ANTIPERSPIRANT RESULTS 181 Table I (continued) Significance levels No. of Standard deviation (p-values) Study Sides no. Treatments Subjects Subjects Axillae effect Side Interaction Normality 801 4 44 0.324 0.070 0.012 0.42 0.54 0.51 802 4 46 0.289 0.090 0.019 0.31 0.12 0.67 803 4 43 0.360 0.070 0.077 0.01 0.49 0.90 804 4 49 0.289 0. 101 0.065 0.34 0.30 0.49 805 4 45 0.410 0. 101 0.065 0.01 0.50 0.35 806 4 54 0.355 0. 163 0.070 0.03 0.50 0.15 807 4 38 0.417 0.112 0.075 0.01 0.32 0.75 809 4 50 0.518 0. 161 0.064 0.05 0.95 0.62 811 4 43 0.407 0.140 0.032 0.29 0.11 0.06 812 4 61 0.417 0. 144 0.089 0.01 0.80 0.15 813 4 55 0.450 0.120 0.114 0.01 0.55 0.15 901 4 48 0.336 0.062 0.025 0.05 0.20 0.91 902 5 59 0.375 0.094 0.052 0.01 0.06 0.15 903 4 64 0.477 0.093 0.059 0.01 0.84 0.15 904 5 67 0.338 0.072 0.085 0.01 0.26 0.15 905 5 62 0.390 0.091 0.019 0.27 0.27 0.10 906 5 62 0.419 0.093 0.036 0.05 0.68 0.93 907 3 36 0.356 0.073 0.034 0.06 0.69 0.11 908 5 63 0.384 0.086 0.052 0.01 0.73 0.18 909 3 71 0.369 0.081 0.059 0.01 0.73 0.18 910 4 70 0.395 0.073 0.080 0.01 0.79 0.84 * n/a, not available. As in the two-treatment case, the analysis of variance procedure is composed of a "whole plot" analysis, with subjects as experimental units (EUs), and a "split plot" analysis, with axillae as EUs. RESULTS OF POSTRT ANALYSIS ON PAST CLINICAL STUDIES Table I summarizes the results of statistical analyses for 70 studies involving from two to seven treatments and panel sizes from 16 to 71. The whole plot standard deviation, a measure of the variation due to differences in sweat rates among subjects, averaged 0.400 in log units. The split plot standard deviation, a measure of variation due to differences in sweat rates between axillae within subjects, averaged 0.091 in log units. This result explains why subjects are not often used as experimental units for treatments (i.e., one treatment per subject applied to both axillae), as the precision of such studies would be much lower than in the usual split plot design for a given number of test subjects. The sides, or laterality, effect was statistically significant at the 0.05 level in 31, or 44%, of the studies. The estimated sides effect varied across studies, ranging in value from -0.02 to 0.13. The magnitude of this effect averaged 0.04 units in the log metric, or slightly below 10% sweat increase of the right over the left axilla, consistent with the estimated average baseline R/L ratio of 1.13. This further confirms the findings of Wooding and Finklestein (6) that the sides term must be included in the model to remove this rather large effect from the error term.

182 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table I also lists results of normality tests that were performed on the residuals (observed minus predicted log sweat rates). These results showed significant departure from normality in only three studies, which lent additional support to the use of the loga- rithmic transformation of the data. The normality tests used were the Shapiro-Wilk test for 50 subjects or less, and the Kolmogoroff-Smirnoff test for greater than 50 subjects. The sides by treatment interaction effect was rarely significant at the 0.05 level, oc- curring in only two studies. These results demonstrate that the interaction effect is negligible. The main plot analysis can therefore be dropped from the analysis, as its only purpose is to test this interaction. Note that if the interaction mean square were mistakenly tested with the split plot error term, instead of the main plot error term, the incidence of "significant" results would greatly increase, giving rise to erroneous con- clusions. A SIMPLER MODEL FOR POST-TREATMENT DATA ANALYSIS Since the treatment by side interaction is nonexistent, the main plot analysis is no longer of interest, and only the split plot analysis need be considered. The split plot analysis is essentially an analysis of the log(R/L) ratios or, equivalently, the differences of the log sweat rates within subjects. To develop the model, consider the two-treatment case where the log ratio of the post- treatment sweat output of right to left axillae, log(R/L) = log(R) - log(L), is: Pij = Yij2m -- Yijlm, for subject j in group i, m • m'. Substituting for the Y terms using the POSTRT model and noting that the whole plot factor levels cancel within a given subject, the resulting model is: P•i = 0t2 - )t•) + (% -q-l) + ½P•i for subjects in group 1, and P2j = ()k2 -- )k l) -- (q-2 -- q'l) •- eP2j for subjects in group 2, where the variability terms EPlj = •j22(1) - •jll(1) and •P2j = •j21(2) -- •j12(2) are N (0, (Tp2), (Tp 2 = 2or 2. Thus the variance of the differences are twice the variance of the split plot variance cr 2. To further simplify the model, we can let )t = k 2 - )t• represent the sides effect and q- = q-2 - q-• represent the treatment effect, and write the model as: P•i = )t + q- + ½P•i for subjects in group 1, and P2j = k - q- + eP2j for subjects in group 2, IfPs. and P2. represent the mean differences over subjects in each group, then the sides effect, )t, and treatment effect, % are estimated, respectively, as: Lp = (P•. + P2.) / 2, and Tp = (PL - P2.) / 2. An estimate of crp 2 is afforded by the pooling the within-group sample variances of the Pii as follows: Sp 2 = -- P1.) + - P2.) 2 (nl • , -3- n 2 -- 2)