100 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table III Antiperspirant Test Results (for Signed Rank Test) VCeight of Moisture Collected, Mg. Control Antiperspirant (Untreated) (Treated) Axilla Axilla Subject No. Left Right Left Right 1 251 ...... 201 2 258 ...... 150 3 ... 1006 1116 ... 4 669 ...... 702 5 ... 150 131 ... 6 ... 484 350 ... 7 ... 397 274 ... 8 783 ...... 503 9 ... 142 99 ... 10 380 ...... 151 11 ... 721 613 ... 12 522 ...... 411 Sums 5763 4701 Averages 480 390 design practice by using the right axilla for the treatment and the left for the control in half of the subjects (selected at random from the available group), and reversing these assignments for the other half. Let us assume that when the test is complete the data shown in Table III have been obtained for a test comprising 12 subjects. In this case, the pair- ing consists of the fact that both treatment and control have been used on each subject. A statistician would point out that differences be- tween the pairs of data will now reflect treatment versus control differ- ences without involving any subject-to-subject variation, and may thus have enhanced the precision of the test, provided that the true differ- ences among subjects, independent of treatments, tend to be larger than the true differences, on the average, between axillae for the average subject.* * There is an assumption implicit in this statmnent: if the magnitude of a given difference is •elated to the variation among the differences, there is a tendency to invalidate the statis- tical assumptions underlying the mathematical basis for the test. In practice, however, although this situation may exist, as a rule it is unlikely to have an important effect upon the conclusions obtained. There are remedies which may be used if it is believed that the problem is severe. The most common one is the use of a "transformation" of the data, such as the conversion of each difference to its logarithm before carrying out the statistical test. This type of transformation tends to reduce or eliminate any correlation between the magnitudes of the differences and their variation.

EASY STATISTICAl. TESTS lol The Wileoxon Signed Rank Test operates on the differences between left and right axillae in Table III rather than upon the original data. The question asked of the data is, "What is the chance that no real differ- ence exists and that the antiperspirant does not give lower values than the control?" This is a "one-tailed" test and is probably the type that would normally be used in this example. The equivalent two-tailed question would be "What is the chance that the real difference is zero, regardless of its direction?" The two-tailed test might have been used, for example, had the comparison been between two antiperspirant for- mulations and if the investigator did not have any knowledge upon which to base a prediction as to their relative efficacy. To carry out the Signed Rank Test, the algebraic difference between the treated and control axillae is obtained for each subject. It is im- material whether the treated items are subtracted from the control figures or vice versa, as long as the procedure used is consistent for all of the data. After these differences and their signs have been obtained, they are ranked in the same manner as for the Rank Sum Test, ignoring the signs. After all of the ranks are gotten, each is assigned the sign of its corresponding difference. Two sums are now possible: one of all the "negative" ranks and one of all the "positive" ones. The smaller of these two is added (Table IV). The smaller of the two rank totals is designated T (here T = --9). Tables can be found (1) which show the probabilities associated with a given smaller rank total. When the number of pairs is 12 as in this example, the following probabilities are given: Smaller Rank Total, 7' Probability 17 0.05 14 O. 025 10 0.01 7 O. 005 Since the actual rank total found was 9 and lies between 7 and 10, it can be claimed that the antiperspirant was effective, with a chance of less than 1% that the conclusion is incorrect. Like the Rank Sum Test, this test (1, 2) is insensitive to the nature of the statistical distribution, so that almost any type of numerical mea- surement or score may comprise the data. For example, instead of determining the milligrams of moisture produced, at some sacrifice of sensitivity of the comparison, each application site could have been scored on a "judgment scale" of, say, 1 to 7, calling one "dry" and seven