Various Improved Approximations to Distributions of Quadratic Test Statistics for Dependent Rank Sums

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Various Improved Approximations to Distributions of Quadratic Test Statistics for Dependent Rank Sums

Journal Title: Biomedical Journal of Scientific & Technical Research (BJSTR) - Year 2018, Vol 9, Issue 2

Abstract

This paper presents a modified chi-square approximation to the distribution of test statistics arising from multivariate ranked data. The modification arises from an improvement to the estimated variance matrix of the responses and from corrections for continuity and skewness and kurtosis of the rank sum statistics. Kawaguchi et al. consider tests of equality of means for multivariate responses, in the presence of covariates, stratification, and tied and missing data. They propose inference based on approximating the joint distribution of Wilcoxon rank sum statistics as multivariate normal, with an estimated covariance matrix. They present a test statistic that may be expressed as a quadratic form of Wilcoxon rank sum statistics, with the variance-covariance matrix estimated using methods derived by Davis and Quade; they also apply this multivariate normal approximation to derive univariate confidence intervals. In this paper, we examine the effect of some alternative variance matrix estimates and also investigate the usefulness of the approximation of Yarnold, correcting for discreteness, skewness and kurtosis [1-3]. Consider subjects on which two or more variables (Yj1 . . . , YjD) are observed on each of M + N subjects, here indexed by j. Assume that the collection of vectors are independent, with a continuous distribution. Suppose that these sub- jects are divided into two groups, with subjects j = 1, . . ., M in the first group and subjects j = M + 1, . . . , M + N in the second group. This paper presents distributional results of use in certain hypothesis tests involving the process generating these data. Under both the null and alternative hypothesis, assume that the collection of vectors {(Yj1 . . . , YjD), j ≤M } have the same distribution, and that the collection of vectors {(Yj1 . . . , YjD), j ≥ M} have the same distribution, and in the entirety of this paper, the null hypothesis specifies that these two common distributions are the same. The alternative hypothesis is that that the common distribution of the vectors in {(j1 . . . , YjD), j ≤ M} is different from the common distribution of {(j1 . . . , YjD), j > M}. Furthermore, alternatives of interest are those for which values of one group are systematically higher than those of another. For example, Kolassa and Seifu apply this to two groups of cancer patients (early vs. advanced) and look for differences in PSA and Gleason score between these two groups [4]. As noted above, we test the null hypothesis that all of these vectors are identically distributed, vs. the alternative hypothesis that those vectors for which j ≤ M have a distribution different from that for which j > M , and again, we choose a test statistic expected to have power when there exists k for which the distribution of Yjk, with j ≤ M , is stochastically larger or smaller than that of Yjk with j > M.

Authors and Affiliations

Xinyan Chen, John E Kolassa

Keywords

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EP ID EP594797
DOI 10.26717/BJSTR.2018.09.001783
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