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A Two-Sample Test for Equality of Means in High Dimension.

ABSTRACT: We develop a test statistic for testing the equality of two population mean vectors in the "large-p-small-n" setting. Such a test must surmount the rank-deficiency of the sample covariance matrix, which breaks down the classic Hotelling T(2) test. The proposed procedure, called the generalized component test, avoids full estimation of the covariance matrix by assuming that the p components admit a logical ordering such that the dependence between components is related to their displacement. The test is shown to be competitive with other recently developed methods under ARMA and long-range dependence structures and to achieve superior power for heavy-tailed data. The test does not assume equality of covariance matrices between the two populations, is robust to heteroscedasticity in the component variances, and requires very little computation time, which allows its use in settings with very large p. An analysis of mitochondrial calcium concentration in mouse cardiac muscles over time and of copy number variations in a glioblastoma multiforme data set from The Cancer Genome Atlas are carried out to illustrate the test.

SUBMITTER: Gregory KB

PROVIDER: S-EPMC4533933 | biostudies-literature | 2015 Jun

REPOSITORIES: biostudies-literature

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A Two-Sample Test for Equality of Means in High Dimension.

Gregory Karl Bruce KB Carroll Raymond J RJ Baladandayuthapani Veerabhadran V Lahiri Soumendra N SN

Journal of the American Statistical Association 20150601 510

We develop a test statistic for testing the equality of two population mean vectors in the "large-p-small-n" setting. Such a test must surmount the rank-deficiency of the sample covariance matrix, which breaks down the classic Hotelling T2 test. The proposed procedure, called the generalized component test, avoids full estimation of the covariance matrix by assuming that the p components admit a logical ordering such that the dependence between components is related to t ...[more]

PMID: 26279594

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Project description:When the distributional assumptions for a t-test are not met, the default position of many analysts is to resort to a rank-based test, such as the Wilcoxon-Mann-Whitney Test to compare the difference in means between two samples. The Wilcoxon-Mann-Whitney Test presents no danger of tied observations when the observations in the data are continuous. However, in practice, observations are discretized due various logical reasons, or the data are ordinal in nature. When ranks are tied, most textbooks recommend using mid-ranks to replace the tied ranks, a practice that affects the distribution of the Wilcoxon-Mann-Whitney Test under the null hypothesis. Other methods for breaking ties have also been proposed. In this study, we examine four tie-breaking methods-average-scores, mid-ranks, jittering, and omission-for their effects on Type I and Type II error of the Wilcoxon-Mann-Whitney Test and the two-sample t-test for various combinations of sample sizes, underlying population distributions, and percentages of tied observations. We use the results to determine the maximum percentage of ties for which the power and size are seriously affected, and for which method of tie-breaking results in the best Type I and Type II error properties. Not surprisingly, the underlying population distribution of the data has less of an effect on the Wilcoxon-Mann-Whitney Test than on the t-test. Surprisingly, we find that the jittering and omission methods tend to hold Type I error at the nominal level, even for small sample sizes, with no substantial sacrifice in terms of Type II error. Furthermore, the t-test and the Wilcoxon-Mann-Whitney Test are equally effected by ties in terms of Type I and Type II error; therefore, we recommend omitting tied observations when they occur for both the two-sample t-test and the Wilcoxon-Mann-Whitney due to the bias in Type I error that is created when tied observations are left in the data, in the case of the t-test, or adjusted using mid-ranks or average-scores, in the case of the Wilcoxon-Mann-Whitney.

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A Two-Sample Test for Equality of Means in High Dimension.

Publications

A Two-Sample Test for Equality of Means in High Dimension.

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