ANTIPERSPIRANT DATA ANALYSIS 17 the DM. The average of the estimates of percent reduction using the WFM are 3.68 larger than the average of the estimates using the DM. In 12 of the 15 studies, the percent reduction produced by the ARM is larger than the corresponding estimate produced by the DM. The ARM produced estimates that averaged 1.22 more than those produced by the DM. There are theoretical reasons that explain the differences in the estimates shown by these empirical results. A basic statistical result is that the arithmetic mean of a sample is an unbiased estima- tor 4. Since the DM uses the mean of the sample of percent reductions in sweating as an estimator, this method will provide an unbiased estimator, one that tends (on the average) to be neither larger nor smaller than the true efficacy of the antiperspirant being tested. It can be mathematically shown that for any given antiperspirant study, the estimated percent reduction produced by the WFM will always be larger than the corresponding estimate produced by the DM 5. Thus the WFM will tend to overestimate the efficacy of an antiperspirant. The estimate produced by the ARM uses the average of ratios. It is for this reason that in Table I the estimates for percent reduction for the ARM tend to be slightly larger than those produced by the DM 6. When comparing estimators, an unbiased estimator is generally preferred to one that is biased. For this reason, the DM is preferable to both the ARM and the WFM. A second point that is often considered when comparing methods of estimation is the variability associated with the estimators 7. This can be thought of as a measure of precision, and may be assessed by examining the widths of the confidence intervals estimates. In Table II we present 95% confidence intervals calculated by the WFM, ARM, and DM for the fifteen antiperspirant studies introduced in Table I. The average width of the ARM confidence intervals is 2.52 less than the average width of the DM confidence intervals. Thus it appears that the ARM estimates are less variable than the DM estimates, and for the same sample sizes, this tends to be true. However, baseline measurements must be collected to obtain the ARM estimates, while this is not necessary for the DM to be used. If baseline measurements were not collected, these resources could be used to collect more posttreatment measurements. Thus, for similar expenditures, the ARM and DM would produce confidence intervals of comparable widths. The average width of the WFM intervals is 0.22 less than the average width of the DM intervals. These empirical results suggest there is little difference in the variability of the two estimators. Finally, for any statistical procedure to be used, it should be statistically valid. By the validity of the procedure we mean certain conditions (assumptions) must be met for any 4 In this context, an unbiased estimator will on the average be equal to the value it is estimating. 5 This is due to the relationship between the geometric mean (calculated for the WFM) and the arithmetic mean (calculated for the DM). 6 When the sample average ratio of two variables is used to estimate the ratio of the true averages of the two variables, the estimate is generally not unbiased. 7 A comparison of the variability of different estimators is of most interest when comparing different unbiased estimators.

18 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table II 95% Confidence Intervals for Mean Percent Reductions Method ARM WFM DM Study Interval Width Interval Width Interval Width 1 - 12.78 2 -0.35 3 12.48 4 13.62 5 10.80 6 15.11 7 20.87 8 25.58 9 33.94 10 35.72 11 38.11 12 39.83 13 44.19 14 49.17 15 50.96 10.46 23.24 - 12.96 19.65 20.00 1.94 26.50 14.02 12.29 26.34 12.72 10.38 34.64 23.84 5.05. 30.43 15.32 13.37 34.49 13.62 23.58 40.34 14.76 27.85. 49.98 16.04 40.32 48.38 12.66 34.43 49.01 10.90 37.83 50.73 10.90 40.19. 56.41 12.22 50.47. 63.93 14.76 45.97 68.88 17.92 56.04. 17.06 30.02 - 17.55, 10.21 27.76 25.60 23.66 -3.04, 20.66 23.70 28.13 15.84 10.39, 25.69 15.30 27.76 17.38 6.57, 25.33 18.76 35.46 30.41 4.81, 33.51 28.70 29.36 15.99 10.63, 26.05 15.42 35.24 11.66 20.54, 32.74 12.20 42.35 14.50 24.60, 39.88 15.28 55.90 15.58 35.38, 53.22 17.84 50.32 15.89 30.29, 47.55 17.26 52.61 14.78 36.13, 49.17 13.04 53.27 13.08 37.24, 50.74 13.50 59.89 9.42 48.48, 58.14 9.66 67.51 21.54 44.04, 64.30 20.26 73.65 17.61 51.47, 73.51 22.04 statistical method to "work properly." These conditions involve how the data are collected, the type of data, the distribution of the data, etc. If the appropriate conditions are not met, a statistical method may yield unreliable results. It has been noted that the distribution of adjusted ratios should have a normal distri- bution for the ARM to be valid. An examination of the distribution of approximately 5000 adjusted ratios calculated from vast amounts of historical data (Figure 1) shows that this distribution is definitely not symmetric. Thus the distribution is not normal. This means that for small samples the ARM is not appropriate. However, when more than thirty subjects are used in antiperspirant studies, the t-statistic is approximately valid and the mentioned criticism is no longer a worry. The WFM performs an analysis of variance to analyze antiperspirant studies, and the data must again be normally distributed. Since the milligrams of sweat collected are not normally distributed (Figure 2), the WFM analyzes log-transformed data. While the transformed data are more nearly normally distributed than the original milligrams of sweat collected, the transformed data are not exactly normal it is somewhat nonsym- metric (Figure 3). Thus the WFM is approximately theoretically valid. The DM analyzes the collection of individual percent reductions in sweating for each subject. The distribution of percent reductions is not normal (Figure 4). Due to the nonsymmetry of this distribution, we recommend that over thirty panelists be used to provide percent reductions in this manner the DM would be approximately theoreti- cally valid. DISCUSSION The adjusted ratio method and direct method agree quite well, with the adjusted ratio