Biostatistics in Practice
Peter D. ChristensonBiostatistician
http://gcrc.LABioMed.org/Biostat
Session 2: Summarization of Quantitative
Information
Topics for this Session
Experimental Units
Independence of Measurements
Graphs: Summarizing Results
Graphs: Aids for Analysis
Summary Measures
Confidence Intervals
Prediction Intervals
Most Practical from this Session
Geometric Means
Confidence Intervals
Reference Ranges
Justify Methods from Graphs
Experimental Units_____
Independence of Measurements
Statistical IndependenceExperimental units are the smallest independent entities for addressing a scientific question in an analysis of an experiment.
“Independent” refers to the measurement that is made and the question, not the units.
Definition: If knowledge of the value for a unit does not provide information about another unit’s value, given other factors (and the overall mean) in the analysis of the experiment, then the units are independent for this measurement.
There may be a hierarchy of units.
Importance of Independence
Many basic statistical methods require that measurements are independent for the analysis to be valid.
Other methods can incorporate the lack of independence.
There can be some subjectivity regarding independence. Statistical methods use models. Models can be wrong.
Example: Units and Independence
Ten mice receive treatment A, each is bled, and blood samples are each divided into 3 aliquots. The same is done for 10 mice on treatment B.
1. A serum hormone is measured in the 60 aliquots and compared between A and B.
The aliquots for a mouse are not independent.
The unit is a mouse.
A summary statistic from a mouse’s 3 aliquots (e.g., maximum or mean) are independent.
N=10 and 10, not 30 and 30.
Example, Continued
2. One of the 30 A aliquots is further divided into 25 parts and 5 different in vitro challenges are each made to a random set of 5 of the parts. The same is done for a single B aliquot.
For this challenge experiment, each part is a unit, the values of challenge response are independent, and N=25+25.
For comparing A and B, there are only N=1+1 experimental units, the two mice.
Experimental Units in Case Study
Experimental Units in Case Study
There is a nested hierarchy of several "levels" of data: Schools, children within the schools, and diets received by every child. What would you use for the "N" for this study?
Which outcomes do you intuitively think are correlated (in common language)? Results from one child's three diets? Results from children in the same school? Schools?
Experimental Units in Case Study
N = Number of children
Results from one child's three diets cannot be modeled as independent.
Results from children in the same school also could be “correlated” (dependent). They can be modeled as independent, if the effect of school is included in the analysis. Knowing one child’s score and the school mean gives no info on another child’s score.
Units and Analysis in the Case Study
N = Number of children
Analysis:
This method is a complex generalization of methods we discuss in Session 3.
For any method, though, you need to inform the software of the correct experimental units. For some experiments, it is obvious and implicit.
Graphs:
Summarizing Results
Common Graphical Summaries
Graph Name Y-axis X-axis
Histogram Count or % Category
Scatterplot Continuous Continuous
Dot Plot Continuous Category
Box Plot Percentiles Category
Line Plot Mean or value Category
Kaplan-Meier Probability Time
Many of the examples are from StatisticalPractice.com
Data Graphical Displays
Histogram Scatter plot
Raw DataSummarized*
* Raw data version is a stem-leaf plot. We will see one later.
Data Graphical Displays
Dot Plot Box Plot
Raw Data Summarized
Data Graphical DisplaysLine or Profile Plot
Summarized - bars can represent various types of ranges
Data Graphical Displays
Kaplan-Meier Plot0.
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250.
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0 5 10 15 20Years
Kaplan-Meier survival estimate
This is not necessarily 35% of subjects
Probability of Surviving 5 years is 0.35
Graphs:
Aids for Analysis
Graphical Aids for Analysis
Most statistical analyses involve modeling.
Parametric methods (t-test, ANOVA, Χ2) have stronger requirements than non-parametric methods (rank -based).
Every method is based on data satisfying certain requirements.
Many of these requirements can be assessed with some useful common graphics.
Look at the Data for Analysis Requirements
What do we look for?
In Histograms (one variable):Ideal: Symmetric, bell-shaped.
Potential Problems:• Skewness.• Multiple peaks.• Many values at, say, 0, and bell-shaped
otherwise.• Outliers.
Example Histogram: OK for Typical* Analyses
• Symmetric.• One peak.• Roughly bell-shaped.• No outliers.
*Typical: mean, SD, confidence intervals, to be discussed in later slides.
876543210
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Intensity
Fre
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Skewed
Need to transform intensity to another
scale, e.g. Log(intensity)
1207020
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Tumor Volume
Fre
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Multi-Peak
Need to summarize with percentiles, not
mean.
Histograms: Not OK for Typical Analyses
Truncated Values
Need to use percentiles for most
analyses.
Outliers
Need to use median, not mean, and
percentiles.
1050
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Assay Result
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LLOQ
Undetectable in 28 samples (<LLOQ)
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Expression LogRatio
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Look at the Data for Analysis Requirements
What do we look for?
In Scatter Plots (two variables): Ideal: Football-shaped; ellipse.
Potential Problems:• Outliers.• Funnel-shaped.• Gap with no values for one or both variables.
Example Scatter Plot: OK for Typical Analyses
Scatter Plot: Not OK for Typical Analyses
Gap and Outlier
Consider analyzing subgroups.
Funnel-Shaped
Should transform y-value to another scale,
e.g. logarithm.
0 100 200 300 400
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EPO
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All Subjects:
r = 0.54 (95% CI: 0.27 to 0.73)
p = 0.0004
EPO < 150:
r = 0.23 (95% CI: -0.11 to 0.52)
p = 0.17
EPO > 300:
r = -0.04 (95% CI: -0.96 to 0.96)
p = 0.96
Ott, Amer J Obstet Gyn 2005;192:1803-9.Ferber et al, Amer J Obstet
Gyn 2004;190:1473-5.
Summary Measures
Common Summary Measures
Mean and SD or SEM
Geometric Mean
Z-Scores
Correlation
Survival Probability
Risks, Odds, and Hazards
Summary Statistics: One Variable
Data Reduction to a few summary measures.
Basic: Need Typical Value and Variability of Values
Typical Values (“Location”):• Mean for symmetric data.• Median for skewed data.• Geometric mean for some skewed data - details in later slides.
Summary Statistics:Variation in Values
• Standard Deviation, SD =~ 1.25 *(Average |deviation| of values from their mean).
• Standard, convention, non-intuitive values.
• SD of what? E.g., SD of individuals, or of group means.
• Fundamental, critical measure for most statistical methods.
Examples: Mean and SD
Mean = 60.6 min.
Note that the entire range of data in A is about 6SDs wide, and is the source of the “Six Sigma” process used in quality control and business.
95857565554535
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SD = 9.6 min.
201510
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OD
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Mean = 15.1 SD = 2.8
A B
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Intensity
Fre
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Skewed
1207020
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Tumor Volume
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Multi-Peak
Mean = 1.0 min.SD = 1.1 min. Mean = 70.3
SD = 22.3
Summary Statistics:Rule of Thumb
For bell-shaped distributions of data (“normally” distributed):
• ~ 68% of values are within mean ±1 SD
• ~ 95% of values are within mean ±2 SD “(Normal) Reference
Range”
• ~ 99.7% of values are within mean ±3 SD
Summary Statistics: Geometric means
Commonly used for skewed data.1. Take logs of individual values.2. Find, say, mean ±2 SD → mean and
(low, up) of the logged values.3. Find antilogs of mean, low, up. Call
them GM, low2, up2 (back on original scale).
4. GM is the “geometric mean”. The interval (low2,up2) is skewed about GM (corresponds to graph).
[See next slide]
Geometric Means
These are flipped histograms rotated 90º, with box plots.
Any log base can be used.
≈ 909.6
≈ 11.6
GM = exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09)
= 11.6
upp2 = exp(4.633+2*1.09)
= 909.6
≈ 102.8
Confidence Intervals
Reference ranges - or Prediction Intervals -are for individuals.
Contains values for 95% of individuals. _____________________________________
Confidence intervals (CI) are for a summary measure (parameter) for an entire population.
Contains the (still unknown) summary measure for “everyone” with 95% certainty.
Z- Score = (Measure - Mean)/SD
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35 45 55 65 75 85 95
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Mean = 60.6 min.SD = 9.6 min.
Z-Score = (Time-60.6)/9.6
-2 0 2
41 61 79
Mean = 0SD = 1
Standardize a measure to have mean=0 and SD=1.
Z-scores make different measures comparable.
Outcome Measure in Case StudyGHA = Global Hyperactivity Aggregate
For each child at each time:Z1 = Z-Score for ADHD from TeachersZ2 = Z-Score for WWP from ParentsZ3 = Z-Score for ADHD in ClassroomZ4 = Z-Score for Conner on Computer
All have higher values ↔ more hyperactive.Z’s make each measure scaled similarly.
GHA= Mean of Z1, Z2, Z3, Z4
Confidence Interval for Population Mean
95% Reference range - or Prediction Interval - or “Normal Range”, if subjects normal, is
sample mean ± 2(SD) _____________________________________
95% Confidence interval (CI) for the (true, but unknown) mean for the entire population is
sample mean ± 2(SD/√N)
SD/√N is called “Std Error of the Mean” (SEM)
Confidence Interval: More Details
Confidence interval (CI) for the (true, but unknown) mean for the entire population is
95%, N=100: sample mean ± 1.98(SD/√N)95%, N= 30: sample mean ± 2.05(SD/√N)90%, N=100: sample mean ± 1.66(SD/√N)99%, N=100: sample mean ± 2.63(SD/√N)
If N is small (N<30?), need normally, bell-shaped, data distribution. Otherwise, skewness is OK. This is not true for the PI, where percentiles are needed.
Confidence Interval: Case Study
Confidence Interval:
-0.14 ± 1.99(1.04/√73) =
-0.14 ± 0.24 → -0.38 to 0.10
Table 2
Prediction Interval:
-0.14 ± 1.99(1.04) =
-0.14 ± 2.07 → -2.21 to 1.93
0.13 -0.12 -0.37
Adjusted CI
close to
CI for the Antibody Example
So, there is 95% assurance that an individual is between 11.6 and 909.6, the PI.
So, there is 95% certainty that the population mean is between 92.1 and 114.8, the CI.
GM = exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09)
= 11.6
upp2 = exp(4.633+2*1.09)
= 909.6
GM = exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09 /√394)
= 92.1
upp2 = exp(4.633+2*1.09 /√394)
= 114.8
Summary Statistics:Two Variables (Correlation)
• Always look at scatterplot.• Correlation, r, ranges from -1 (perfect
inverse relation) to +1 (perfect direct). Zero=no relation.
• Specific to the ranges of the two variables.
• Typically, cannot extrapolate to populations with other ranges.
• Measures association, not causation.
We will examine details in Session 5.
Correlation Depends on Range of Data
Graph B contains only the points from graph A that are in the ellipse.
Correlation is reduced in graph B.
Thus: correlation between two quantities may be quite different in different study populations.
BA
Correlation and Measurement Precision
A lack of correlation for the subpopulation with 5<x<6 may be due to inability to measure x and y well.
Lack of evidence of association is not evidence of lack of association.
B
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0 5 10 15 20Years
Kaplan-Meier survival estimate
Actually uses finer subdivisions than 0-2, 2-4, 4-5 years, with exact death times.
Example: 100 subjects start a study. Nine subjects drop out at 2 years and 7 drop out at 4 yrs and 20, 20, and 17 died in the intervals 0-2, 2-4, 4-5 yrs.
Then, the 0-2 yr interval has 80/100 surviving.
The 2-4 interval has 51/71 surviving; 4-5 has 27/44 surviving.
So, 5-yr survival prob is (80/100)(51/71)(27/44) = 0.35.
Summary Statistics: Survival Probability
Don’t know vital status of 16 subjects at 5 years.
Summary Statistics:Relative Likelihood of an Event
Compare groups A and B on mortality.
Relative Risk = ProbA[Death] / ProbB[Death]where Prob[Death] ≈ Deaths per 100 Persons
Odds Ratio = OddsA[Death] / OddsB[Death] where Odds= Prob[Death] / Prob[Survival]
Hazard Ratio ≈ IA[Death] / IB[Death]where I = Incidence
= Deaths per 100 PersonDays