ANTIPERSPIRANT RESULTS 193 D2)/2, as in the previous section, •rm 2 = •r 2 (1 + p)/2, with p the correlation coefficient between days, we found that the variance of the averages over the two days was 0.87 of the single-day variance. However, multiple baselines can only be run at the expense of fewer treatment applications or extending the test period beyond five days. It may be more cost-effective to increase the number of panelists than to disturb the test logistics, but again this is a question that must be decided by individual circumstances. The correlation of repeated measures data appears to decrease over time. In one test with 25 subjects, seven consecutive daily pre-treatment hot-room measurements were con- ducted. The correlation coefficients of the log ratios of right over left axillae sweat rates as a function of lag time in days were as follows: Lag, days Lag correlation coefficient 1 0.71 2 0.69 3 0.69 4 0.63 5 0.59 6 0.45 The magnitude of these coefficients is in general agreement with similar work by Wooding (5), where he found correlation coefficients for pre-treatment measurements to vary in the range 0.50 to 0.87 for lag times of 1, 3, and 21 days. Multiple post-treatment measurements are different from the pre-treatment case in that each is preceded by differing numbers of treatment applications. Averaging these results is not strictly valid, since each day measures a different phenomenon. The multiple measures can be used to follow a time course, and in these cases the correlation must be taken into account in the analysis. Winer (8) outlines multivariate procedures for these repeated measures applications. TEST POWER OR THE PANEL SIZE REQUIREMENT Perhaps the most frequently asked question of a statistician is: "How many subjects do we need for this study?" To answer this question, we need to know the estimate of the test variability, usually expressed as the standard deviation, and the difference in efficacy to be detected. When the efficacy difference is expressed in terms of percent reduction, the sample size also depends on the efficacy levels. Table III has been constructed to give estimated sample sizes needed to declare two treatments significantly different at the 0.05 significance level at 80% power. The difference is specified in terms of the higher and lower percent reduction levels. By 80% power we mean that this difference would be declared statistically significant in 80% of future studies. This table is based on a standard deviation estimate of 0. ! (log units), but this value may vary for different laboratories. The sample sizes were estimated using the approximate method given by Kupper and Kafner (!5), their formula 4. General formulae can be developed to show the relationship of treatment comparison precision with panel size in multiple-treatment studies. Let n represent the number of

194 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table III Panel Size: Required Number of Subjects in Panel for Detecting a True Difference in Percent Reduction at p = 0.05 With 80% Power and Standard Deviation of 0.1 Percent reduction Percent reduction Number of Number of Higher Lower subjects Higher Lower subjects 9O 85 6 45 25 9 85 8O 11 45 2O 6 85 75 4 40 35 130 80 75 17 40 30 36 80 70 6 40 25 17 75 70 26 40 20 11 75 65 8 40 15 7 75 60 4 35 30 152 70 65 36 35 25 41 70 60 11 35 20 20 70 55 6 35 15 12 70 50 4 35 10 8 65 60 47 30 25 175 65 55 14 30 20 47 65 50 7 30 15 23 65 45 5 30 10 14 60 55 60 30 5 9 60 50 17 30 0 7 60 45 9 25 20 200 60 40 6 25 15 54 55 50 75 25 10 26 55 45 21 25 5 15 55 40 11 25 0 11 55 35 7 20 15 227 55 30 5 20 10 60 50 45 92 20 5 29 50 40 26 20 0 17 50 35 13 15 10 255 50 30 8 15 5 68 50 25 6 15 0 32 45 40 110 10 5 285 45 35 30 10 0 75 45 30 15 5 0 317 panelists per group, and assume all groups have equal numbers of panelists. Then for the RRB design the total panel size N = nt(t - 1) and for the EVC design N = 2n(t - 1). Let 0 '2 be the variance of the log(R/L) responses. Then the variances for the relative treatment effects will be: 0.2/nt = (t - 1)0.2/N for RRB estimates 0.2/2n = (t -- 1)0.2/N for EVC direct estimates (test-versus-control) 0.2/n = 2(t - 1)0.2/N for EVC indirect estimates (test-versus-test) In the EVC design, the direct (test-versus-control) comparisons have variances half as large as those of the indirect (test-versus-test) comparisons and, therefore, require half the sample size to yield a given level of power. Comparisons requiring half the sample size to attain equivalent power are termed twice as "efficient." In the RRB design, all