Comparing K Populations With Linear Rank Statistics

原文摘要

There are many nonparametric tests available for the analysis of k independent samples. For example, the Kruskal—Wallis test (Wilcoxon rank sum test when k = 2) is widely used for detecting location differences, and Mood or Ansari—Bradley or Sigel—Tukey rank statistics are often suggested for detecting scale differences. If more general alternatives are of interest, then one can use omnibus tests such as the generalized Kolmogorov—Smirnov and Cramér—von Mises statistics [see Conover (1980, secs. 6.3 and 6.4) or Kiefer (1959)]. Unfortunately, “significant” results from these omnibus tests are hard to interpret. Instead, I would like to propose some new tests based on a 4×k table of linear rank statistics that are simple to interpret because the four rows of the table are aimed at specific alternatives: location, scale, skewness, and kurtosis. Row summary statistics consist of weighted sums of squares of the individual entries. The first two of these are the Kruskal—Wallis statistic and Mood's k-sample statistic for testing scale differences, and the last two are new k-sample test statistics relating to skewness and kurtosis alternatives. An overall omnibus statistic GLOBE is then formed by adding all four of these row summary statistics. GLOBE is also a weighted sum of column summary statistics that are computed by taking the sum of squares of each column of the 4×k table. The proposed statistics are easy to use, because the null distributions are approximately normal or chi squared, and Monte Carlo results show that they are reasonably powerful. In some cases the samples must be standardized before interpreting the nonlocation components, and new linear rank statistics theory is developed to cover these situations. An example should help illustrate some of the main ideas. Nation, Bourgeois, Clark, Baker, and Hare (1984) studied the effect of diets containing cadmium (Cd) on the neurobehavior of adult rats. The data consist of the number of platform descents during a passive-avoidance training scheme for 27 rats randomly assigned to three groups: control—82, 80, 77, 75, 72, 68, 59, 47, 42; Cd1—86, 66, 60, 51, 44, 41, 38, 29, 10; Cd5—81, 67, 38, 36, 32, 29, 20, 17, 14. The control group had no Cd in the diet, and Cd1 and Cd5 refer to daily diets containing 1 milligram and 5 milligrams, respectively, of Cd per kilogram of body weight. The omnibus test GLOBE shows that the groups are not identically distributed. More important, in what ways are the groups different? The sample means (66.9, 47.2, 37.1) suggest decreasing performance with increasing Cd levels. This result is confirmed by the first row of our linear rank table (T 11 = 2.67, T 21 = −.46, T 31 = −2.21, KW = 8.17), where T 11, T 21, and T 31 are standardized Wilcoxon statistics and KW is the usual Kruskal—Wallis statistic with p value <.02 read from a χ2 2 table. Because location differences are strong, a group mean is subtracted from each observation and then all of the data are reranked. The second, third, and fourth rows of the linear rank table are then (−.58, .26, .33, MOOD = .34), (−2.01, .04, 1.97, SKEW = 5.28), and (−1.11, .37, .73, KURT = 1.27), where MOOD, SKEW, and KURT are the row summaries that have approximate χ2 2 distributions under the null hypothesis of identical populations, and the other entries are approximately standard normal. Thus the second and fourth rows show no differences in scale or kurtosis, but the third row reveals an interesting monotonic change in skewness across samples. In fact, a simple plot shows that the control group is skewed to the left, the Cd1 group is approximately symmetric, and the Cd5 group is skewed to the right.

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