Abstract
The data for the tests considered here may be presented in two-way contingency tables with all marginal totals fixed. We show that Pearson's test statistic XP2 (P for Pearson) may be partitioned into useful and informative components. The first detects location differences be tween the treatments, and the subsequent components detect dispersion and higher order moment differences. For Kruskal-Wallis-type data when there are no ties, the location component is the Kruskal-Wallis test. The subsequent components are the extensions. Our approach enables us to generalise to when there are ties, and to when there is a fixed number of categories and a large number of observations. We also propose a generalisation of the well-known median test. In this situation the location-detecting first component of XP2 reduces to the usual median test statistic when there are only two categories. Subsequent components detect higher moment departures from the null hypothesis of equal treatment effects
Highlights
The idea of decomposing a test into orthogonal contrasts, as in the analysis of variance, has long been appreciated by statisticians as a way of making hypothesis tests more informative
Omnibus test statistics are partitioned into smooth components
We de ne the components of a test statistic to be asymptotically pairwise independent, with each asymptotically having the chi-squared distribution, and such that their sum gives the original test statistic
Summary
The idea of decomposing a test into orthogonal contrasts, as in the analysis of variance, has long been appreciated by statisticians as a way of making hypothesis tests more informative. Data for a generalisation of the median test that we subsequently propose, and for the Kruskal-Wallis test both with and without ties, may be presented in the form of two-way tables with xed marginal totals. The model for twoway contingency derived as a test tables with statistic for txheednmulal rhgyinpaolthteostiaslsofisligkievernow, as.ndInPseeacrtsioonn'sthXreP2e is a mwiullltbipelefamofilX iaP2r is to partitioned into components. Section six introduces a generalisation of the usual median X2 test, which is identi ed as a location detecting test the extensions permit dispersion and other e ects to be detected. N:j=n:: See the Appendix for the de nitions of the rst two polynomials and the derivation of subsequent polynomials This approach results, when there are no ties, in the rst component bDeeinngetGhe. Write gs for the c by vector with elements gs(j). The rst instructor is less variable be formalised by a 0.75, indicating no LftuhSraDtnheatrnhaeelytehsciitsrs.diTnwhhtehoeriesdsialdetusasa. lvX arP2ia;bleVtT1hVan[1] t;heVsT2ecVo2ndh.asTPh-isvaclaune
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