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ENGR 610Applied Statistics
Fall 2007 - Week 9
Marshall University
CITE
Jack Smith
Overview for Today Review Design of Experiments, Ch 10
One-Factor Experiments Randomized Block Experiments
Go over homework problems: 10.27, 10.28 Design of Experiments, Ch 11
Two-Factor Factorial Designs Factorial Designs Involving Three or More Factors Fractional Factorial Design The Taguchi Approach
Homework assignment
Design of Experiments R.A. Fisher (Rothamsted Ag Exp Station)
Study effects of multiple factors simultaneously Randomization Homogeneous blocking
One-Way ANOVA (Analysis of Variance) One factor with different levels of “treatment” Partitioning of variation - within and among treatment groups Generalization of two-sample t Test
Two-Way ANOVA One factor against randomized blocks (paired treatments) Generalization of two-sample paired t Test
One-Way ANOVA ANOVA = Analysis of Variance
However, goal is to discern differences in means One-Way ANOVA = One factor, multiple treatments (levels) Randomly assign treatment groups Partition total variation (sum of squares)
SST = SSA + SSW SSA = variation among treatment groups SSW = variation within treatment groups (across all groups)
Compare mean squares (variances): MS = SS / df Perform F Test on MSA / MSW
H0: all treatment group means are equal H1: at least one group mean is different
Partitioning of Total Variation Total variation
Within-group variation
Among-group variation
SST (X ij X)2
i1
n j
j1
c
SSW (X ij X j )2
i1
n j
j1
c
SSA n j (X j X)2
j1
c
X 1
nX ij
i1
n j
j1
c
X j 1
n jX ij
i1
n j
(Grand mean)
(Group mean)
c = number of treatment groupsn = total number of observationsnj = observations for group jXij = i-th observation for group j
Mean Squares (Variances) Total mean square (variance)
MST = SST / (n-1) Within-group mean square
MSW = SSW / (n-c) Among-group mean square
MSA = SSA / (c-1)
F Test F = MSA / MSW Reject H0 if F > FU(,c-1,n-c) [or p<]
FU from Table A.7
One-Way ANOVA SummarySource Degrees of
Freedom (df)Sum of Squares (SS)
Mean Square (MS) (Variance)
F p-value
Among groups
c-1 SSA MSA = SSA/(c-1) MSA/MSW
Within groups
n-c SSW MSW = SSW/(n-c)
Total n-1 SST
Tukey-Kramer Comparison of Means
Critical Studentized range (Q) test
qU(,c,n-c) from Table A.9
Perform on each of the c(c-1)/2 pairs of group means Analogous to t test using pooled variance for
comparing two sample means with equal variances
X i X j qUMSW
2
1
ni
1
n j
One-Way ANOVA Assumptions and Limitations Assumptions for F test
Random and independent (unbiased) assignments Normal distribution of experimental error Homogeneity of variance within and across group
(essential for pooling assumed in MSW) Limitations of One-Factor Design
Inefficient use of experiments Can not isolate interactions among factors
Randomized Block Model Matched or repeated measurements assigned to a
block, with random assignment to treatment groups Minimize within-block variation to maximize treatment
effect Further partition within-group variation
SSW = SSBL + SSE SSBL = Among-block variation SSE = Random variation (experimental error) Total variation: SST = SSA + SSBL + SSE
Separate F tests for treatment and block effects Two-way ANOVA, treatment groups vs blocks, but the
focus is only on treatment effects
Partitioning of Total Variation Total variation
Among-group variation
Among-block variation
SST (X ij X)2
i1
r
j1
c
SSAr (X j X)2
j1
c
X 1
rcX ij
i1
r
j1
c
X j 1
rX ij
i1
r
(Grand mean)
(Group mean)
SSBLc (X i X)2
i1
r
X i 1
cX ij
j1
c
(Block mean)
Partitioning, cont’d Random error
SSE SST SSA SSBL (X ij X i X j X)2
i1
r
j1
c
c = number of treatment groupsr = number of blocks n = total number of observations (rc)Xij = i-th block observation for group j
Mean Squares (Variances) Total mean square (variance)
MST = SST / (rc-1) Among-group mean square
MSA = SSA / (c-1) Among-block mean square
MSBL = SSBL / (r-1) Mean square error
MSE = SSE / (r-1)(c-1)
F Test for Treatment Effects F = MSA / MSE Reject H0 if F > FU(,c-1,(r-1)(c-1))
FU from Table A.7
Two-Way ANOVA SummarySource Degrees of
Freedom (df)Sum of Squares (SS)
Mean Square (MS) (Variance)
F p-value
Among groups
c-1 SSA MSA = SSA/(c-1) MSA/MSE
Among blocks
r-1 SSBL MSBL = SSBL/(r-1) MSBL/MSE
Error (r-1)(c-1) SSE MSE = SSE/(r-1)(c-1)
Total rc-1 SST
F Test for Block Effects F = MSBL / MSE Reject H0 if F > FU(,r-1,(r-1)(c-1))
FU from Table A.7
Assumes no interaction between treatments and blocks
Used only to examine effectiveness of blocking in reducing experimental error
Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision
Estimated Relative Efficiency Relative Efficiency
Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design. nj (without blocking) RE*r (with blocking)
RE (r 1)MSBL r(c 1)MSE
(rc 1)MSE
Tukey-Kramer Comparison of Means
Critical Studentized range (Q) test
qU(,c,(r-1)(c-1)) from Table A.9 Where group sizes (number of blocks, r) are equal
Perform on each of the c(c-1)/2 pairs of group means Analogous to paired t test for the comparison of two-
sample means (or one-sample test on differences)
X i X j qUMSE
r
Factorial Designs Two or more factors simultaneously Includes interaction terms Typically
2-level: high(+), low(-) 3-level: high(+), center(0), low(-)
Replicates Needed for random error estimate
Partitioning for Two-Factor ANOVA(with Replication)
Total variation
Factor A variation
Factor B variation
SST (X ijk X)2
k1
n'
j1
c
i1
r
SSAcn' (X i X)2
i1
r
X 1
rcn 'X ijk
k1
n '
j1
c
i1
r
X i 1
cn'X ijk
k1
n'
j1
c
(Grand mean)
(Mean for i-th level of factor A)
SSBrn' (X j X)2
j1
c
X j 1
rn'X ijk
k1
n'
i1
r
(Mean for j-th level of factor B)
Partitioning, cont’d Variation due to interaction of A and B
Random error
SSE SST SSA SSB SSAB (X ijk X ij )2
k
n '
j1
c
i1
r
r = number of levels for factor A c = number of levels for factor Bn’ = number of replications for eachn = total number of observations (rcn’)Xijk = k-th observation for i-th level of factor A and j-th level of factor B
SSABn' (X ij X i X j X)2
j1
c
i1
r
'
1'
1 n
kijkij X
nX
(Mean for replications of i-j combination)
Mean Squares (Variances) Total mean square
MST = SST / (rcn’-1) Factor A mean square
MSA = SSA / (r-1) Factor B mean square
MSB = SSB / (c-1) A-B interaction mean square
MSAB = SSAB / (r-1)(c-1) Mean square error
MSE = SSE / rc(n’-1)
F Tests for Effects Factor A effect
F = MSA / MSE Reject H0 if F > FU(,r-1,rc(n’-1))
Factor B effect F = MSB / MSE Reject H0 if F > FU(,c-1,rc(n’-1))
A-B interaction effect F = MSAB / MSE Reject H0 if F > FU(,(r-1)(c-1),rc(n’-1))
Two-Way ANOVA (with Repetition) Summary Table
Source Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance)
F p-value
A r-1 SSA MSA = SSA/(r-1) MSA/MSE
B c-1 SSB MSB = SSB/(c-1) MSB/MSE
AB (r-1)(c-1) SSAB MSAB = SSAB/(r-1)(c-1) MSAB/MSE
Error rc(n’-1) SSE MSE = SSE/rc(n’-1)
Total rcn’-1 SST
Tukey-Kramer Comparisons Critical range (Q) test for levels of factor A
qU(,r,rc(n’-1)) from Table A.9 Perform on each of the r(r-1)/2 pairs of levels
Critical range (Q) test for levels of factor B
qU(,c,rc(n’-1)) from Table A.9 Perform on each of the c(c-1)/2 pairs of levels
X i X i' qUMSE
rn'
X j X j ' qUMSE
cn'
Main Effects and Interaction Effects
No interaction Interaction Crossing Effect
Three-Way ANOVA (with Repetition) Summary Table
Source Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance) F p-value
A i-1 SSA MSA = SSA/(i-1) MSA/MSE
B j-1 SSB MSB = SSB/(j-1) MSB/MSE
C k-1 SSC MSC = SSC/(k-1) MSC/MSE
AB (i-1)(j-1) SSAB MSAB = SSAB/(i-1)(j-1) MSAB/MSE
BC (j-1)(k-1) SSBC MSBC = SSBC/(j-1)(k-1) MSBC/MSE
AC (i-1)(k-1) SSAC MSAC = SSAC/(i-1)(k-1) MSAC/MSE
ABC (i-1)(j-1)(k-1) SSABC MSABC = SSABC/(i-1)(j-1)(k-1) MSABC/MSE
Error ijk(n’-1) SSE MSE = SSE/ijk(n’-1)
Total Ijkn’-1 SST
Main and Interaction Effects For a k-factor design
Number of main effects
Number of 2-way interaction effects
Number of 3-way interaction effects
See text (p 529) for sample plots
k
1
k!
1!(k 1)!k
k
2
k!
2!(k 2)!k(k 1) /2
k
3
k!
3!(k 3)!k(k 1)(k 2) /6
3-Factor 2-Level Design Notation
ABC(1) = a-lo, b-lo, c-lo - - -a = a-hi, b-lo, c-lo + - -b = a-lo, b-hi, c-lo - + -c = a-lo, b-lo, c-hi - - +ab = a-hi, b-hi, c-lo + + -bc = a-lo, b-hi, c-hi - + +ac = a-hi, b-lo, c-hi + - +abc = a-hi, b-hi, c-hi + + +
Contrasts and Estimated Effects
A = (1/4n’)[a + ab + ac + abc - (1) - b - c - bc]B = (1/4n’)[b + ab + bc + abc - (1) - a - c - ac]C = (1/4n’)[c + ac + bc + abc - (1) - a - b - ab]AB = (1/4n’)[abc - bc + ab - b - ac + c - a + (1)]BC = (1/4n’)[(1) - a + b - ab - c + ac - bc + abc]AC = (1/4n’)[(1) + a - b - ab - c - ac + bc + abc]ABC = (1/4n’)[abc - bc - ac + c - ab + b + a - (1)]
Effect = (1/n’2k-1)ContrastSS = (1/n’2k)(Contrast)2
Sum over replications
k = number of factorsn’ = number of replicates
3-Factor 2-Level Contrast Table
Notation A B C AB AC BC ABC
(1) - - - + + + -
a + - - - - + +
b - + - - + - +
c - - + + - - +
ab + + - + - - -
ac + - + - + - -
bc - + + - - + -
abc + + + + + + +
Using Normal Probability Plots Cumulative percentage for i-th ordered effect
pi = (Ri - 0.5)/(2k - 1)
Ri = ordered rank of I-th effect k = number of factors
Plot on normal probability paper, or use PHStat
Note deviations from zero and from the nearly straight vertical line for normal random variation
See example in text (p 535)
Fractional Factorial Design Choose a defining contrast
Typically highest interaction term Halves the number of combinations But introduces confounding interactions
Aliasing
Resolution III, IV, V designs based on types of confounding interactions
remaining in design
==> http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm==> http://www.statsoft.com/textbook/stexdes.html
Taguchi Approach Parameter design Quadratic Loss Function
Loss = k(Yi-T)2
Partition into design parameters (inner array) and noise factors
Use Signal-to-Noise (S/N) ratios to meet target or minimize/maximize response
Use of orthogonal arrays
Homework Work through Appendix 11.1 Work through Problems
11.36-38 Review for Exam #2
Chapters 8-11 Take-home “Given out” end of class Oct 25 Due beginning of class Nov 1