TY - JOUR

T1 - An evaluation of some tests of trend in contingency tables

AU - Cohen, Arthur

AU - Sackrowitz, Harold B.

N1 - Funding Information:
* Arthur Cohen and Harold B. Sackrowitz are Professors, Department of Statistics, Rutgers University, New Brunswick, NJ 08903. This research was supported by National Science Foundation Grant DMS-9112784.

PY - 1992/6

Y1 - 1992/6

N2 - Consider an r × c contingency table under the full multinomial model in which each classification is ordered. The problem is to test the null hypothesis of independence against the alternative that all local log odds ratios are nonnegative with at least one local log odds ratio positive. A number of tests have been proposed for this problem, including the Goodman–Kruskal gamma test; a family of linear tests studied by Agresti, Mehta, and Patel; and a test based on “C–D,” the difference of concordant and discordant pairs in the table. In this article we show that all of these tests can be improved on in some sense for most cases. In fact the preceding tests sometimes are inadmissible in a strict sense. Furthermore, we show by example that in some cases improved tests can yield substantially improved power functions. We suggest a test based on a linear statistic similar to that presented by Agresti, Mehta, and Patel but that is followed up with a test that orders points by their probabilities on sample points where the linear test would randomize. This latter test compares favorably with competitors and has optimal theoretical properties. Exact tests, which entail auxiliary randomization, are discussed, as are the p values of the test procedures, which do not use auxiliary randomization.

AB - Consider an r × c contingency table under the full multinomial model in which each classification is ordered. The problem is to test the null hypothesis of independence against the alternative that all local log odds ratios are nonnegative with at least one local log odds ratio positive. A number of tests have been proposed for this problem, including the Goodman–Kruskal gamma test; a family of linear tests studied by Agresti, Mehta, and Patel; and a test based on “C–D,” the difference of concordant and discordant pairs in the table. In this article we show that all of these tests can be improved on in some sense for most cases. In fact the preceding tests sometimes are inadmissible in a strict sense. Furthermore, we show by example that in some cases improved tests can yield substantially improved power functions. We suggest a test based on a linear statistic similar to that presented by Agresti, Mehta, and Patel but that is followed up with a test that orders points by their probabilities on sample points where the linear test would randomize. This latter test compares favorably with competitors and has optimal theoretical properties. Exact tests, which entail auxiliary randomization, are discussed, as are the p values of the test procedures, which do not use auxiliary randomization.

KW - Concordant–discordant pair

KW - Contingency table

KW - Exact test

KW - Goodman-Kruskal gamma test

KW - Inadmissible test

KW - Ordered category

KW - P value

KW - Unbiased test

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U2 - 10.1080/01621459.1992.10475228

DO - 10.1080/01621459.1992.10475228

M3 - Article

AN - SCOPUS:0001208452

SN - 0162-1459

VL - 87

SP - 470

EP - 475

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 418

ER -