Logan, Baron, and Kohout (1995) and Guo et al. (2013) study the effect of a dental intervention on the memory of pain after root canal therapy. The intervention is a sensory focus strategy, in which patients are instructed to pay attention only to the physical sensations in their mouth during the root canal procedure.
Suppose you are interested in the longterm effects of this sensory focus intervention, because avoidance behavior has been shown to build along with memory of pain. You are planning a study to compare sensory focus to standard of care over a period of a year, asking patients to selfreport their memory of pain immediately after the procedure and then again at 1 week, 6 months, and 12 months. You use a scale from 0 (no pain remembered) to 5 (maximum pain remembered).
The betweensubject factor in your model is treatment, with two levels (sensory focus versus standard of care), and you allocate each treatment equally for a balanced design. The withinsubject factor is time, with four levels (0, 1, 26, and 52 weeks).
You want to determine the number of patients who are needed in order to achieve a power of 0.9 at significance level for the test of the interaction between time and treatment, where the contrast over time contains all pairwise comparisons. You also want to generate a plot of power versus sample size that covers the power range of 0.05 to 0.99.
The default HotellingLawley test is appropriate for this study, especially because it is the same as the Wald test in PROC MIXED with the DDFM=KR KenwardRoger degreesoffreedom method and an unstructured covariance model.
You conjecture that the mean memory of pain for each treatment follows the information in Table 46.13.
Table 46.13: Mean Memory of Pain by Treatment
Time Since Root Canal Therapy 


Treatment 
Later on Same Day 
1 Week 
6 Months 
12 Months 
Sensory Focus 
2.40 
2.38 
2.05 
1.90 
Standard of Care 
2.40 
2.39 
2.36 
2.30 
The following statements create a data set named Pain
that is to contain these means over treatment and time:
data Pain; input Treatment $ PainMem0 PainMem1Wk PainMem6Mo PainMem12Mo; datalines; SensoryFocus 2.40 2.38 2.05 1.90 StandardOfCare 2.40 2.39 2.36 2.30 ;
The variable Treatment
specifies the two treatments. The four variables PainMem0
, PainMem1Wk
, PainMem6Mo
, and PainMem12Mo
specify the mean memory of pain scores in Table 46.13.
To characterize the variability, you must specify a set of parameters that defines the entire covariance matrix of the residuals. You conjecture that the error standard deviation is the same at all four time points, with a value somewhere between and , and you account for your uncertainty by including both the lower and upper ends of this range in the sample size analysis. You believe that the correlation has a linear exponent autoregressive (LEAR) structure, with a correlation of about between measurements one week apart and a decay rate of about over oneweek intervals. The correlation matrix that contains these LEAR parameters, rounded to three decimal places, is shown in Table 46.14.
Table 46.14: Conjectured Correlation Matrix

0 
1 
26 
52 
0 
1 
0.6 
0.491 
0.399 
1 
0.6 
1 
0.495 
0.402 
26 
0.491 
0.495 
1 
0.491 
52 
0.399 
0.402 
0.491 
1 
Use the DATA= option in the PROC GLMPOWER statement to specify Pain
as the exemplary data set. Specify the between and withinsubject factors and the model by using the CLASS, MODEL, and REPEATED statements just as you would in PROC GLM for the repeated measures data analysis. Use the POWER statement to indicate sample size as the result parameter and specify the other analysis parameters, and use the PLOT statement to generate the power curves. The following statements perform the sample size analysis:
ods graphics on; proc glmpower data=Pain; class Treatment; model PainMem0 PainMem1Wk PainMem6Mo PainMem12Mo = Treatment; repeated Time contrast; power mtest = hlt alpha = 0.01 power = .9 ntotal = . stddev = 0.92 1.04 matrix ("PainCorr") = lear(0.6, 0.8, 4, 0 1 26 52) corrmat = "PainCorr"; plot y=power min=0.05 max=0.99 yopts=(ref=0.9) vary (linestyle by stddev, symbol by dependent source); run; ods graphics off;
The STDDEV= option specifies the two scenarios for the common residual standard deviation, and . The MATRIX= option defines the LEAR correlation structure, and the CORRMAT= option specifies it as the correlation matrix of the residuals. The Y=POWER option in the PLOT statement requests a plot that has power on the Y axis. (The result parameter—in this case, total sample size—is always plotted on the other axis.) The MIN= and MAX= options in the PLOT statement specify the power range. The YOPTS=(REF=) option adds a reference line at the target power value of . The VARY option specifies that the line style vary by the residual standard deviation and that the plotting symbol vary by the combination of withinsubject and betweensubject effects. The ODS GRAPHICS ON statement enables ODS Graphics.
Output 46.3.1 shows the output, and Output 46.3.2 shows the plot.
Output 46.3.1: Sample Size Analysis for Repeated Measures
Fixed Scenario Elements  

Wilks/HLT/PT Method  O'BrienShieh 
F Test  HotellingLawley Trace 
Alpha  0.01 
Correlation Matrix  PainCorr 
Nominal Power  0.9 
Computed N Total  

Index  Transformation  Source  Std Dev  Effect  Num DF  Den DF  Actual Power  N Total 
1  Time  Intercept  0.92  Time  3  176  0.900  180 
2  Time  Intercept  1.04  Time  3  226  0.903  230 
3  Time  Treatment  0.92  Time*Treatment  3  346  0.901  350 
4  Time  Treatment  1.04  Time*Treatment  3  442  0.901  446 
5  Mean(Dep)  Intercept  0.92  Intercept  1  4  0.960  6 
6  Mean(Dep)  Intercept  1.04  Intercept  1  4  0.907  6 
7  Mean(Dep)  Treatment  0.92  Treatment  1  950  0.900  952 
8  Mean(Dep)  Treatment  1.04  Treatment  1  1214  0.900  1216 
Output 46.3.1 reveals that the required sample size to achieve a power of 0.9 for the test of the Time*Treatment
interaction is 350 for the error standard deviation of 0.92 and 446 for the error standard deviation of 1.04.