-binomial 245.243 (980.9) 27.114 (108.5) 7.949 (31.8)
Binomial 630.008 (2520.0) 40.631 (162.5) 10.295 (41.2)
nm,j = 10
1 sub-sample at a time J-binomial 231.817 (2318.0) 15.093 (151.0) 3.708 (37.0)
Binomial 40.037 (400.4) 7.320 (73.2) 3.143 (31.4)
J sub-samples together J-binomial 154.249 (6169.9) 5.656 (226.2) 1.397 (55.9)
Binomial 395.561 (15822.5) 7.062 (282.5) 1.658 (66.3)
nm,j = 25
1 sub-sample at a time J-binomial 248.792 (6219.8) 8.716 (217.9) 2.122 (53.1)
Binomial 8.233 (205.8) 2.798 (69.9) 2.049 (51.2)
J sub-samples together J-binomial 672.567 (67256.7) 3.975 (397.5) 1.044 (104.4)
Binomial 652.478 (65247.8) 4.052 (405.2) 1.063 (106.3)
53
Figure 2-16: Case I: Impact of sample size on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4,� α = β = 0.05, δ = 1.5,
H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
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54
Figure 2-17: Case II: Impact of sample size on the performance of RSPT chart for 1 binomial event
available at a time versus J separate binomial events known simultaneously: J = 4,� α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76),
ANI: average number of items, ARL: average run length
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55
It is not surprising to observe also that as the deviation between the null and alternate
hypotheses increases, a change in the process parameters can be detected faster,
especially when the correct (heterogeneous) process distribution is used. In general,
smaller ARLs are obtained for larger sample sizes although this naturally corresponds to
larger ANIs.
Tables 2-17 and 2-18 and Figures 2-18 and 2-19 summarize the impact of sample size
and the amount of separation between the null and alternate hypotheses, δ, when only the
total count Tm is available. In general, as δ increases the binomial approximation, again,
produces larger ANIs and ARLs than the normal approximation and the correct
J-binomial distribution. Although the normal approximation and J-binomial results are
similar for large sample sizes and large δ, faster false alarm rates occur for the former
approach in most cases. Another interesting observation is that the difference in the
performance of an RSPRT chart becomes negligible as sample sizes increase, when Xm,j
terms are available, all J Xm,j terms are observed at one time, and only Tm terms are
known.
Table 2-17: Case I: Impact of sample size on the performance of RSPRT for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4)
versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
Sample size Method ARL (ANI)
H0 true Halfway H1 true
nm,j = 1
J-binomial 645.933 (2584.0) 79.846 (319.4) 24.658 (98.6)
Binomial 1424.591 (5698.4) 111.517 (446.1) 29.622 (118.5)
Normal 403.648 (1614.6) 68.120 (272.5) 22.615 (90.4)
nm,j = 10
J-binomial 158.313 (6332.5) 11.833 (473.3) 3.286 (131.4)
Binomial 341.700 (13668.0) 14.822 (592.9) 3.748 (149.9)
Normal 130.978 (5239.1) 11.337 (453.5) 3.124 (124.9)
nm,j = 25 J-binomial 146.968 (14696.8) 6.527 (652.7) 1.685 (168.5)
Binomial 296.808 (29680.9) 7.834 (783.4) 1.863 (186.3)
Normal 143.287 (14328.7) 6.437 (643.7) 1.692 (169.2)
56
Table 2-18: Case II: Impact of sample size on the performance of RSPRT for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4)
versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length
Sample size Method ARL (ANI)
H0 true Halfway H1 true
nm,j = 1
J-binomial 268.365 (1073.5) 28.517 (114.1) 8.640 (34.5)
Binomial 630.008 (2520.0) 40.631 (162.5) 10.495 (41.1)
Normal 184.179 (736.7) 25.125 (100.5) 7.981 (31.9)
nm,j = 10
J-binomial 128.656 (5146.2) 5.419 (216.8) 1.425 (56.9)
Binomial 397.720 (15908.8) 6.927 (277.1) 1.644 (65.7)
Normal 127.326 (5093.0) 5.246 (209.8) 1.423 (56.9)
nm,j = 25 J-binomial 638.530 (63853.0) 3.978 (397.8) 1.065 (106.5)
Binomial 656.352 (65635.2) 4.052 (405.2) 1.065 (106.5)
Normal 611.293 (61129.3) 3.716 (371.6) 1.062 (106.2)
57
Figure 2-18: Case I: Impact of sample size on the performance of RSPRT chart for normal and binomial approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.5, H0: p = (0.02, 0.1, 0.25, 0.4)
versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
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58
Figure 2-19: Case II: Impact of sample size on the performance of RSPRT chart for normal and binomial
approximations versus exact J-binomial data: J = 4, α = β = 0.05, δ = 1.9, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.038, 0.19, 0.475, 0.76), ANI: average number of items, ARL: average run length
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59
2.3.2.3. Impact of a Change in One or Some of the Rate Parameters
The previous sections explored the performance of SPRTs and RSPRTs for the cases
when the same amount of change is applied to all of the rate parameters. This section
investigates the impact when a change in only the smallest, largest, or a random 50% of
the rate parameters occurs. A null hypothesis of H0: p = (0.02, 0.1, 0.25, 0.4) versus the
alternate H1: p = (0.03, 0.15, 0.375, 0.6), and parameters with J = 4, α = β = 0.05, and
constant nm,j = 10 are assumed.
Table 2-19 and Figure 2-20 illustrate the impact of a shift in different rate parameters on
the performance of RSPRT chart when individual Xm,j terms are known while Table 2-20
and Figure 2-21 summarize the impact for when only the total, Tm, is available. As
expected, a shift in the smallest parameter has the least effect on ARL regardless of its
magnitude (assuming equal sample sizes), and a change in all process parameters can be
detected faster than the same percent shift in only one or half the rate parameters.
Interestingly, the impact of a particular change in the largest or random 50% of the rate
parameters, on the average, can be similar to the impact of a smaller shift in all of the
process parameters simultaneously.
The relative performance of the correct approaches versus the homogeneity assumption is
similar to the results observed in the previous sections. The SPRTs based on the correct
and the approximate approaches behave similarly for larger shifts.
60
Table 2-19: Impact of shift in different rate parameter values on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4, nm,j = 10,
α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6)
Changing rate
parameters Sampling Method
ARL (ANI)
H0 true Halfway H1 true
All
1 sub-sample at a time J-binomial 371.848 (3718.5) 36.576 (365.8) 10.299 (102.9)
Binomial 127.512 (1275.1) 26.096 (260.9) 9.495 (94.9)
J sub-samples together J-binomial 154.734 (6189.3) 11.593 (463.7) 3.142 (125.7)
Binomial 338.359 (13534.4) 15.243 (609.7) 3.779 (151.2)
Random 50%
1 sub-sample at a time J-binomial 371.848 (3718.5) 119.298 (1198.3) 59.934 (599.3)
Binomial 127.512 (1275.1) 58.906 (589.1) 34.587 (345.8)
J sub-samples together J-binomial 154.734 (6189.3) 46.927 (1877.1) 23.821 (952.8)
Binomial 338.359 (13534.4) 72.644 (2905.7) 28.248 (1129.9)
Smallest
1 sub-sample at a time J-binomial 371.848 (3718.5) 350.978 (3509.8) 328.208 (3282.1)
Binomial 127.512 (1275.1) 125.333 (1253.3) 122.674 (1226.7)
J sub-samples together J-binomial 154.734 (6189.3) 146.127 (5845.1) 138.037 (5521.5)
Binomial 338.359 (13534.4) 298.677 (11947.2) 264.234 (10569.4)
Largest
1 sub-sample at a time J-binomial 371.848 (3718.5) 80.341 (803.4) 24.362 (243.6)
Binomial 127.512 (1275.1) 44.986 (449.8) 20.316 (203.2)
J sub-samples together J-binomial 154.734 (6189.3) 28.314 (1132.5) 7.919 (316.8)
Binomial 338.359 (13534.4) 59.118 (2364.7) 14.962 (598.5)
61
(a) All change (b) Random 50% change
(c) Smallest change (d) Largest change
Figure 2-20: Impact of shift in different rate parameter values on the performance of RSPRT chart for 1 binomial event available at a time versus J separate binomial events known simultaneously: J = 4,
nm,j = 10, α = β = 0.05, H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
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62
Table 2-20: Impact of shift in different rate parameter values on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, α = β = 0.05,
H0: p = (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
Changing rate
parameters Method
ARL (ANI)
H0 true Halfway H1 true
All
J-binomial 158.313 (6332.5) 11.833 (473.3) 3.286 (131.4)
Binomial 338.360 (13534.4) 14.822 (592.9) 3.748 (149.9)
Normal 130.978 (5239.1) 11.337 (453.5) 3.124 (124.9)
Random 50%
J-binomial 158.313 (6332.5) 42.921 (1716.8) 18.878 (755.1)
Binomial 338.360 (13534.4) 72.644 (2905.7) 28.248 (1129.9)
Normal 130.978 (5239.1) 38.984 (1559.4) 17.368 (694.7)
Smallest
J-binomial 158.313 (6332.5) 142.799 (5711.9) 129.106 (5164.2)
Binomial 338.360 (13534.4) 298.677 (11947.1) 264.234 (10569.3)
Normal 130.978 (5239.1) 119.172 (4766.9) 108.365 (4334.6)
Largest
J-binomial 158.313 (6332.5) 37.445 (1497.8) 11.725 (469.0)
Binomial 338.360 (13534.4) 59.118 (2364.7) 14.961 (598.5)
Normal 130.978 (5239.1) 34.787 (1391.5) 11.400 (456.0)
(a) All change
(c) Smallest change
Figure 2-21: Impact of shift in different rate parameter values normal and binomial approximations versus the exact
H0: p = (0.02, 0.1, 0.25, 0.4) versus ANI: average number of items, ARL: average run length
2.4. Discussion
Exact risk-adjusted SPRTs and RSPRT methods are developed
processes where the data either have unique rates or stratified into
populations. Three possible ways by which data can become available
namely (i) for each individual outc
individuals that belong to different risk
statistic (sum of data coming from non
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63
(b) Random 50% change
(d) Largest change
Impact of shift in different rate parameter values on the performance of RSPRT chart for normal and binomial approximations versus the exact J-binomial data: J = 4, nm,j = 10, α
= (0.02, 0.1, 0.25, 0.4) versus H1: p = (0.03, 0.15, 0.375, 0.6), ANI: average number of items, ARL: average run length
adjusted SPRTs and RSPRT methods are developed in this chapter
processes where the data either have unique rates or stratified into homogeneous sub
populations. Three possible ways by which data can become available are considered
) for each individual outcome with unique failure rate, (ii) for a group of
individuals that belong to different risk-groups (sub-populations), or (iii) just as a count
statistic (sum of data coming from non-identical sub-populations).
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Random 50% change
Largest change
on the performance of RSPRT chart for α = β = 0.05,
in this chapter to monitor
homogeneous sub-
are considered,
) for a group of
) just as a count
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64
The analysis in the preceding sections illustrates that the manner by which data are
collected has negligible effect on false alarm rates and the amount of sampling until a test
concludes. In contrast, the sample size, specified type I and type II error rates, and
separation between the null and alternate hypotheses each can significantly impact
performance. More generally, these results underscore the importance of using the correct
distribution of heterogeneity to ensure detection of process changes in a timely manner.
Naively assuming homogeneity when data from sub-populations are available results in
more false alarms and detection delays for all cases examined. Normal and binomial
approximations when only the total count statistic Tm is available has similar impacts,
with the normal approximation producing more false alarms and the binomial
approximation causing delays in detection of shifts.
Additionally, an analyst’s selection of the separation between the null and alternate
hypotheses significantly affects ARLs and detection ability. Some care therefore should
be taken in any particular application to ensure the SPRT is designed with acceptable
ability to detect process differences of practical importance. Table 2-21 and Figure 2-22
summarize the impact of the size of δ under the scenario for which J = 4, for a given null
hypothesis H0: p = (0.02, 0.1, 0.25, 0.4) versus four different alternate hypotheses
H11: p = (0.03, 0.15, 0.375, 0.6), H1
2: p = (0.035, 0.175, 0.4375, 0.7),
H13: p = (0.04, 0.2, 0.5, 0.8), and H1
4: p = (0.045, 0.225, 0.5625, 0.9); i.e. values of δ
equal to 1.5, 1.75, 2, and 2.25, respectively. The values in the left hand column of Table
2-21 and along the horizontal axes in Figure 2-22 now are the magnitude of increase in
each pm,j term from their null values. These results similarly indicate that for the same
increase in pm,j, larger differences between the null and alternate hypotheses result in
smaller type I errors and ARLs but also significantly larger type II errors, resulting in an
inability to detect true differences.
65
Table 2-21: Impact of delta on the performance of SPRT for J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1
1: p = (0.03, 0.15, 0.375, 0.6), H1
2: p = (0.035, 0.175, 0.4375, 0.7), H13: p = (0.04, 0.2, 0.5, 0.8), H1
4: p = (0.045, 0.225, 0.5625, 0.9)): P(H0): P (Accept H0), ANI: average number of individual items, ANS: average number of samples needed
until a decision is made
Amount of increase in each pj
H11 (δδδδ = 1.5) H1
2 (δδδδ = 1.75) H13 (δδδδ = 2.0) H1
4 (δδδδ = 2.25) ANS P(H0) ANS P(H0) ANS P(H0) ANS P(H0)
0.00 3.204 0.980 1.833 0.985 1.279 0.991 1.125 0.999 0.05 3.785 0.955 2.063 0.973 1.360 0.981 1.176 0.996 0.10 4.506 0.905 2.291 0.956 1.439 0.972 1.229 0.991 0.15 5.260 0.811 2.570 0.922 1.538 0.957 1.280 0.982 0.20 5.861 0.664 2.939 0.861 1.696 0.926 1.357 0.973 0.25 6.097 0.480 3.246 0.777 1.853 0.897 1.437 0.959 0.30 5.775 0.301 3.519 0.668 1.988 0.833 1.525 0.937 0.35 5.083 0.170 3.689 0.531 2.168 0.773 1.631 0.905 0.40 4.363 0.089 3.552 0.395 2.273 0.688 1.760 0.857 0.45 3.722 0.045 3.371 0.261 2.344 0.586 1.845 0.794 0.50 3.204 0.022 3.310 0.166 2.390 0.480 1.885 0.725
Although for convenience most results presented herein assumed the same percent
increase in all J parameters, this may not be the case in some applications. Different step
sizes (δj) by risk category also might be considered in alternate hypotheses, such as to
detect small changes faster for more severe or costly strata. While there are a near infinite
number of combinations of manners by which H0 cannot be true, the performance of
RSPRT charts are explored under four cases that represent this spectrum: 100%, 50%, the
minimum and the maximum of all parameters change. Performance results suggest that a
shift in one or a few of the parameters might equate to a different percentage increase in
all parameters. For example, the ARL of a 70% increase in the largest or a random 50%
of all risk parameters might correspond to that of a 30% shift in all parameters. Different
percentage shifts also might occur in different rate parameters simultaneously. Further
analysis thus might consider each risk category separately to determine the real cause in
case of an out of control signal.
66
Figure 2-22: Impact of delta on the performance of SPRT for J-binomial data (J = 4, α = β = 0.05, ln(A) = 2.944439, ln(B) = -2.944439, H0: p = (0.02, 0.1, 0.25, 0.4), H1
1: p= (0.03, 0.15, 0.375, 0.6), H1
2: p = (0.035, 0.175, 0.4375, 0.7), H13: p = (0.04, 0.2, 0.5, 0.8), H1
4: p = (0.045, 0.225, 0.5625, 0.9)): ANI: average number of items, ANS: average number of samples needed until a decision is made
Finally, the resetting SPRT charts described in this chapter mathematically correspond to
one-sided CUSUM charts with the resetting value set to the lower limit instead of zero
(Woodall, 2006). Although herein set to Wald’s approximate lower bound β/(1-α), the
lower resetting value instead could be chosen as zero or another value, to increase
sensitivity to process shifts. While a larger lower limit would prevent a chart from
building up credit against failures (Aylin et al., 2003, Spiegelhalter et al., 2003), it also
will increase the possibility of false alarms. Choice of limits, however, should not affect
the relative performance of the methods presented here versus the conventional methods
based on homogeneity assumptions.
00.10.20.30.40.50.60.70.80.9
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67
Chapter 3 – Risk-adjusted Bernoulli and Binomial Scan Statistics
3.1. Background Although examples of clusters can be found in many areas such as genetics, sociology,
reliability, and quality control and so on (Glaz and Balakrishnan, 1999), detection of
clusters is especially important in epidemiology in terms of detecting time periods or
geographical areas with higher than expected disease rates. Spatial (geographical) or
temporal (through time) clusters exist when the occurrence rate of events in a certain part
of a study region or time period is significantly different than that of events in the rest of
that geographical area or portion of time (Jacquez et al., 1996a, Wen and Kedem, 2009).
Jacquez et al (Jacquez et al., 1996a, Jacquez et al., 1996b) present an overview of the
field of cluster detection and the basic methods to detect temporal, spatial, or spatio-
temporal (both through space and time) clusters. A detailed review of tests for spatial
randomness also is provided by Kulldorff (Kulldorff, 2006). A common assumption for
temporal cluster detection methods are that cases occur randomly in time as the null
hypothesis versus that an excess number of cases occurs in adjacent time intervals as the
alternate hypothesis. Some of the early developed temporal cluster detection methods
include empty cells test, Larsen’s test, Grimson’s method, Dat’s method, and scan
statistics (Jacquez et al., 1996b), where all but the scan test also can be used to test
clusters in multiple time series simultaneously.
Spatial cluster detection techniques seek an answer to the question of whether or not
events are distributed randomly over a geographical region. Global type of spatial cluster
detection tests investigate if there is clustering in the study region but not the specific
68
location of the cluster. Grimson’s method, Besag and Newell’s R (Besag and Newell,
1991), Cuzick and Edwards’ K nearest neighborhood method (Cuzick and Edwards,
1990), Moran’s I (Moran, 1950) are some of the more common global cluster detection
methods. Focused cluster detection tests are concerned with clusters that occur around a
focus area such as a hazardous waste site. Besag and Newell (Besag and Newell, 1991)
and Cuzick and Edwards (Cuzick and Edwards, 1990) also propose focused versions of
their tests. Local cluster detection tests focus simultaneously on determining the location
and the statistical significance of the clusters. Kulldorff’s scan statistic (Kulldorff, 1997),
which is of particular interest here, is a local cluster detection test that detects the most
likely cluster and its location and then determines its significance via Monte Carlo
estimation, as further explained in the following paragraphs and sections.
The history of scan statistics dates back to the one dimensional scan statistic, which has
been studied by Naus (Naus, 1965), Wallenstein (Wallenstein, 1980), and Glaz et al
(Glaz et al., 2001). This statistic, denoted by Sw, is the maximum number of cases
counted in a scanning window of a predefined size w as it is moved along consecutive
time periods, where data are assumed to be randomly distributed according to a uniform
or Poisson process over the study interval. The test is based on the idea that the maximum
number of cases in the scanning window will be large when there is a cluster and small
when the cases are spread randomly among the time periods. A generalized scan statistic,
proposed by Weinstock (Weinstock, 1981), captures changes in the population at risk or
the risk factors due to uncontrollable reasons such as seasonal effects on time series data.
Scan statistics have further been improved so as to detect spatial and spatio-temporal
clusters, such as Kulldorff’s scan statistic (Kulldorff, 1997), flexibly shaped spatial
(Tango and Takahashi, 2005), spatio-temporal scan statistics (Takahashi et al., 2008),
semi-parametric scan statistics (Wen and Kedem, 2009), expectation-based spatial
statistics (Neill, 2009), and multiple window discrete scan statistics (Glaz and Zhang,
2004).
Kulldorff’s model, which is one of the most popular methods of detecting temporal or
spatio-temporal clusters, is an extension of the one-dimensional local scan statistic to the
69
two or three dimensional scale. It detects spatial (spatio-temporal) clusters by scanning
through a geographic area (plus a time period) with typically a circular (or cylindrical)
window and identifying the region and window size that maximizes a likelihood ratio
(LR) test statistic. The model is first developed based on Bernoulli or Poisson processes
(Kulldorff, 1997), and since has been extended to other data types such as normal
(Kulldorff et al., 2009), exponential (Huang et al., 2006b), and ordinal (Jung et al., 2007)
which are briefly discussed in Section 3.2.1.1.
The scan method has been widely applied (Mostashari et al., 2003, Kulldorff et al., 1997,
Heffernan et al., 2004, Klassen et al., 2005, Lian et al., 2007, Sheehan et al., 2004) and
compared to other spatial and spatio-temporal cluster detection methods in epidemiology
(Hanson and Wieczorek, 2002, Ozdenerol et al., 2005, Ozonoff et al., 2005, Wheeler,
2007). Although mostly used in a restrospective context, the prospective use of scan
statistics also is recommended and discussed for surveillance in health care (Naus and
Wallenstein, 2006, Kulldorff, 2001, Joner Jr et al., 2008). Woodall et al. (Woodall et al.,
2008) provides a summary of prospective scan statistics to monitor temporal and spatio-
temporal data and a comparison of the performance measure of time to signal in
industrial processes.
While Kulldorff’s Poisson and space-time permutation models allow adjusting for
multiple categorical covariates such as age and gender, there is no risk-adjustment for the
Bernoulli model (Kulldorff, 2010). This study investigates a new approach of adjusting
Kulldorff’s scan statistic Bernoulli model for heterogeneous dichotomous data when the
population either consists of individuals each with a unique risk value, in the sense that
each Bernoulli trial has a different failure probability, or is stratified into within-
homogeneous categories of a covariate, i.e. the individuals within any sub-population are
independent and identically distributed (i.i.d.) but not identical between sub-populations.
70
3.2. Methodology
This section presents an overview of Kulldorff’s scan statistic, describes the conventional
Bernoulli model, discusses risk-adjustment in Kulldorff’s scan statistic, and proposes two
risk-adjustment approaches for the Bernoulli model.
3.2.1. Kulldorff’s Scan Statistic 3.2.1.1. Overview Kulldorff’s scan statistic detects the most possible spatial clusters by scanning over the
study region using a circular scan window. If testing for spatio-temporal clusters, the
scanning window becomes cylindrical, where the height of the cylinder is the time
dimension. For each particular coordinate in space and/or time, the scan window is
iteratively expanded to include neighboring areas (and time periods) until the maximum
size (50% of total population at-risk) is reached, and the window that maximizes LR
overall center coordinates is identified, as illustrated in Figure 3-1.
Due to the lack of a closed form for the reference distribution of this LR test statistic,
statistical significance is determined via Monte Carlo simulation, also called
“randomization testing” in the scan literature (Dwass, 1957). Given an observed total
number of cases in a study region S, data are generated under the null hypothesis of equal
rates inside and outside the scanning window and used to calculate the LR value,
repeating this calculation for a large number of replications to obtain the reference
distribution percentile. If the LR test statistic falls into the top five percent of simulated
LR values, for example, the null hypothesis is rejected with a significance of α = 0.05.
71
Radius
Size LR p-value
1 2.46 0.031
2* 2.65 0.027
3 1.12 0.171
*Maximum
Figure 3-1: Conceptual illustration of Kulldorff’s scan statistic
The Bernoulli and Poisson models of Kulldorff’s scan statistic are explained in the
following sections. Another model is developed for continuous data based on the normal
distribution (Kulldorff et al., 2009). Evaluating the statistical significance of the clusters
via a permutation based Monte Carlo simulation assures that the alpha level is maintained
and the model is still valid for non-normal distributions. However, use of this model for
survival data is not recommended since there is an exponential model for that type. An
important practical issue is to exclude outliers from the data set before the analysis since
the model is sensitive to their existence.
The exponential model (Huang et al., 2006b) is designed to analyze geographical survival
data. It is possible to test for clusters with shorter survival times as well as with longer
survival times or both. The likelihood function is computed by products of exponential
distributions where the mean survival time is estimated from the data. This method has
50%
72
been shown to perform well in analyzing continuous data coming from populations
having exponential, gamma or lognormal distributions as underlying processes. The
exponential model also enables one to distinguish between censored and uncensored data
with the use of a variable having 0/1 values and also can be adjusted for covariates.
In the ordinal model (Jung et al., 2007), every observation is considered as an individual
case. The null hypothesis is defined as the probability of a case falling into category k is
the same inside and outside the region. That is, if pj and qj are the probabilities of falling
into category j inside and outside the region, respectively, the null hypothesis is
H0: p1 = q1, p2 = q2,…, pk = qk and H1: p1/q1 ≤ p2/q2 ≤ … ≤ pk/qk , with at least one being a
strict inequality (Jung et al., 2007). The categories here, for example, may be stages of a
disease, where the first category is the lowest stage and the last category is the highest
stage of the disease. Hence, the alternate hypothesis implies that the selected area is a
cluster having higher probabilities of being in the higher risk category than the rest of the
study region. Note that the ordinal model for two categories corresponds to the Bernoulli
model as explained in detail below.
3.2.1.2. Bernoulli Model Suppose there are totally T(S) cases (events or diseases) and the size of the overall
population is N(S) in the study region S. The null hypothesis states that the cases occur
with equal probability, denoted by pS, across the whole region S; i.e. if pR is the rate
within a certain scanning window R within S and pS-R is the rate for the rest of the region
S - R, then H0: pR = pS-R = pS. Under the null hypothesis, the likelihood function of the
Bernoulli model is
)()()(0 )1( STSN
SST
S ppL −−= , (3-1)
where pS is estimated from the data using the maximum likelihood of the parameter; i.e.
Sp̂ = T(S)/N(S). The alternate hypothesis states that the probability of the occurrence of
the cases within the scanning window, R, is larger than that of the cases in the rest of the
region S-R, i.e. H1: pR > pS-R. Suppose the size of the population in the scanning window
73
R and the rest of the region S-R are N(R) and N(S-R), respectively, such that N(R) +
N(S-R) = N(S). Suppose further that there are T(R) cases in the scanning window R and
T(S-R) cases in the rest of the study region S-R, such that T(R) + T(S-R) = T(S). Then the
likelihood function under the alternate hypothesis is
)()()()()()(1 )1()1( RSTRSN
RSRST
RSRTRN
RRT
R ppppL −−−−
−−
− −−= (3-2)
such that pR > pS-R and can be estimated by the maximum likelihood estimates of pR and
pS-R, which are Rp̂ = T(R) / N(R) and RSp −ˆ = T(S-R) / N(S-R). Hence the LR becomes
)()()(
)()()()()()(
)1(
)1()1(
)),(),((
),),(),((STSN
SST
S
RSTRSNRS
RSTRS
RTRNR
RTR
S
RSR
pp
pppp
pSTSNL
ppRTRNLLR
−
−−−−
−−
−−
−
−−== (3-3)
and
)()()(
)()()()()()(
)(
)(1
)(
)(
)(
)(1
)(
)(
)(
)(1
)(
)(
STSNST
RSTRSNRSTRTRNRT
SN
ST
SN
ST
RSN
RST
RSN
RST
RN
RT
RN
RT
LR−
−−−−−
−
−−
−
−−
−
= ,(3-4)
if T(R)/N(R) > T(S-R)/N(S-R), and 1 otherwise.
3.2.1.3. Risk-adjustment in Kulldorff’s Scan Statistic
It may be necessary to adjust the model parameters for covariates when it is required to
eliminate some of the well-known causes for increased rates from the analysis of cluster
detection or the covariate is not randomly distributed geographically (Kulldorff, 2010).
For example, if the geography of traffic accidents is studied in a region in which alcohol
consumption differs in different parts of that region and the detection of clusters due to
factors other than high alcohol consumption is of interest, an adjustment for alcohol
consumption should be made. As mentioned earlier, in Kulldorff’s scan approach it is
possible to risk-adjust the underlying probability model parameters for risk factors
(covariates) known to influence occurrence rates for Poisson, space-time permutation,
74
normal, and exponential models (Kulldorff, 2010). In the Poisson model, for example, the
likelihood ratio function is given by
)()(
0
1
)]([)(
)(
)]([
)(RSTRT
RTEST
RST
RTE
RT
L
LLR
−
−−
== (3-5)
where E[T(R)] = N(R) * T(S)/N(S) (the expected number of cases in the region R under
the null hypothesis) if T(R)/N(R) > T(S-R)/N(S-R) and 1, otherwise (Kulldorff, 1997). In
this case, the expected number of cases in the study region used in Equation (3-5) can be
adjusted for each covariate j as
E[T(R)] = Σ N(R)j* [T(S)j / N(S)j], (3-6)
essentially a weighted average of the individual Poisson rates, where j is the index for the
covariate category and T(S)j/N(S)j is an estimate of the occurrence rate for category j in
the study region S, which might also be estimated via logistic regression (Kleinman et al.,
2005). In the continuous case, for example, Huang et al. (Huang et al., 2006b) estimate
the mean survival time via exponential regression and adjust each survival time based on
the estimated mean by multiplying a survival time with the ratio of the mean survival for
the highest and lowest value of the factor to eliminate the risk factors’ effects.
Adjustments with respect to individual case attributes such as age, race, year of diagnosis,
census block group, and county-level socioeconomic measures are similarly made in a
study by Klassen et al. (Klassen et al., 2005) on geographical clustering of prostate
cancer grade and stage at diagnosis. The expected number of aggressive grade and later
stage cases of prostate cancer in Maryland between years 1992 and 1997 are estimated by
a multivariate logistic regression model where the individual characteristics, such as age
and race, are the explanatory variables in one analysis and the area-level characteristics as
county-level socioeconomic measures are added to the explanatory variables in a second
risk-adjustment analysis.
This study assumes that the risk (event rate) already is known or estimated by such
statistical methods in the studies mentioned above uniquely for each individual, or the
75
population is categorized into J heterogeneous strata each with a different risk, again
estimated in a similar way.
3.2.2. Risk-adjusted Bernoulli and Binomial Scan Statistics
An important underlying assumption of Kulldorff’s Bernoulli scan statistic is that every
individual inside the scanning window has a common rate and every individual outside
the window has a different common rate (although potentially different from the within-
window rate). In fact, however, the risk might not be the same for every individual
throughout either area even if there is no clustering. Thus, in the proposed approach it is
assumed that there might be heterogeneity among the individuals within the study region
and the null and alternate hypotheses are restated accordingly as follows:
H0: The total number of cases is distributed according to a J-binomial distribution
having parameters ),...,,;,...,,;( 2121SJ
SSSJ
SS pppnnnJ through the study area S.
H1: The total number of cases in scanning window R and in the rest of the
region S-R are distributed according to J-binomial distributions with
parameters ),...,,;,...,,;( 2121RJ
RRRJ
RR pppnnnJ and ),,...,,;,...,,;( 2121RS
JRSRSRS
JRSRS pppnnnJ −−−−−−
respectively, where RSj
Rj pp −> ∀ j = 1, 2, …, J.
Figure 3-2 and Table 3-1 illustrate heterogeneous data in and outside a scanning window
according to this alternate hypothesis for any size of the scanning window. The
individuals of the population in the study region S belong to one of the J distinct
categories according to their risk of being a case. There are Sjn people in category j, with
a risk Sjp where j = 1, 2, …, J, )(
1SNn
J
j
Sj =∑ =
, and N(S) is the overall population size.
Consequently, for a scanning window, R, and the rest of the study region S-R outside the
scanning window, the number of people in each category j are Rjn and RS
jn − , and the
number of cases (events) in each category are Rjx and RS
jx − , respectively. The following
subsections develop two specific models based on this context and explain the models
proposed for the type of heterogeneity herein described.
Figure 3-2: Graphical illustration of
Table 3-1: Spatial heterogeneity in and outside the scanning window Rr
Rr nnn 2,1, ≤≤
Radius Size (r)
Inside the scanning window
Rrn 1,
Rrx 1, … R
Jrn , R
Jrx ,
1 Rn 1,1 Rx 1,1
… RJn ,1 R
Jx ,1
2 Rn 1,2 Rx 1,2
... RJn ,2 R
Jx ,2
3 Rn 1,3 Rx 1,3
... RJn ,3 R
Jx ,2
…
3.2.2.1. J-Bernoulli Model Because the risk of a case is not the same within each category
likelihood function cannot be calculated using a Bernoulli distribution. The first proposed
model calculates the LR as a product of Bernoulli probabilities as described below. If
cases are observed in each category, the likelihood function under the null hypothesis
now becomes
SJ
SS pnnnJL ;,...,,;( 1210
,;( 21SRS nnJ −
76
subsections develop two specific models based on this context and explain the models
proposed for the type of heterogeneity herein described.
Graphical illustration of spatial heterogeneity in and outside the scanning window
Spatial heterogeneity in and outside the scanning window R with radius rR
JrRr n ,3, ≤⋅⋅⋅≤ , and RS
JrRS
rRS
rRS
r nnnn −−−− ≥⋅⋅⋅≥≥≥ ,3,2,1,
Inside the scanning window Outside the scanning window
Totals RSrn −
1, RS
rx −1,
… RSJrn −
, RS
Jrx −,
N(R) T(R)
N1(R) T1(R) RSn −1,1
RSx −1,1
… RSJn −
,1 RS
Jx −,1
N2(R) T2(R) RSn −1,2
RSx −1,2
... RSJn −
,2 RS
Jx −,2
N3(R) T3(R) RSn −1,3
RSx −1,3
... RSJn −
,3 RS
Jx −,3
...
...
Because the risk of a case is not the same within each category under heterogeneity
likelihood function cannot be calculated using a Bernoulli distribution. The first proposed
as a product of Bernoulli probabilities as described below. If
cases are observed in each category, the likelihood function under the null hypothesis
Sj
Sj
Sj
xnSj
xJ
j
Sj
SJ
SS pppp−
=
−= ∏ )1()(),...,,1
21
),,...,,;,..., 212RS
JRSRSRS
JRS pppn −−−−−
),...,,;,...,,;( 2121RJ
RRRJ
RR pppnnnJ
subsections develop two specific models based on this context and explain the models
spatial heterogeneity in and outside the scanning window R
r, where
Outside the scanning window
Totals N(S-R) T(S-R)
N1(S-R) T1(S-R)
N2(S-R) T2(S-R)
N3(S-R) T3(S-R)
...
...
under heterogeneity, the
likelihood function cannot be calculated using a Bernoulli distribution. The first proposed
as a product of Bernoulli probabilities as described below. If Sjx
cases are observed in each category, the likelihood function under the null hypothesis
(3-7)
77
The alternate hypothesis of the risk adjusted scan statistic states that for a certain
scanning window R within the study region S, the risk for one category of individuals
within this window is larger than that for the corresponding category of individuals in the
rest of the space S-R. If in a specified scanning window R, there are Rjn people in
category j, each with a risk of Rjp , where j = 1, 2, …, J and )(
1RNn
J
j
Rj =∑ =
, then there
are RSjn − people in category j, each with a risk of RS
jp − , where j = 1, 2, …, J and
)(1
RSNnJ
j
RSj −=∑ =
− in the rest of the study area, S - R. If Rjx cases are observed in each
category in the scanning window R, and RSjx − cases are observed in each category in the
rest of the study area S - R, the likelihood function under the alternative hypothesis
becomes
),...,,;,...,,;,...,,;,...,,;( 212121211RS
JRSRSRS
JRSRSR
JRRR
JRR pppnnnpppnnnJL −−−−−−
RSj
RSj
RSj
Rj
Rj
Rj xnRS
j
xRSj
J
j
xnRj
xRj pppp
−−− −−−
=
− −−= ∏ )1()()1()(1
. (3-8)
Hence, the likelihood ratio is
Sj
Sj
Sj
RSj
RSj
RSj
Rj
Rj
Rj
xnSj
J
j
xSj
xnRSj
xRSj
J
j
xnRj
xRj
pp
pppp
LR−
=
−−−
=
−
−
−−
=
∏
∏−−−
)1()(
)1()()1()(
1
1
(3-9)
when ,RSj
Rj pp −> j = 1, 2, …, J, and LR = 1 otherwise. In practice, the value of this
likelihood ratio is estimated from the data using the estimates of the risks, Sj
Sj
Sj nxp =ˆ ,
Rj
Rj
Rj nxp =ˆ , and RS
jRS
jRS
j nxp −−− =ˆ .
3.2.2.2. J-binomial Model An alternate approach is based on the total number of cases across all categories, where
now the likelihood function under the null hypothesis is calculated by J-binomial
distribution accounting for different categories,
78
))(())((),...,,;,...,,;( 002121 RSMTPRMTPpppnnnJL SJ
SSSJ
SS −=== , (3-10)
where T has a J-binomial distribution and P(T = t) is a J-binomial probability. Similarly,
the likelihood function under the alternate hypothesis is
),...,,;,...,,;,...,,;,...,,;( 21212121RS
JRSRSRS
JRSRSR
JRRR
JRR pppnnnpppnnnJL −−−−−−
.))(())(( 11 RSMTPRMTP −=== (3-11)
The likelihood ratio then becomes
))(())((
))(())((
00
11
RSMTPRMTP
RSMTPRMTPLR
−==
−=== , (3-12)
where the risks again are estimated from the data as Sj
Sj
Sj nxp =ˆ , R
jRj
Rj nxp =ˆ , and
RSj
RSj
RSj nxp −−− =ˆ .
3.3. Results The performance of the proposed versus conventional methods is investigated for
different scenarios (sample sizes and parameter values) using simulated data, considering
a fixed sized scanning window in all cases. For each initial data generated, the p-value is
calculated via Monte Carlo estimation of the likelihood ratio distribution under the null
hypothesis (Dwass, 1957, Kulldorff, 1997), with the p-value equal to the probability that
the test statistic takes any value greater than or equal to the initial LR value. The
empirical p-value is found in the following way. Given the simulated total number of
cases in the whole study region S, the numbers of these cases falling inside and outside
the scanning window can be generated either by (1) randomly sampling from all possible
permutations of the cases in the study region and counting the number of cases in and
outside the scanning window (Dwass, 1957), or (2) generating the number of cases inside
the scanning window based on a hypergeometric random variate and finding the number
of cases outside the scanning window by simply subtracting the number of cases inside
from the total number of cases in the study region. The LR estimate then can be computed
79
from these values and among the R number of replications, the number of simulated LRs
greater than or equal to the value obtained using the initial “real” data is recorded. If this
number is r, then the empirical p-value is (r + 1)/(R + 1). In this study, a total of 9999
replications is used to estimate one p-value, this process then is repeated 1000 times to
find the empirical distribution and average value of p-values. The logic of the Monte
Carlo procedure is described further in the last chapter.
The power analysis herein assumes that the most likely cluster is obtained and its size and
location are known. The aim of this study is to assess if the proposed risk-adjusted
models perform any better than Kulldorff’s conventional Bernoulli model in terms of
detecting the difference in the risk-parameters inside versus outside the scanning window.
Figure 3-3 illustrates the average p-values of all three approaches while Table 3-2
summarizes the results for the scenario where J = 4, pS-R = (0.2, 0.05, 0.15, 0.35), and
three different sample sizes njR = 15, nj
S-R = 30; njR = 30, nj
S-R = 60, and njR = 60,
njS-R = 100. The zero percent difference indicates that the null hypothesis is true and the
corresponding probabilities correspond to type I error rates α = P(Reject H0|H0 is True)
whereas all other percent differences represent levels of increase in the rates inside the
scanning window and indicate that the alternate hypothesis is true. The corresponding
probabilities in this case show the power of the tests, 1 – β = P(Reject H0|H1 is True). The
probability values P(p-value < 0.01), P(p-value < 0.05), and P(p-value < 0.1) respectively
correspond to the probabilities of rejecting the null hypothesis at the levels of
significance 0.01, 0.05, and 0.1. While the J-Bernoulli model produces the most accurate
type I error values in most of the cases, it also has the worst power of all three models.
Although the J-binomial model produces slightly bigger type I error rates than the
conventional Bernoulli model, these are closer to the intended error probabilities.
J-binomial model also is the most powerful test in detecting the potential clusters. While
the relative performance of the methods does not change, the power of the tests increases,
and consequently the average p-value drops as the difference inside and outside the
scanning window or the sample sizes increase for all models, as would make intuitive
sense.
80
(a)
(b)
(c)
Figure 3-3: Average p-values (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35)) (a) njR = 15, nj
S-R = 30, (b) njR = 30,
njS-R = 60, (c) nj
R = 60, njS-R = 100
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0 10 20 30 40 50 60 70 80 90
Ave
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e p
-val
ue
Percent difference
J-binomialBernoulliJ-Bernoulli
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-val
ue
Percent difference
J-binomialBernoulliJ-Bernoulli
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-val
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Percent difference
J-binomialBernoulliJ-Bernoulli
81
Table 3-2: Error analysis (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35)) with sample sizes njR = 15, nj
S-R = 30; nj
R = 30, njS-R = 60; and nj
R = 60, njS-R = 100
Percent difference between rates in and outside the scanning region Model 0 10 20 30 40 50 60 70 80 90
n jR =
15,
njS
-R =
30
P(p-value < 0.01)
J-binomial 0.003 0.017 0.027 0.059 0.104 0.152 0.198 0.336 0.401 0.543 Bernoulli 0.002 0.009 0.014 0.033 0.075 0.115 0.145 0.276 0.339 0.456 J-Bernoulli 0.008 0.011 0.018 0.036 0.060 0.087 0.117 0.191 0.235 0.351
P(p-value < 0.05)
J-binomial 0.034 0.077 0.096 0.188 0.286 0.359 0.463 0.598 0.674 0.792 Bernoulli 0.020 0.053 0.071 0.135 0.229 0.286 0.381 0.537 0.601 0.722 J-Bernoulli 0.040 0.064 0.085 0.147 0.190 0.254 0.326 0.438 0.492 0.624
P(p-value < 0.1)
J-binomial 0.072 0.130 0.180 0.306 0.418 0.493 0.622 0.735 0.810 0.875 Bernoulli 0.051 0.100 0.137 0.252 0.348 0.422 0.530 0.672 0.737 0.826 J-Bernoulli 0.070 0.116 0.167 0.242 0.303 0.380 0.457 0.576 0.643 0.750
n jR =
30,
njS
-R =
60
P(p-value < 0.01)
J-binomial 0.009 0.025 0.064 0.126 0.223 0.372 0.546 0.712 0.806 0.9 Bernoulli 0.004 0.015 0.053 0.099 0.179 0.333 0.488 0.66 0.765 0.872 J-Bernoulli 0.011 0.021 0.038 0.061 0.116 0.205 0.344 0.494 0.615 0.737
P(p-value < 0.05)
J-binomial 0.042 0.107 0.198 0.318 0.467 0.629 0.782 0.876 0.935 0.975 Bernoulli 0.023 0.069 0.168 0.267 0.404 0.573 0.737 0.853 0.91 0.959 J-Bernoulli 0.039 0.093 0.144 0.199 0.302 0.436 0.607 0.765 0.838 0.908
P(p-value < 0.1)
J-binomial 0.096 0.184 0.317 0.449 0.607 0.755 0.87 0.932 0.971 0.991 Bernoulli 0.074 0.145 0.271 0.403 0.558 0.707 0.84 0.913 0.956 0.985 J-Bernoulli 0.092 0.17 0.229 0.333 0.472 0.591 0.735 0.853 0.907 0.956
n jR =
60,
njS
-R =
100
P(p-value < 0.01)
J-binomial 0.006 0.032 0.116 0.248 0.467 0.670 0.808 0.949 0.980 0.991 Bernoulli 0.004 0.024 0.092 0.210 0.408 0.624 0.786 0.933 0.975 0.989 J-Bernoulli 0.017 0.024 0.053 0.106 0.248 0.432 0.641 0.805 0.922 0.976
P(p-value < 0.05)
J-binomial 0.055 0.135 0.289 0.509 0.720 0.877 0.950 0.990 0.996 1.000 Bernoulli 0.040 0.111 0.253 0.464 0.681 0.850 0.938 0.984 0.996 1.000 J-Bernoulli 0.042 0.091 0.192 0.340 0.518 0.702 0.851 0.947 0.987 0.997
P(p-value < 0.1)
J-binomial 0.100 0.220 0.435 0.668 0.822 0.936 0.979 1.000 0.998 1.000 Bernoulli 0.085 0.191 0.387 0.626 0.801 0.917 0.973 0.997 0.998 1.000 J-Bernoulli 0.085 0.170 0.316 0.479 0.675 0.824 0.929 0.980 0.994 0.999
A similar analysis for the scenario in which J = 2 and pS-R = (0.0183, 0.048) illustrates
that the model based on the correct distribution of heterogeneity improves the power even
when the number of categories is as small as two. Increases in the sample size and the
percent difference of rates inside and outside the scanning window have similar effects
on the power and average p-values, as the previous scenario explored. Again, higher
sample sizes or differences in the rates produce higher test power.
82
(a)
(b)
(c)
Figure 3-4: Average p-values (J = 2, pS-R = (0.0183, 0.048)) (a) njR = 15, nj
S-R = 30, (b) njR = 30, nj
S-R = 60, (c) nj
R = 60, njS-R = 100
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-val
ues
Percent difference
J-binomialBernoulliJ-Bernoulli
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e p
-val
ue
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J-binomialBernoulliJ-Bernoulli
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-val
ue
Percent difference
J-binomialBernoulliJ-Bernoulli
83
Table 3-3: Error analysis (J = 2, pS-R = (0.0183, 0.048)) with sample sizes njR = 15, nj
S-R = 30; njR = 30,
njS-R = 60; and nj
R = 60, njS-R = 100
Percent difference between rates in and outside the scanning region Model 0 10 20 30 40 50 60 70 80 90
n jR =
15,
njS
-R =
30
P(p-value < 0.01)
J-binomial 0.005 0.007 0.007 0.023 0.025 0.033 0.044 0.053 0.081 0.102 Bernoulli 0.002 0.002 0.007 0.017 0.016 0.025 0.031 0.032 0.061 0.076 J-Bernoulli 0.005 0.005 0.008 0.021 0.020 0.029 0.045 0.047 0.076 0.071
P(p-value < 0.05)
J-binomial 0.027 0.035 0.057 0.077 0.110 0.13 0.172 0.184 0.256 0.281 Bernoulli 0.019 0.027 0.040 0.060 0.091 0.101 0.138 0.141 0.197 0.221 J-Bernoulli 0.027 0.035 0.054 0.071 0.111 0.123 0.136 0.157 0.215 0.238
P(p-value < 0.1)
J-binomial 0.052 0.070 0.106 0.140 0.189 0.22 0.290 0.294 0.383 0.412 Bernoulli 0.036 0.052 0.070 0.111 0.154 0.18 0.234 0.245 0.335 0.351 J-Bernoulli 0.058 0.075 0.108 0.139 0.189 0.201 0.244 0.278 0.342 0.371
n jR =
30,
njS
-R =
60
P(p-value < 0.01)
J-binomial 0.008 0.01 0.018 0.04 0.042 0.062 0.117 0.162 0.235 0.257 Bernoulli 0.005 0.008 0.014 0.032 0.032 0.045 0.095 0.145 0.190 0.216 J-Bernoulli 0.007 0.012 0.017 0.029 0.037 0.038 0.095 0.130 0.188 0.199
P(p-value < 0.05)
J-binomial 0.033 0.050 0.084 0.138 0.169 0.219 0.310 0.408 0.469 0.526 Bernoulli 0.026 0.031 0.061 0.112 0.136 0.184 0.262 0.342 0.413 0.457 J-Bernoulli 0.043 0.049 0.075 0.109 0.143 0.164 0.254 0.324 0.397 0.426
P(p-value < 0.1)
J-binomial 0.073 0.100 0.162 0.208 0.276 0.338 0.456 0.553 0.614 0.670 Bernoulli 0.056 0.08 0.128 0.182 0.232 0.281 0.393 0.496 0.563 0.606 J-Bernoulli 0.090 0.086 0.139 0.208 0.242 0.288 0.373 0.486 0.545 0.585
n jR =
60,
njS
-R =
100
P(p-value < 0.01)
J-binomial 0.006 0.014 0.03 0.091 0.118 0.162 0.234 0.359 0.454 0.518 Bernoulli 0.005 0.008 0.024 0.073 0.093 0.131 0.206 0.303 0.397 0.472 J-Bernoulli 0.012 0.014 0.019 0.057 0.08 0.108 0.187 0.267 0.325 0.413
P(p-value < 0.05)
J-binomial 0.036 0.071 0.155 0.216 0.31 0.394 0.511 0.634 0.726 0.786 Bernoulli 0.028 0.046 0.128 0.177 0.277 0.348 0.459 0.589 0.674 0.746 J-Bernoulli 0.041 0.08 0.109 0.179 0.246 0.308 0.424 0.546 0.634 0.699
P(p-value < 0.1)
J-binomial 0.075 0.145 0.244 0.348 0.434 0.54 0.666 0.754 0.818 0.873 Bernoulli 0.056 0.115 0.205 0.298 0.38 0.489 0.61 0.719 0.796 0.845 J-Bernoulli 0.084 0.132 0.215 0.308 0.376 0.453 0.554 0.674 0.76 0.815
Figure 3-5 further illustrates the comparative performance of the conventional versus the
J-binomial models, while displaying the empirical distributions of the p-values for the
particular scenarios pS-R = (0.2, 0.05, 0.15, 0.35) and pR = (0.2, 0.05, 0.15, 0.35) (null
hypothesis is true) versus pR = (0.375, 0.075, 0.225, 0.525) (alternate hypothesis is true).
The distributions of p-values under H0 and H1 both exhibit greater separation, and
although there is a slight increase in the type I error, an increase also is observed in the
power as the sample sizes increases for both models.
84
Conventional Model J-binomial Model n j
R =
15,
njS
-R =
30
n jR =
30,
njS
-R =
60
n jR =
60,
njS
-R =
100
Figure 3-5: Empirical distribution of p-values (J = 4, pS-R = (0.2, 0.05, 0.15, 0.35), pR = (0.375, 0.075, 0.225, 0.525)) for sample sizes nj
R = 15, njS-R = 30; nj
R = 30, njS-R = 60; and nj
R = 60, njS-R = 100
Figure 3-6 further illustrates the empirical distributions of 1000 LR test statistic values for
the conventional versus the J-binomial models when pS-R = (0.2, 0.05, 0.15, 0.35) and
pR = (0.2, 0.05, 0.15, 0.35) (null hypothesis is true) versus pR = (0.375, 0.075, 0.225,
0.525) (alternate hypothesis is true). The risk-adjusted model produces estimated LR
values with larger variability (longer and heavier right-hand tails) than the conventional
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
fre
qu
en
cie
s
p-values
Ho True
H1 True
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
fre
qu
en
cie
s
p-values
Ho True
H1 True
Max: 0.152
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
fre
qu
en
cie
s
p-values
HoTrue
H1 True
Max: 0.333
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
fre
qu
en
cie
s
p-values
Ho True
H1 True
Max: 0.372
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
Fre
qu
en
cie
s
p-values
Ho True
H1 True
Max:0.624
0
0.02
0.04
0.06
0.08
0.1
0.12
0.0
1
0.0
5
0.0
9
0.1
3
0.1
7
0.2
1
0.2
5
0.2
9
0.3
3
0.3
7
0.4
1
0.4
5
0.4
9
0.5
3
0.5
7
Re
lati
ve
Fre
qu
en
cie
s
p-values
Ho True
H1 True
Max: 0.67
α = 0.02 ± 0.01328 1-β = 0.286 ± 0.04287
α = 0.034 ± 0.01719 1-β = 0.359 ± 0.04551
α = 0.023 ± 0.01422 1-β = 0.573 ± 0.04692
α = 0.042 ± 0.01903 1-β = 0.629 ± 0.04583
α = 0.04 ± 0.01859 1-β = 0.85 ± 0.07575
α = 0.055 ± 0.02163 1-β = 0.877 ± 0.03116
85
model, both when either H1 or H0 is true. The peak at the value 1 when the null
hypothesis is true is mostly due to the rule that sets LR equal to 1 when the estimated rate
of events inside is smaller than that outside the scanning window (Kulldorff, 1997). Note
that the LRs can only take certain values due to the discrete nature of the random
variables, and the distributions of LRs under H0 and H1 again exhibit greater separation as
the sample sizes increase for both models.
Conventional Model J-binomial Model
n jR =
15,
njS
-R =
30
n jR =
30,
njS
-R =
60
n jR =
60,
njS
-R =
100
Figure 3-6: Empirical distribution of LR values for pS-R = (0.2, 0.05, 0.15, 0.35) and pR = (0.2, 0.05, 0.15, 0.35) (null hypothesis is true) versus pR = (0.375, 0.075, 0.225, 0.525) (alternate hypothesis is true).
0
100
200
300
400
500
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
0
100
200
300
400
500
600
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
0
100
200
300
400
500
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
0
100
200
300
400
500
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
0
50
100
150
200
250
300
350
400
450
500
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
0
50
100
150
200
250
300
350
400
450
500
1
1.5 2
2.5 3
3.5 4
4.5 5
5.5 6
6.5 7
7.5 8
8.5 9
9.5 10
Fre
qu
en
cy
Likelihood ratio
Ho True
H1 True
86
3.4. Discussion This study proposes and explores the performance of two alternate risk-adjusted models
for Kulldorff’s Bernoulli scan statistic in the case of heterogeneity when the population is
either stratified into J sub-populations each with a different risk or each individual has a
unique risk.
The performance study illustrates that different assumptions regarding the underlying
probability models impact the inferences obtained by Kulldorff’s scan statistic. In
particular, the proposed risk-adjusted model based on the J-binomial distribution
produces slightly inflated type I error rates than assumed (e.g. P(Reject H0|α =.05) > .05)
but larger power than the conventional non-adjusted method. The J-Bernoulli model,
conversely, has larger error rates than both the conventional Bernoulli and J-binomial
models. This power study is limited in the sense that it does not involve the scanning
feature of Kulldorff’s scan statistic (Kedem and Wen, 2007). That is, this work assumes
that the scanning window with the maximum likelihood is already given with the models
are investigated to determine if risk-adjusted methods outperform the conventional
Bernoulli model.
Timely detection of unwanted events such as spread of diseases or excess cancer or
mortality rates in a spatial area is particularly important since the cost of detection failure
is not only limited to finance but concerns lives. The implementation of the J-binomial
scan statistic model in fields such as health care, therefore, is particularly important to
increase the ability to detect clusters. The computational complexity of J-binomial
probabilities, however, can make the proposed risk-adjusted scan method impractical,
especially given the need to repeatedly compute probabilities in the likelihood ratio test
statistic. Several cumulant based orthogonal polynomial expansions and SPA are
therefore investigated in the following chapter. As described, a normalized Gram-
Charlier expansion (NGCE) produces fast and accurate estimates and can be used in the
MC test to calculate likelihood ratios.
87
Chapter 4 – Approximating J-binomial Distributions
4.1. Background The problem of estimating probabilities when the underlying PDF of a statistic is not
obvious or calculation of a convolution is not straightforward has been studied
extensively in the literature. General approaches for approximating convolutions include
integral transforms such as Laplace and Fourier transforms (Davies, 1973, Schlenker,
1986), orthogonal polynomial expansions (Badinelli, 1996), cumulant based
approximations using orthogonal polynomials (Zhang and Lee, 2004, Jorgensen, 1991,
Tian et al., 1989, Stremel and Rau, 1979), and saddle point approximations (Daniels,
1954, Guotis and Casella, 1999, Rubin and Zidek, 1965, Matis and Guardiola, 2006).
Numerical inversion of probability or moment generating functions (Abate and Whitt,
1992) is another possible fast and accurate approach. Cumulant based expansions using
orthogonal polynomials and saddle point approximations (SPA) are chosen as the foci of
the present study because other methods either are more suitable for continuous PDFs or
often are computationally difficult. A brief summary of cumulant based orthogonal
polynomial approximations and SPA is provided below to provide context for their
application to J-binomial probabilities.
Piecewise approximation of PDFs using orthogonal polynomial expansions is discussed
by Badinelli (Badinelli, 1996). The standard normal, beta, and gamma densities and
expected values and convolutions of these densities are approximated using the
orthogonal polynomials, Tchebyshev-Type I, Tchebyshev-Type II, Legendre, Hermite,
and Laguerre. Badinelli’s study leads to the conclusions that no particular choice of one
polynomial is better than the others in all cases. Badinelli also shows that whereas an
88
approximation of low order may not be sufficient to capture the complexity of the shape
of the function to be approximated, higher order polynomials might be volatile. The order
of the approximation, therefore, is investigated carefully in this study.
A different approach to approximating a PDF can be achieved by using both the
orthogonality property of certain polynomials and the statistical properties of cumulants
of random variables. Although not well-known in many disciplines, the cumulant based
method of probability approximation has been applied to a variety of problems. Examples
include estimating the power flow, loss of load probabilities, or capacity outage density
functions in electric power and energy systems (Zhang and Lee, 2004, Jorgensen, 1991,
Tian et al., 1989, Schellenberg et al., 2005, Stremel and Rau, 1979).
Gram-Charlier expansion (GCE) - which is an approximation using Hermite polynomials
as described in Section 4.2.1- of order 5 is used by Stremel and Rau (Stremel and Rau,
1979) to estimate the loss of load probability of a system with 49 units having different
capacities and availability probabilities. In terms of both accuracy and speed, the GCE
method is found to perform better than the Calabrese method, which expresses the loss of
load probabilities as the fraction of total time periods the system load may be expected to
exceed the available capacity (AIEE Committee, 1961, Calabrese, 1950). Zhang and Lee
(Zhang and Lee, 2004) also compute the transmission line flow probability and
cumulative distribution functions via GCE. In comparisons of GCE of orders 3 to 9 with
Monte Carlo Estimation (MCE), the GCE method is much faster (e.g. GCE of order 7
takes 12.08 CPU seconds while MCE with 753 iterations takes 203.44 CPU seconds and
with 5000 iterations takes 941.59 CPU seconds). Although GCE of order 6 produces the
best approximate cumulative distribution function curve, order 7 is recommended since it
provides better tail probability estimations. As also explained in Section 4.3.1, order 6 is
preferred since orders 6 and 7 have the same accuracy level and order 6 is sufficient to
capture the complexity of the J-binomial distribution. Schellenberg et al. (Schellenberg et
al., 2005) compare GCE to MCE in estimating the optimal power flow with Gaussian and
Gamma distributions. GCE is found to provide a substantial reduction in the computation
while highly increasing the level of accuracy. Another application is described in
89
communications by Nazarathy (Nazarathy, 2006), in which the PDF and CDF of optical
communication systems are estimated by GCE.
Although GCE - constructed based on Hermite polynomials - is the most commonly used
method, cumulant based expansions using other orthogonal polynomials also have been
investigated in the literature. Jorgensen (Jorgensen, 1991) uses Legendre polynomials to
approximate the distribution function of load and equivalent load in electric power
industry, whereas Tian et al (Tian et al., 1989) develop a cumulant based approximation
using Laguerre polynomials to estimate the PDF of load duration. The study by Jorgensen
claims the expansion with Legendre polynomials is expected to be superior to GCE
methods, the latter of which would require many terms to capture the multi-modality of
the electric system. However, a performance comparison is not provided. The method
developed by Tian et al., on the other hand, is shown to produce more accurate results
than the GCE approach. The common conclusion of all studies mentioned here is that
cumulant based orthogonal polynomial approximations provide accurate results and are
more time-efficient than MCE.
An alternate method of approximating probability densities is to use saddle point
approximations (SPA), developed by Daniels (Daniels, 1954). The author argues that
although it is possible to approximate a probability density by using GCE type formulas
or inverse transformation of characteristic functions of densities, the former might not
give satisfactory results in the tails and the latter may not be analytically easy to achieve.
Reid (Reid, 1988) reviews statistical applications of SPA in the literature, while Guotis
and Casella (Guotis and Casella, 1999) provide a general explanation of the method.
Huzurbazar (Huzurbazar, 1999) illustrates univariate and conditional saddle-point density
and distribution function approximations. Giles (Giles, 2001) finds SPA to perform well
especially in the lower tail of the distribution of Anderson-Darling goodness-of-fit test
statistic. Lugannani and Rice (Lugannani and Rice, 1980) develop a SPA formula based
on normal distribution that does not require integration to find the tail probabilities,
whereas Wood, et al. (Wood et al., 1993) present a generalized version of the Lugannani-
90
Rice formula for an arbitrary base distribution instead of normal. Rubin and Zidek (Rubin
and Zidek, 1965) develop SPAs both (1) to calculate the distribution and moment
generating function of sums of independent Chi random variables, and (2) compare to
Edgeworth series expansion and Cramer approximations. SPA is found to perform better
than the former two techniques. Matis and Guardiola (Matis and Guardiola, 2006)
develop truncated saddle point approximations (TSPA) to estimate small tail probabilities
of convoluted Bernoulli, normal, and exponential random variables and claim to achieve
satisfactory results while estimating cumulative tail probabilities. Other recent
applications of SPA include geographical (Tiefelsdorf, 2002), financial (Gordy, 2002),
and reliability analyses (Huang et al., 2006a, Du, 2008, Du, 2010, Yuen et al., 2007, Du
and Sudjianto, 2004).
4.2. Methodology
4.2.1. Cumulant Based Approximation Using Orthogonal Polynomials Let Ψ be any set of orthogonal polynomials, ...},,,{ 21 ψψ=Ψ where a polynomial of
order r of a variable z can be expressed as
∑=
=r
i
iir zz
0
,)( πψ (4-1)
where iπ is the i th coefficient. The orthogonality property specifies that the inner product
of any two polynomials is equal to their normalization constant only when the two
polynomials are the same and zero when they are different. That is, for arbitrary mψ and
rψ ,
=
=∫ ..,0
,)()()(
wo
rmifξdzzωzz r
z
rm ψψ , (4-2)
91
where )(zω and rξ respectively are the weighting function and normalization constant,
both of which are specific to the polynomial used (Badinelli, 1996). A function f(z) then
can be approximated by a series expansion of orthogonal polynomials as
)()()(0
zωzCzfr
rr∑∞
=
= ψ , (4-3)
where the rC coefficients in the polynomial expansion are determined by using the
orthogonality property as follows (Badinelli, 1996). Multiplying both sides of Equation
(4-3) with )(zmψ and integrating over the domain of the polynomial )(zrψ produces
rrr z
mrr
z
m ξCdzzωzzCdzzzf ∑ ∫∫∞
=
==0
)()()()()( ψψψ , (4-4)
where the summation terms in the middle expression equal rr ξC when rm = , and 0
otherwise, and hence
∫=z
rr
r dzzzfξ
C )()(1
ψ . (4-5)
Truncation of the infinite series in Equation (4-3) after R polynomial terms yields an
approximation “of order R”
)()()(0
zωzCzfR
r rr∑ =≈ ψ . (4-6)
This method herein is illustrated using Hermite polynomials, the rth order of which are
defined as
2222
)1()(Hz
r
rzrr e
dz
dez
−−= , ∞<<∞− z . (4-7)
The first few orders of Hermite polynomials are listed in the left hand side of Table 4-1.
The orthogonality property of these polynomials is defined as (Cramer, 1963, Badinelli,
1996)
92
mr
z
z
rm Irdzezz π2!)(H)(H 22
=∫−
, (4-8)
i.e. Hermite polynomials are orthogonal with respect to the weighting function
)(zω = 22z
e−
and their normalization function is mrIr π2! , where 1=mrI if m = r, or 0
otherwise. The approximation then can be written as
2
00
2
)(H)()(H)(z
rrr
rrr ezCzzCzf
−∞
=
∞
=∑∑ =≈ ω , (4-9)
where
∫=z
rr dzzzfr
C )(H)(2!
1
π (4-10)
from Equation (4-5). Given the functional form of Hr(z), the coefficients can be expressed
in terms of the moments of the variable z as shown in the right hand side of Table 4-1.
Table 4-1: First six Hermite polynomials and expansion coefficients in terms of moments µi of random variable z
Hermite polynomial, order r Coefficients
1)(H0 =z ππ 2
1)(
2!0
10 == ∫
∞
∞−
dzzfC ,
zz =)(H1 112
1)(
2!1
1µ
ππ== ∫
∞
∞−
dzzzfC
1)(H 22 −= zz ]1[
22
1)()1(
2!2
12
22 −=−= ∫
∞
∞−
µππ
dzzfzC
zzz 3)(H 33 −= ]3[
26
1)()3(
2!3
13
33 µµ
ππ−=−= ∫
∞
∞−
dzzfzzC
36)(H 244 +−= zzz ]36[
224
1)()6(
2!4
124
244 +−=−= ∫
∞
∞−
µµππ
dzzfzzC
zzzz 1510)(H 355 +−= ]1510[
2120
1)()1510(
2!5
1135
355 µµµ
ππ+−=+−= ∫
∞
∞−
dzzfzzzC
154515)(H 2466 −+−= zzzz ]154515[
2720
1)()154515(
2!6
1246
2466 −+−=−+−= ∫
∞
∞−
µµµππ
dzzfzzzC
93
Standardization of the variable being approximated might be required, in such a way that
the domain of the function maps onto the domain of the polynomial in the expansion.
Hermite polynomials cover the entire real number domain ),( ∞−∞ , the random variable
T, therefore, is standardized as t
tTZ
σµ−
= . The left hand side of Table 4-2 summarizes
the first few moments of the standardized variable Z and their relation to the cumulants of
the random variable T, and the right hand side summarizes the coefficients of the Hermite
approximations in terms of these cumulants. Similar to how moments are obtained from
moment generating functions, the cumulants of T can be computed by taking consecutive
derivatives of its cumulant generating function (CGF) KT(s), which is the log of the
moment generating function MT(s), namely KT(s) = ln(MT(s)), with respect to s and then
setting s = 0.
Table 4-2: First six moments of random variable z and Hermite expansion coefficients in terms of cumulants
iK of random variable T
Moments Coefficient
0)(0 =zµ π
=2
10C
0)(
)(1 =−
=σ
µµ
TEz 01 =C
1
)()(
22
2
2
2 ==−
=σσ
µµ
KTEz 02 =C
33
3
3
3
)()(
σσµ
µKTE
z =−
= 33
3 6
1
2
1
σπ=
KC
4
224
4
4
4
3)()(
σσµ
µKKXE
z+
=−
=
−
σ
+
π= 9
3
24
1
2
14
224
4
KKC
5235
5
5
5
10)()(
σσµ
µKKKXE
z+
=−
=
σ
−σ
+
π=
33
5235
5 1010
120
1
2
1 KKKKC
6
32
23246
6
6
6
151015)()(
σσµ
µKKKKKXE
z+++
=−
=
+
σ
+−
σ
+++
π= 30
315
151015
720
1
2
14
224
6
32
23246
6
KKKKKKKC
The multiplication of the coefficients with the weighting function ω(z) = 22z
e−
of the
Hermite polynomials in the expansion Equation (4-1) produces the standard normal
probability density function )(zφ (Cramer, 1963, Kendall et al., 1987). Combining the
94
terms and assuming that the remaining are negligible, the series expansion up to the 6th
order using the first six cumulants then is
.)(H10
720
1)(H
120
1)(H
24
1)(H
6
11)()( 6
2
33
66
555
444
333
+++++≈ z
KKz
Kz
Kz
Kzφzf
σσσσσ(4-11)
Equation (4-11) often is called the Gram-Charlier expansion (Cramer, 1963, Kendall et
al., 1987).
Even when no such standardization of a variable is necessary, as in Laguerre polynomial
expansions, or if it is easier to estimate the cumulants from the data, one might need to
use the following recursive relationship to moments to compute the coefficients in the
approximation (Cramer, 1963, Tian et al., 1989)
∑−
=−+=µ
1
1
k
iikikk KµK . (4-12)
To approximate the J-binomial PDF using GCE, one first needs to calculate the
cumulants of a J-binomial random variable. In general, the CGF of any sum of random
variables is equal to the sum of the CGFs of the random variables in the convolution (the
log results in summations rather than the products). Since the CGF of a binomial random
variable X is
)1ln()1ln()(ln)( snsXX pepnpepsMsK +−=+−== , (4-13)
the CGF of a J-binomial random variable T then is
∑∏==
+−=
+−==
J
j
sjjj
J
j
nsjjTT eppneppsMsK j
11
)1ln(])1[(ln)(ln)( . (4-14)
The cumulants of a J-binomial random variable can be calculated by taking the
consecutive derivatives of the CGF in Equation (4-14), or similar to finding the moments
of a convolution, by simply adding the cumulants of the binomial random variables in the
95
convolution (Kendall et al., 1987). The first few J-binomial cumulants are summarized in
Table 4-3.
Table 4-3: First six cumulants of J-binomial distribution
∑=
=J
jjj pnK
11
)1(1
2 j
J
jjj ppnK −= ∑
=
∑=
+−=J
jjjjj pppnK
1
323 )23(
∑=
−+−=J
jjjjjj ppppnK
1
4324 )6127(
∑=
+−+−=J
jjjjjjj pppppnK
1
54325 )24605015(
∑=
−+−+−=J
jjjjjjjj ppppppnK
1
654326 )12036039018031(
To illustrate the GCE method, consider the scenario where J = 3, n = (5, 3, 7), p = (0.05,
0.25, 0.15), and P(T = 4) is of interest. For an approximation of order 6, the first six
cumulants of the J-binomial distribution are found as
K1 = µ = 5*(0.05) + 3*(0.25) + 7*(0.15) = 2.05,
K2 = σ2 = 5*(0.05)*(0.95) + 3*(0.25)*(0.75) + 7*(0.15)*(0.85) = 1.6925,
and similarly K3 = 1.11975, K4 = 0.309237, K5 = -0.590768, and K6 = -1.045418. When
t = 4, the corresponding standardized variable is z = (t – µt)/σt = (4 - 2.05) / 1.30 =
1.49889, with which the first six Hermite polynomials can be evaluated as H0(z) = 1,
H1(z) = z = 1.498890, H2(z) = z2 - 1 = 1.246670, H3(z) = z3 - 3z = -1.129150 and similarly
H4(z) = -5.432504, H5(z) = -3.626136, and H6(z) = 21.727335. Using these results and
Equation (4-3) the GCE for P(T = 4) is f(4) ≈ 0.124049. As explained in Section 4.3.1,
one first needs to calculate f(t) values for all values of T to then compute an improved
normalized probability estimate for any given t.
96
4.2.2. Saddle Point Approximations Guotis and Casella (Guotis and Casella, 1999) provide a general explanation of saddle
point PDF approximations using inverse Fourier transforms and Edgeworth expansions,
the first pursued below. Given the moment generating function ,)()( dttfesM stT ∫
∞
∞−=
provided that it is finite for real s in some open neighborhood of the origin (Guotis and
Casella, 1999), the PDF f(t) can be approximated from MT(s) using the Fourier inversion
formula as
dsei
dsedseisMtfi
i
stsKistisKistT
TT ∫∫∫∞+
∞−
−∞
∞−
−∞
∞−
− ===τ
τπππ])([])([
2
1
2
1)(
2
1)( , (4-15)
where i is the complex variable 1− and KT(s) = ln(MT(s)) is the CGF of the random
variable T. The path of the integral can be parallel to the imaginary axis in a
neighborhood of zero where MT(s) exists. It is chosen to pass through a saddlepoint of the
integrand; i.e. τ is chosen as the saddlepoint such that the integrand is negligible outside
its immediate neighborhood (Daniels, 1954). In Equation (4-15) this is the point that
equates the derivative of the exponent stsKT −)( with respect to s to zero (Daniels, 1954,
Giles, 2001). In other words, the saddle point is defined as a point which is neither a
maximum nor a minimum, but for which the function stsKT −)( is constant in the
imaginary direction and has an extrema in the real direction (Guotis and Casella, 1999).
This point, herein denoted by s*(t), is found by solving the equality
0)( =−′ tsKT (4-16)
for s, where )(sKT′ is the first derivative of the CGF. The saddle point s*(t), thus, is a
function of t and a new saddle point value is obtained for every value t.
By expanding the exponent around the saddle point and ignoring the higher order terms,
one can write
))((2
))(()())(()( *
2*** tsK
tssttstsKstsK TTT ′′
−+−≈− , (4-17)
97
where ))(( * tsKT′′ is the second order derivative of the CGF of T evaluated at the saddle
point s*(t). Defining s = s*(t) + iy, and expanding the integrand in Equation (4-5)
produces
∫∞+
∞−
′′−
+−≈
its
its
tsKtss
ttstsKdse
itf
TT)(
)(
))((2
)()())((
*
*
**
**
2
1)(
π (4-18)
≈ ∫
∞
∞−
′′−− dyetsK
tsKe
tsKy
T
T
ttstsK TT
))((2
*"
*"
})())(({*
2
**
2
))((
))((
2
2
1
π
ππ
. (4-19)
Due to the normal kernel in the integral, the right hand side of Equation (4-19) equals to 1
and the final saddle point approximation of a PDF in general is
})())(({
*
**
))((2
1)( ttstsK
T
TetsK
tf −
′′≈
π. (4-20)
It is not possible to obtain a closed form saddle point approximation expression for the
J-binomial distribution, since the cumulant generating function for J-binomial
distribution is
)ln()(1
sjj
J
jjT epqnsK += ∑
=
, (4-21)
and the equality
0)(1
=−+
=−′ ∑=
stepq
epnstsK
sjj
sj
J
jjT
(4-22)
cannot be solved analytically for s. A numerical solution such as the Newton-Raphson or
secant method, therefore, is needed to find the saddle point. The Newton-Rhapson
method finds the solution to g(s) = 0, by replacing the function g(s) -here the very left
hand side of Equation (4-22)- with its tangent line approximation at the point sn (Bradie,
2006, Pozrikidis, 2008) as
))(()()( nnn sssgsgsg −′+= , (4-23)
98
and then taking the x-intercept of the tangent line as the next approximation, sn+1, to the
root s. At the nth iteration, after setting g(s) = 0, the value sn+1 is found as
)(
)(1
n
nnn sg
sgss
′−=+ . (4-24)
The algorithm stops when the absolute value of the difference (sn+1 – sn) is less than a
tolerance value. The convergence of the Newton-Raphson method is stated by the
following theorem (Bradie, 2006). If the function g(s) is twice continuously differentiable
on the interval [a, b] with s* ∈ (a, b) and that ,0)( * ≠′ sg then there exists a value δ > 0
such that for any s0 ∈ I = [s*- δ, s*+ δ], the sequence generated by Newton-Rhapson
methods converges to s*. This theorem implies that the method converges for any starting
point s0∈I, i.e. guarantees the existence of a δ, which, however, can be very small. The
convergence of Newton-Rhapson method, thus, is highly dependent on the choice of this
starting value s0. As an alternative, the secant method approximates the derivative by the
slope of the straight (secant) line passing through two consecutive points in the Newton-
Rhapson algorithm (Thisted, 1988, Pozrikidis, 2008, Bradie, 2006),
1
1)()()(
−
−
−
−≈′
nn
nnn ss
sgsgsg . (4-25)
The recurrence relation of the Newton-Rhapson method given by Equation (4-24) then
can be approximated as
)()()(
1
11
−
−+ −
−−≈
nn
nnnnn sgsg
sssgss . (4-26)
While the secant method does not require calculation of the derivative )(sg ′ , which can
be difficult in some cases, it now requires two initial values instead of one. Similar to
Newton-Rhapson, the secant method also is very sensitive to these initial values, both of
which have to be sufficiently close to the true root of the function for the algorithm to
converge. Both methods can be used to find the saddle point in the SPA for J-binomial
distribution.
99
The results obtained in this study are compared to the truncated SPA (TSPA) of 3rd order
by Matis and Guardiola (Matis and Guardiola, 2006) which proposes an alternate way of
expressing the CGF in its Rth order truncated form, ∑ ==
R
r
rrR
T r
sKsK
1 !)( , in order to avoid
long or intractable saddle point expressions. Since the derivative of the expansion of a
CGF truncated at order R = 3 has the form K1 + K2s + K3s2/2, it is possible to find an
analytical solution to the equation 0)( =−′ stsKT using the quadratic formula
3
13222 )(2
K
tKKKK −−− µ . For orders above 3, however, a numerical method again is
needed to find the saddle point. For this reason, a TSPA of order 3 is used for
performance comparisons.
Matis and Guardiola (Matis and Guardiola, 2006) treat the SPA as an estimator for a
continuous probability density function and find the cumulative tail probabilities P(T > t)
by numerical integration. Based on this approach J-binomial probabilities are further
estimated by using continuity correction such that P(T = t) = P(t - 0.5 < T < t + 0.5),
where the interval probabilities are computed using numerical integration and accuracy is
compared to probability estimates found using full SPAs and NGCE. A Mathematica
algorithm provided by Matis and Guardiola (Matis and Guardiola, 2006) is used to
compute the continuity correction estimates.
The SPA computation is illustrated for the same earlier example in Section 4.2.1 where
J = 3, n = (5, 3, 7), and p = (0.05, 0.25, 0.15). To estimate P(T = 4), first Equation (4-22)
is evaluated as follows,
Solving this equality by the secant method produces the saddle point s* = 0.884069.
Evaluating the J-binomial distribution CGF )log()(1
sjj
J
jjT epqnsK += ∑
=
and its second
0 415 .085.0
15. 07
25 .075.025. 0
3 05 .095.0
05. 0544*
1
=−+
++
++
=−+
= − ′ ∑ =
se
e
e e
e e
se p q
epn s(s)K
s
s
s
s
s
s
sjj
sj
J
jjT
100
derivative )(sKT′′ at this saddle point produces 2.067251 and 2.710721, respectively.
P(T = 4) then is estimated as
,095697.0710721.2
394398.0
}4*884069.0)884069.0(exp{)884069.0(2
1)4(
=
=
−′′
≈ T
T
KK
fπ
which is accurate to the second digit (the actual probability is 0.093736).
4.3. Results Two performance measures are used to compare the accuracy of different approximation
methods, a modified Kullback-Leibler (M-KL) statistic that totals weighted ratio
comparison of all probabilities,
∑=
=
===−
N
t App
JBJB tTP
tTPtTPKLM
0 )(
)(log)( (4-27)
and the total absolute deviation (TAD) between the exact and approximate PDFs,
∑=
=−==N
tAppJB tTPtTPTAD
0
,)()( (4-28)
where PJB(T = t) and PApp(T = t) denote the exact J-binomial probability and whichever
approximation is being assessed, respectively. In both cases, a result closer to zero
indicates a better overall agreement.
4.3.1. Orthogonal Polynomial Expansions The accuracy of cumulant based expansions for six orthogonal polynomials - Hermite,
Laguerre, Legendre, Shifted Legendre, Tchebyshev Type I and Type II - up to order 14
101
initially are compared to the exact J-binomial probabilities for the 4 cases summarized in
Table 4-4. Properties of each polynomial are given in Table 4-5. Approximations using
Legendre, Shifted Legendre, Tchebyshev Type I and Type II polynomials often produced
negative values and are eliminated from further detailed analysis. The accuracy of the
approximations with Laguerre and Hermite polynomials are investigated in further detail
and the results are presented in the rest of this sub-section.
Table 4-4: J-binomial distributions used for approximation analysis
Test Cases Parameters (J; nj; pj), j = 1, 2, …, J
1 J = 4; n = (3, 6, 2, 7); p = (0.016, 0.071, 0.093, 0.035)
2 J = 4; n = (26, 40, 18, 16); p = (0.04, 0.084, 0.056, 0.025)
3 J = 10; n = (12, 14, 4, 2, 20, 17, 11, 1, 8, 11); p = (0.074, 0.039, 0.095, 0.039, 0.053, 0.043, 0.067, 0.018, 0.099, 0.045)
4 J = 15; nj = 837
102
Table 4-5: Properties of orthogonal polynomials
Type of Polynomial First Few Polynomials Orthogonality Property
Transformation of the Random
Variable
Hermite
36)(
3)(
1)(
)(
1)(
244
33
22
1
0
+−=
−=
−=
=
=
zzzH
zzzH
zzH
zzH
zH
=
=
=∫∞
∞−
−
..,0
,1
2!)()( 22
wo
rnifIwhere
IndzezHzH
nr
nr
z
rn π
X
XXZ
σµ−
=
Laguerre
6
)6189()(
2
)24()(
1)(
1)(
23
3
2
2
1
0
+−+−=
+−=
+−=
=
xxxxL
xxxL
xxL
xL
=
=
=∫∞
−
..,0
,1
)()(0
wo
rnifIwhere
IdxexLxL
nr
nrx
rn
None
Shifted Legendre
1209014070)(
133020)(
166)(
12)(
1)(
2344
233
22
1
0
+−+−=
−+−=
+−=
−=
=
xxxxxL
xxxxL
xxxL
xxL
xL
=
=
+=∫ −
..,0
,1
12
1)()(
1
0
wo
rnifIwhere
In
dxexLxL
nr
nrx
rn
maxX
XZ =
Legendre
2
)35()(
2
)13()(
)(
1)(
3
3
2
2
1
0
xxxL
xxL
xxL
xL
−=
−=
=
=
=
=
+=∫
−
−
..,0
,1
12
2)()(
1
1
wo
rnifIwhere
In
dxexLxL
nr
nrx
rn
2
)0(
,
max +=
−=
Xm
wherem
mXZ
Tchebyshev - Type I
188)(
34)(
12)(
)(
1)(
244
33
22
1
0
+−=
−=
−=
=
=
xxxT
xxxT
xxT
xxT
xT
==
≠=
=
=−
∫−
..,0
0,
0,2
1
1
1)()(
1
12
wo
rnif
rnif
Iwhere
Idxx
xTxT
nr
nrrn
π
π
2
)0(
,
max +=
−=
Xm
wherem
mXZ
Tchebyshev - Type II
11216)(
48)(
14)(
2)(
1)(
244
33
22
1
0
+−=
−=
−=
=
=
xxxT
xxxT
xxT
xxT
xT
≠=
=
=−∫−
..,0
0,2
1
1)()(1
1
2
wo
rnifIwhere
IdxxxTxT
nr
nrrn
π
2
)0(
,
max +=
−=
Xm
wherem
mXZ
103
As summarized in Table 4-6 and illustrated in Figure 4-1a for parameter sets 2 and 3
from Table 4-4, the estimated probabilities obtained by GCE do not sum to one, as
required of any PDF, an artifact due to applying this method to a discrete random
variable. All GCE results, therefore, are proportionally scaled by dividing each f(t) value,
by the total,
.)(
)()(
0∑ =
==N
i i
ii
tf
tftTP (4-29)
To the best of our knowledge, this normalization correction has not been discussed in
previous studies and herein is referred to as normalized Gram-Charlier Expansion
(NGCE). Figure 4-1b illustrates the decrease in relative error of the normalized versus
conventional GCEs. Figure 4-2 and Table 4-7 indicate that this improvement is consistent
for other expansion orders and parameter cases.
Table 4-6: Sum of all estimated probabilities using GCE of order 6, without and with normalization
Test Cases Sum of Probability Estimates
GCE-o6 NGCE-o6 1 0.913 1.0 2 2.33 1.0 3 2.32 1.0 4 9.071 1.0
104
P
aram
eter
Set
2
Par
amet
er S
et 3
(a) (b)
Figure 4-1: Illustration of the need for normalization, (a) Exact probability distribution versus GCE of order 6, (b) Relative error of normalized versus conventional GCE of order 6,
Relative Error = (Exact – Approximate)/Exact
(a) M-KL (b) TAD
Figure 4-2: Impact of normalization and order on GCE accuracy: (a) Modified Kullback-Leibler statistic, (b) Total absolute deviation. NGCE: normalized GCE; JB1, JB2, JB3: parameter sets 1, 2 and 3 from
Table 4-4, respectively
0
0.1
0.2
0.3
0.4
1 3 5 7 9 11 13 15 17 19
P(T
= t
)
t
GCE-o6
Exact
-1
-0.5
0
0.5
1
1.5
2
1 3 5 7 9 11 13 15 17 19
Re
lati
ve
Err
or
t
GCE
NGCE
0
0.1
0.2
0.3
0.4
1 3 5 7 9 11 13 15 17 19
P(T
= t
)
t
GCE-o6
Exact
-0.2
0.3
0.8
1.3
1.8
1 3 5 7 9 11 13 15 17 19
Re
lati
ve
Err
or
t
GCE
NGCE
-0.4
0.1
0.6
1.1
1.6
2.1
2 3 4 5 6 7 8 9 10 11 12 13
Mo
dif
ied
-Ku
llb
ack
Le
ible
r
Order of Approximation
GCE-JB1
GCE-JB2
GCE-JB3
NGCE-JB1
NGCE-JB2
NGCE-JB3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
2 3 4 5 6 7 8 9 10 11 12 13
To
tal
Ab
solu
te D
ev
iati
on
Order of Approximation
GCE-JB1
GCE-JB2
GCE-JB3
NGCE-JB1
NGCE-JB2
NGCE-JB3
Σ fGCE (t) = 2.33
Σ fGCE (t) = 2.32
105
Table 4-7: Impact of normalization and expansion order on the accuracy of cumulant based expansions using Hermite polynomials (GCE and NGCE) and Laguerre polynomials, for the parameter sets given in
Table 4-4
Parameter Set
Expansion Order
GCE Normalized GCE Laguerre M-KL TAD M-KL TAD M-KL TAD
1
2 -0.085030 0.216526 0.040324 0.263258 1.127876 0.810854 3 -0.088160 0.111749 0.006477 0.103540 1.437092 1.013501 4 -0.074830 0.180500 0.018757 0.186032 2.071174 1.374350 5 -0.076720 0.183178 0.018954 0.189147 4.150684 2.263000 6 -0.081050 0.091568 0.002442 0.037884 4.150684 2.263000 7 -0.081540 0.092715 0.002620 0.041282 5.457262 2.745423 8 -0.080750 0.092428 0.002924 0.039556 6.861853 3.189455 9 -0.080750 0.092428 0.002923 0.039556 8.374088 3.631009 10 -0.080420 0.092398 0.003025 0.040042 10.002780 4.026364 11 -0.080410 0.093674 0.003177 0.041343 11.717700 4.423677 12 -0.080450 0.093550 0.003175 0.041180 13.51734 4.965981 13 0.125502 0.303297 0.148667 0.322909 15.37738 5.572507 14 0.348388 0.793539 0.377812 0.808187 17.30583 6.160421
2
2 1.976453 1.321897 0.010153 0.093612 40.50597 7.781359 3 1.971266 1.329440 0.000652 0.017171 1.422602 2.533877 4 1.969367 1.328068 0.000910 0.027525 1.421565 2.532469 5 1.969491 1.328117 0.000923 0.027689 3.393619 0.768662 6 1.968407 1.328570 7.74E-05 0.004427 5.243802 1.097578 7 1.968418 1.328576 7.75E-05 0.004408 0.337441 0.163104 8 1.968416 1.328575 7.73E-05 0.004410 0.016300 0.406555 9 1.968414 1.328574 7.73E-05 0.004410 0.011960 0.266461 10 1.968418 1.328576 7.73E-05 0.004411 0.080858 0.041994 11 1.968416 1.328575 7.73E-05 0.004411 0.498779 0.156102 12 1.968416 1.328575 7.73E-05 0.004411 0.280655 0.097348 13 1.968415 1.328575 7.72E-05 0.004393 0.173978 0.065481 14 1.968416 1.328575 7.72E-05 0.004393 -0.001600 0.090918
3
2 1.953227 1.309257 0.010262 0.095541 37.73056 7.387179 3 1.948760 1.317198 0.000680 0.017829 1.141602 2.199152 4 1.946728 1.315745 0.000945 0.027729 1.329203 2.463932 5 1.946870 1.315802 0.000959 0.028010 2.563544 0.621007 6 1.945930 1.316363 8.42E-05 0.004573 4.847163 1.036741 7 1.945940 1.316368 8.42E-05 0.004552 0.533610 0.206055 8 1.945938 1.316367 8.43E-05 0.004553 0.014918 0.354551 9 1.945938 1.316367 8.43E-05 0.004553 0.012097 0.260063 10 1.945940 1.316368 8.43E-05 0.004553 0.025281 0.022965 11 1.945938 1.316367 8.43E-05 0.004554 0.399209 0.131616 12 1.945938 1.316367 8.43E-05 0.004554 0.112215 0.046530 13 1.945936 1.316366 8.36E-05 0.004535 -0.002430 0.055590 14 1.945938 1.316367 8.41E-05 0.004535 0.024551 0.435213
106
Additionally, each polynomial empirically appears to have an optimal expansion order
that gives the best approximation. Beyond that order, the accuracy can fluctuate, as also
mentioned by Badinelli (Badinelli, 1996). Figure 4-3 further illustrates the impact of
order on accuracy via the M-KL and TAD measures. As shown, orders of 6 and 7 for
Hermite polynomials (NGCE) and an order of 10 for Laguerre polynomials seem to be
the most effective. The approximations with Laguerre polynomials are not normalized
since the expansion produces reasonable estimates when the optimal order is used.
(a) Hermite polynomials (b) Laguerre polynomials
Figure 4-3: Impact of order on accuracy of the cumulant based approximations using (a) Hermite polynomials (NGCE) and (b) Laguerre polynomials. JB1, JB2, JB3: parameter sets 1, 2 and 3 from
Table 4-4, respectively
Table 4-7 also summarizes the accuracy of the NGCE and Laguerre polynomial
expansions, respectively, illustrating that NGCE is slightly better in all examined cases.
Figure 4-4 illustrates the approximate distributions versus the exact PDF and MCE.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
2 3 4 5 6 7 8 9 10 11 12 13 14Order of the approximation
M-K
L
JB1JB2JB3
0
5
10
15
20
25
30
35
40
45
2 3 4 5 6 7 8 9 10 11 12 13 14
Order of approximation
M-K
L
JB1JB2JB3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2 3 4 5 6 7 8 9 10 11 12 13 14Order of the approximation
TA
D
JB1JB2JB3
0
1
2
3
4
5
6
7
8
9
2 3 4 5 6 7 8 9 10 11 12 13 14Order of approximation
TA
D
JB1JB2JB3
107
Although all approximations perform poorly for small sample sizes, in these cases the
exact J-binomial distribution can easily be computed in a reasonable amount of time. In
addition to the exact PDF, MCE results also are shown since in some cases the exact
computation is not possible. All MCEs for J-binomial distribution approximations are run
until all probabilities are estimated with confidence interval half-width less than or equal
to 0.001.
Figure 4-4 also indicates that the tail probability estimations can be poor, relative to the
body of the distribution. For the cases when T = 0 and T = N, however, computing the
exact probabilities directly from the PDF is straightforward since they reduce to P(T = 0)
= ∏∏ ==−==
J
j
nj
J
j jjpXP
11)1()0( and P(T = N) = ∏∏ ==
==J
j
nj
J
j jjjpnXP
11)( .
108
(a) Parameter Set 1
(b) Parameter Set 2
(c) Parameter Set 3
(d) Parameter Set 4
Figure 4-4: Accuracy of cumulant based approximation via NGCE up to order 6 and Laguerre polynomials up to order 10 versus exact J-binomial probabilities and Monte Carlo estimates
00.050.1
0.150.2
0.250.3
0.350.4
0.45
0 1 2 3 4 5 6 7 8
P(T
=t)
t
ExactMonte CarloNGCE-o6Laguerre-o10
00.020.040.060.080.1
0.120.140.160.18
0 1 2 3 4 5 6 7 8 9 1011121314151617181920
P(T
=t)
t
ExactMonte CarloNGCE-o6Laguerre-o10
00.020.040.060.080.1
0.120.140.160.18
0 1 2 3 4 5 6 7 8 9 1011121314151617181920
P(T
=t)
t
ExactMonte CarloNGCE-o6Laguerre-o10
00.005
0.010.015
0.020.025
0.030.035
0.040.045
0.05
P(T
=t)
t
NGCE-o6
Monte Carlo
109
4.3.2. Saddle Point Approximations
This section compares the accuracy of SPA, TSPA of order 3, and probability estimates
with continuity correction such that P(T = t) = P(t - 0.5 < T < t + 0.5) to that of NGCE,
exact J-binomial calculations, and MCE. Figure 4-5 illustrates the approximate versus the
exact PDFs. As mentioned in Section 4.2.2, TSPA of order 3 is used to avoid the need for
a numerical method to find the saddle points. When the CGF is truncated at order 3,
however, the equation 0)( =−′ stsKT does not always have a real root for the first few
values of T. For case 1 of Table 4-4, for example, the saddle point when T = 0 could not
be found and for cases 2 and 3 it is not possible to find the roots when T = 0, 1, and 2.
The TSPA and continuation correction estimates for those cases, therefore, are omitted
from Figure 4-5 and Table 4-8. Likewise, neither the Newton-Rhapson nor secant
methods converge for the value T = 0. It is, however, possible to easily compute the
probability P(T = 0) from the actual PDF, as discussed in the previous section.
Table 4-8 further summarizes the M-KL and TAD accuracy measures for each
approximation method for the four parameter sets in Table 4-4. Both Figure 4-5 and
Table 4-8 suggest that poor accuracy is obtained with all methods for small sample sizes.
The near exact overlay of the probability estimate curves in Figure 4-5d further indicates
that all methods perform well with large sample sizes, with the Newton-Rhapson and
secant methods producing exactly the same results. The Newton-Rhapson method,
however, requires evaluation of both the function to be approximated and its derivative at
each iteration, and thus the secant method might be preferred when the function
evaluations are costly, although this approach needs two initial points (Pozrikidis, 2008).
Figure 4-6 illustrates the relative accuracy of NGCE of order 6 to SPA. While SPA
performs better at the tails, NGCE produces slightly better estimates in the body of the
distributions. Table 4-8 also suggests that NGCE performs better over the entire PDF for
all cases investigated, although the difference seems negligible.
110
(a) Parameter Set 1
(b) Parameter Set 2
(c) Parameter Set 3
(d) Parameter Set 4
Figure 4-5: Comparison of accuracy of NGCE-o6: Normalized Gram-Charlier expansion of order 6, SPA-NR: Saddle point approximations using Newton-Raphson method, SPA-S: Saddle point
approximations using secant method, TSPA-o3: Truncated saddle point approximation of order 3, and Cont.Corr.: Estimating the probabilities via numerical integration using continuation correction such that
P(T = t) = P(t - 0.5 < T < t + 0.5)
00.050.1
0.150.2
0.250.3
0.350.4
0.45
1 2 3 4 5 6 7 8P
(T=t
)t
ExactNGCE-o6SPA-NRSPA-STSPA-o3Cont.Corr.
00.020.040.060.080.1
0.120.140.160.180.2
0 1 2 3 4 5 6 7 8 9 1011121314151617181920
P(T
=t)
t
ExactNGCE-o6SPA-NRSPA-STSPA-o3Cont.Corr.
00.020.040.060.080.1
0.120.140.160.180.2
0 1 2 3 4 5 6 7 8 9 1011121314151617181920
P(T
=t)
t
ExactNGCE-o6SPA-NRSPA-STSPA-o3Cont.Corr.
00.0050.01
0.0150.02
0.0250.03
0.0350.04
0.0450.05
57 62 67 72 77 82 87 92 97 102
107
112
117
122
127
132
137
P(T
=t)
t
Monte CarloNGCE-o6SPA-NRSPA-STSPA-o3Cont.Corr.
111
Table 4-8: Comparison of accuracy of the approximation methods. NGCE-o6: Normalized Gram-Charlier expansion of order 6; SPA-S: Saddle point approximations using secant method; SPA-NR: Saddle point
approximation using Newton-Raphson method; TSPA-o3: Truncated saddle point approximation of order 3; Cont.Corr.: Estimating the probabilities via numerical integration using continuation correction such that
P(T = t) = P(t - 0.5 < T < t + 0.5)
Parameter Set 1 Parameter Set 2 Parameter Set 3 Parameter Set 4 Method M-KL TAD M-KL TAD M-KL TAD M-KL TAD
NGCE-o6 0.01119 0.02903 -0.00087 0.00304 -0.00087 0.00305 0.00018 0.01218 SPA-S 0.04220 0.04072 0.01473 0.01460 0.01482 0.01469 0.00105 0.01242
SPA-NR 0.04220 0.04072 0.01473 0.01460 0.01482 0.01469 0.00105 0.01242 TSPA-o3 0.03921 0.03780 0.02046 0.02003 0.02013 0.01972 0.00107 0.01243 Cont.Corr. 0.06610 0.06338 0.01995 0.01988 0.01965 0.01962 0.00150 0.01256
(a) Parameter Set 2 (b) Parameter Set 3
Figure 4-6: Performance of NGCE of order 6 versus SPA in the body and tails of the distribution Relative Error = (Exact – Approximated) / Exact
The SPA and NGCE are compared versus the exact J-binomial distribution in a SPRT
setting in order to assess the impact of both methods on the inferences made. Figure 4-7
illustrates the SPRT for the case of J = 4, nj = 10, j = 1, 2, 3, 4, H0: p = (0.02, 0.1, 0.25,
0.4), H1: p = (0.03, 0.15, 0.375, 0.6), and data is generated from a J-binomial distribution
with rate parameters p = (0.025, 0.125, 0.3125, 0.5). The overlaying lines imply
agreement between SPRT results obtained using NGCE, SPA, and the exact probability
values when computing the likelihood ratio.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1 3 5 7 9 11 13 15 17 19
Re
lati
ve
Err
or
t
NGCE-o6
SPA
-1
-0.5
0
0.5
1
1 3 5 7 9 11 13 15 17 19
Re
lati
ve
Err
or
t
NGCE-o6
SPA
112
Figure 4-7: Illustration of SPRT using the exact J-binomial probabilities, Normalized Gram-Charlier expansion of order 6 and saddle-point approximations: J = 4, nj = 10, j = 1, 2, 3, 4,
H0: p = (0.02, 0.1, 0.25, 0.4), H1: p = (0.03, 0.15, 0.375, 0.6), and p = (0.025, 0.125, 0.3125, 0.5)
4.4. Discussion This chapter provides the reader with a practical and computationally effective cumulant-
based methodology that produces J-binomial probability estimates highly close to the
exact values and has the advantage of being easy to integrate in other algorithms used for
statistical analysis. Cumulant based orthogonal polynomial expansions and saddle point
approximations both can produce fast when J-binomial probabilities can be intractable to
compute directly, although the former is recommended for the reasons summarized at the
end of this section. As shown in Table 4-9, all examined approximation methods are
significantly faster than the exact calculations and MCE. NGCE and TSPA of order 3 are
the fastest since neither algorithm requires a numerical method, whereas SPA (non-
truncated) becomes slower for larger distributions (larger J and nj).
None of the approximations are very accurate for small sample sizes, although it is
relatively easy to calculate the exact J-binomial probabilities for smaller J and sample
sizes. The accuracy of all methods investigated except for Laguerre polynomial
expansions improves as sample size increases. Expansion with Laguerre polynomials of
-5
-4
-3
-2
-1
0
1
2
3
4
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43
Ln l
ike
lih
oo
d r
ati
o
Sample number
Exact
NGCE-o6
SPA
Accept H0
Reject H0
113
order 10 also takes longer than any other approximation method even with small sample
sizes.
Table 4-9: CPU times (in seconds) to calculate the entire J-binomial PDF
Test Cases
Parameters
Exact MCE
(HW = 0.001) Laguerre
(o10) NGCE
(o6) SPA
TSPA (o3)
1 J = 4, N = 18 0.000 4.985 0.218 0.000 0.015 0.000 2 J = 4, N = 100 0.406 13.750 0.219 0.015 0.016 0.015 3 J = 10, N = 100 3829.430 14.000 0.266 0.016 0.016 0.015 4 J = 25, N = 1278 >one week 45.172 --- 0.016 0.047 0.016 5 J = 40, N = 634 >one week 25.500 --- 0.016 0.078 0.016
Furthermore, since the numerical methods used to find saddle points necessitate initial
values that are sufficiently close to the true solution, SPA does not guarantee
convergence in all cases. NGCE, conversely, does not rely on a numeric algorithm and
hence always produces at least some result. This phenomenon is illustrated in Table 4-10
for the example taken from Matis and Guardiola (Matis and Guardiola, 2006) where the
entire PDF of the sum of 20 Bernoulli random variables each with p = 0.4 is estimated.
The numbers highlighted in bold face correspond to the probability estimates for which
the secant or Newton-Rhapson method does not converge or cannot find the saddle point.
SPA therefore produces either much larger values than the correct probabilities or does
not find a value at all. While the secant method does not converge for the values T = 0
and T = 20, the Newton-Rhapson method cannot find the saddle point for even more
values of T.
Due to the convergence problem of numerical methods required to find the saddle points,
SPA therefore may not be recommendable for the particular problem of estimating all
J-binomial distributions. In general, however, it is a fast and effective approximation
algorithm when a closed form for the first derivative of CGF in Equation (4-6) exists.
Although TSPA of order 3 performs well in this particular example, there are situations
where it does not, as illustrated in the previous sections. It is not always possible to find
saddle points for the tails with a TSPA of order 3 and higher orders of TSPA requires a
numerical method to find saddle points which again may not always converge.
114
As discussed above, the NGCE and SPA methods produce slightly different probability
estimates, with SPA performing better in the tails and NGCE being more close to the
actual values in the body of the distribution. The differences in accuracy, however, are
not substantial enough to affect the performance of sequential probability ratio tests.
Moreover, although NGCE requires computing the entire probability distribution for
normalization even if only one probability value is needed, implementation of the
algorithm is fast enough to absorb these extra calculations. For these reasons, NGCE is
recommended as a fast and accurate method for estimating J-binomial probabilities.
Although NGCE works well for the distributions considered in this study, future work
can include assessment of its accuracy and effectiveness for further applications and
providing the reader with a general guide as to when to use the algorithm versus direct
calculation via the exact J-binomial PDF.
Table 4-10: Comparison of accuracy of NGCE-o6: Normalized Gram-Charlier expansion of order 6, TSPA-o3: Truncated saddle point approximation of order 3, SPA-NR: Saddle point approximation using Newton-Raphson method, SPA-S: Saddle point approximations using secant method for the case J = 20,
nj = 1, pj = 0.4
T Exact NGCE-o6 TSPA-o3 SPA-NR SPA-S 0 0.000036 0.000035 0.000058 -1.#IND00 -1.#IND00 1 0.000487 0.000532 0.000543 0.000527 0.000529 2 0.003087 0.003165 0.003154 0.037241 0.003219 3 0.012349 0.012303 0.012436 1.185953 0.012706 4 0.034991 0.034741 0.035330 2.630576 0.035763 5 0.074647 0.074523 0.075645 1.454898 0.076005 6 0.124411 0.124733 0.126230 0.643596 0.126375 7 0.165882 0.166259 0.168227 0.352398 0.168244 8 0.179705 0.179569 0.182091 0.23915 0.182091 9 0.159738 0.159319 0.161779 0.174842 0.161774 10 0.117141 0.117027 0.118622 0.120496 0.118614 11 0.070995 0.071194 0.071890 0.072041 0.071900 12 0.035497 0.035648 0.035961 0.035953 0.035969 13 0.014563 0.014551 0.014799 0.014752 0.014770 14 0.004854 0.004798 0.004988 0.004925 0.004931 15 0.001294 0.001271 0.001370 0.001318 0.001318 16 0.000269 0.000271 0.000305 0.000276 0.000276 17 4.230E-05 4.701E-05 0.000055 0.000043 0.000044 18 4.700E-06 6.846E-06 0.000008 0.000005 0.000005 19 3.298E-07 9.128E-07 0.000001 0.000000 0.000000 20 1.099E-08 0.00000 0.00000 0.000000 0.308849
115
Chapter 5 – Conclusions
5.1. Summary of Major Results Many manufacturing, healthcare, and service systems exist for which individual data
either have unique failure rates or can be stratified into inter-heterogeneous but intra-
homogeneous sub-populations or categories. This dissertation emphasizes the fact that in
such cases it is important to properly adapt existing statistical methods to avoid false
alarms, losses in test power, or delays in detection of process changes. Performance
analyses illustrate that assuming a common rate parameter when heterogeneity exists can
affect the accuracy or timeliness of the decisions made. The normal assumption usually
leads to faster decisions but at the cost of increased false alarm rate. Thus the three
general results obtained and tools developed in this research would be of great use to any
researcher or practitioner who work with heterogeneous data herein considered.
The first part of this research develops and investigates SPRTs and RSPRTs based on an
underlying heterogeneous J-binomial probability model. The results show that the design
parameters α and β and difference between the null and alternate hypotheses can
significantly affect performance. It is, therefore, important to ensure the tests or charts are
designed to perform within acceptable thresholds.
The second part of this research develops two models, J-Bernoulli and J-binomial, for
heterogeneity for Kulldorff’s scan statistic. Of the two proposed methods, the one based
on J-binomial distribution is shown to have larger power, thereby allowing timely
detection of clusters and earlier interventions than the conventional method.
116
The third part of this research addresses the computational difficulty of J-binomial
distribution and investigates feasible ways to accurately approximate these probabilities.
A normalized GCE is developed and shown to produce fast and highly accurate
estimates. Computer codes for computing the saddle-point approximations, NGCE, and
MCE of J-binomial probabilities are developed in the C programming language and are
described below. The NGCE algorithm is easily adaptable to other computer languages
and is used in the VBA code developed for Monte Carlo simulation analysis of risk-
adjusted scan statistics.
5.2. Description of Computer Codes This section describes the computer programs developed for Monte Carlo simulation
analyses and approximating J-binomial probabilities.
Risk-adjusted SPRT and RSPRT
To assess performance of the risk-adjusted SPRT and RSPRT charts, a Monte Carlo
simulation procedure is developed using the C computer programming language. The
code prompts the user for the number of replications, number of categories, design
parameters α and β, and the type of test the user wants to perform. The program reads
these parameters from an input file, illustrated by Figure 5-1, and produces an output file,
presented by Figure 5-2, with the performance measures P(Accept H0), ARL or ANS,
ANI where appropriate, and half-widths of 95% confidence intervals for the performance
measures. In the risk-adjusted SPRT study, 100,000 replications are used to compute all
performance measures. Figure 5-1a, Figure 5-1b and Figure 5-1c illustrate the different
input options for the performance analysis of risk-adjusted SPRTs. When the Xj terms are
known (Figure 5-1a), user is prompted the manner which the samples are drawn. If only
the total count T is known (Figure 5-1b), the user is prompted how to construct the
SPRTs; based on J-binomial distribution, binomial or normal approximation. If the user
picks J-binomial distribution (Figure 5-1c), the user is prompted how to compute the
J-binomial probabilities.
117
(a)
(b)
(c)
Figure 5-1: The input window for SPRT simulation performance analysis (a) when the Xj terms are known
(b) when only T total counts are known (c) different choices of calculating J-binomial probabilities
118
The program reads the null and alternate hypotheses values, the actual population rate
parameters (to generate the data from) and sample sizes from two different data files. The
file in Figure 5-3a includes the rate parameters for null and alternate hypotheses and the
actual population. Sample sizes are read from a separate file as shown in Figure 5-3b.
Figure 5-2: The output window for SPRT simulation performance analysis
(a) (b)
Figure 5-3: Format of the input data files for SPRT simulation performance analysis
119
In addition to the performance analysis, a program that constructs risk-adjusted SPRTs
and RSPRTs for real data is also developed. Figure 5-4 illustrates an example of the input
window and Figure 5-5 presents the format of the input data file that the program reads.
As well as the data, the program also reads the null and alternate parameters from a data
file that is similar to the one given in Figure 5-3a except for the actual parameters
column.
Figure 5-4: An example of the input window for constructing SPRTs
Figure 5-5: Format of the data file for constructing SPRTs
120
Risk-adjusted Scan Statistic
A Visual Basic code is developed to explore the comparative performance of the risk-
adjusted Bernoulli scan statistic models J-Bernoulli and J-binomial versus Kulldorff’s
Bernoulli scan statistic. Figure 5-6 illustrates the logic of the general MCE procedure,
which has 2 stages.
Stage 1: The first step of the MC simulation is to generate the initial data inside and
outside the cluster candidate from J-binomial distributions. Assuming that this is the
information in the most likely cluster, and given the number of cases in the whole study
region, the numbers of these cases falling inside and outside the scanning window can be
generated by either (1) randomly sampling from all possible permutations of the cases in
the study region and counting the number of cases in and outside the scanning window
(Dwass, 1957), or (2) generating the number of cases inside the scanning window based
on a hypergeometric random variate and finding the number of cases outside the scanning
window by simply subtracting the number of cases inside from the total number of cases
in the study region. A LR estimate then can be computed from these values. For the
results reported in Chapter 3, 9999 replications of the LR are used to estimate the p-
values with each p-value estimated by the fraction of simulated LR estimates greater than
or equal to the value obtained using the “real” data.
Stage 2: The above process then is repeated 1000 times to find the empirical distribution
and average of all p-values.
121
Yes
Yes
No
Yes
No
Generate J-binomial data (# of cases (events))
inside & outside the scanning window
r = 1, Count = 0
Given the total number of cases in the
whole study region, generate data inside
& outside the scanning window under H0
Estimate rates and calculate the
likelihood ratios (LR(MC)),
r = r +1
Estimate rates from the generated data
and calculate the likelihood ratios
(LR(INIT))
Count = Count + 1LR(MC) ≤ LR(INIT)?
r < Rep2?
Calculate p-value and update statistics,
k = k + 1
k < Rep1?
Print the statistics
No
User enters the number of replications Rep1
and Rep2 for the inner and outer loops,
Read the parameters from the worksheet,
k = 1,
Figure 5-6: Illustration of the risk- adjusted scan statistic procedure Monte Carlo simulation
122
Approximating J-binomial Distribution
The C programming language is used to develop a program that computes the exact,
NGCE, saddle point approximations, and MCE of the J-binomial PDF. This program
reads the parameters from a data file. Figure 5-7 and Figure 5-8 illustrate the input
window and the parameter file of this program.
Figure 5-7: The input window for calculation of J-binomial probabilities
Figure 5-8: The input parameter file for calculation of J-binomial probabilities
123
Because the calculation of NGCE and SPA are straightforward evaluations of the
formulae, only the logic of the MCE program is explained in this sub-section. Figure 5-9
illustrates the logic of MCE of J-binomial probabilities for the case of only a single value
probability. The program first prompts the user for the value of T to consider and whether
the PDF or CDF should be evaluated and continues until the halfwidth (in the given case)
or the maximum of all halfwidths (in case of the whole PDF is calculated) of the 95%
confidence interval(s) of the probability estimate(s) is less than a user-specified value. A
high value is initially assigned to the confidence interval halfwidth for the probability
estimate, the program then continues generating J-binomial random variates and updating
the probability estimates and the confidence interval halfwidth until the halfwidth is less
than the allowed threshold.
124
Figure 5-9: The logic of MCE of J-binomial probabilities
125
5.3. Future Possible Work This dissertation addresses the two problems of (1) handling heterogeneous dichotomous
events in the context of SPRTs, RSPRTs, and scan statistics, and (2) approximating the
PDF of sums of heterogeneous dichotomous events. The insight gained and the
limitations encountered in this study lead to the following suggestions for future work.
One possible extension of this research is to study the impact of estimating the rate
parameters in the construction of the SPRTs and RSPRT charts. While the rate
parameters are assumed to be known in this study, this is usually not the case.
Performance and relative performance might be affected by parameter estimation or the
accuracy of the risk prediction method. Risk-adjustment methods such as logistic
regression typically might be used to estimate each probability with estimation error
impacting the performance of the tests or charts developed (Grigg et al., 2003). However,
to the best of our knowledge, this impact has not been studied extensively in the
literature. For this reason, future research should investigate the impact of estimation on
statistical surveillance and identify those methods with minimum error.
Given the combined importance of shift size and the alternate hypothesis on detection,
further investigation also might include work to optimize performance across a range of
step sizes, such as taking a max-SPRT approach (Kulldorff et al., 2007) or otherwise.
If the individual Xm,j data themselves are known and if only one or a few before-after
parameters are different, it may be beneficial to test each category separately so that
signals of these differences are not diluted by a majority of data being sampled still from
the remaining population. However, it is important to note that this would not be possible
in the case for which every Bernoulli trial has a unique failure probability, such as
logistic-regression risk adjusted mortality. Even in the categorical case, such stratification
may reduce detection performance (time, power), especially for cases in which all or a
majority of parameters change.
126
Similarly, the number of categories or sub-populations also is assumed to be known.
Similar to how assuming homogeneity can alter conclusions when there actually exists
heterogeneity, erroneous stratification of the population or assuming heterogeneity when
none truly exists also can negatively impact the results. The effects of such errors are also
unexplored and potential future research areas.
This dissertation also highlights the impact of approximating the J-binomial distribution
with its binomial and normal counterparts. Although the well-known central limit
theorem states that a convolution of random variables can be approximated by a normal
distribution, in particular when the number of random variables in the convolution and
sample sizes are large, it is interesting that the risk-adjusted non-resetting and resetting
SPRTs presented here suggest that even large sample sizes can lead to different
inferences or ARLs. Therefore it does not appear straightforward to determine when it is
acceptable in these types of applications to use a normal approximation. Further study,
therefore, could attempt to develop guidance for when a normal approximation is
reasonable.
Other possible extensions deal with the risk-adjusted scan statistics models. The power
analysis provided in this study is somewhat limited in the sense that it lacks the scanning
feature of the original spatial scan statistic methodology. Thus, future work could
incorporate a scanning module over the geographical area and determine the window
with the maximum likelihood ratio, to further investigate the relative performance of
J-binomial scan statistic versus the Kulldorff’s Bernoulli scan statistic.
127
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