ATest for Homogeneity of Odds Ratios in Ordered 2  2 Tables Arthur Cohen *; **; 1 , John Kolassa ***; 1 , and H. B. Sackrowitz **; 1 1 Department of Statistics, Rutgers University, Piscataway NJ, USA Received 3 February 2004, revised 10 July 2004, accepted 23 August 2004 Summary Consider K ordered 2  2 contingency tables. A new test of the null hypothesis that the odds ratios of these tables are equal vs the alternative hypothesis that the odds ratios are nondecreasing, is recom- mended. The test is exact (non-asymptotic), is easily carried out (software is available), and has other favorable properties. Key words: Directed chi-square test; Conditional P-values; Linear test; Pool-adjacent-viola- tors algorithm. 1 Introduction Consider K 2  2 contingency tables. Let X ijk , i ¼ 1; 2; j ¼ 1; 2; k ¼ 1; ... ; K be the cell frequency in the ij-th cell of the k-th table. Let q k be the odds ratio for table k. Test the hypothesis H 0 : q 1 ¼¼ q K vs. H 1 : q 1  q K , with at least one strict inequality. Let X ij: ¼ P K k¼1 X ijk , X i:k ¼ P 2 j¼1 X ijk , X :jk ¼ P 2 i¼1 X ijk , X ::k ¼ P i;j X ijk . To test H 0 vs. H 1 we will condition on all marginal totals, i.e., we condi- tion on X :jk , X i:k , and X 11: . Conditioning on these variables leaves ðX 111 ; X 112 ; ... ; X 11ðK1Þ Þ 0 ¼ X ð1Þ , the subvector of X ¼ðX 111 ; X 112 ; ... ; X 11K Þ 0 as the random vector upon which to base a test statistic. This is so, since X 11K ¼ X 11:  P K1 k¼1 X 11k . This testing problem has been studied by Hirotsu [1982]. Hirotsu considered a large sample test, based on a statistic which is a linear function of the efficient scores evaluated at ^ q, the maximum likelihood estimator of the common log odds ratio parameter. Hirotsu’s test statistic is of the form u 0 X where u 1  u k are constants. Agresti [1990, Section 7.5.3] discusses this problem. He notes that if there is a concomitant variable and the alternative to H 0 is that log q k changes linearly across the strata (k tables), then Zelen [1971] gives an exact test and Breslow and Day [1980, p. 142] give an asymptotic chi-square test. In this paper we offer an exact test (non-asymptotic) for testing H 0 vs. H 1 . The test is a directed chi-square test based on the methodology of Cohen et al. [2003]. The test has desirable properties. Namely, it is exact (non-asymptotic), it is admissible, it is easily carried out (software is available), and it has a somewhat more robust power function than the linear test. The new test is described in the next section. Section 3 contains some power comparisons with Hirotsu’s test and for contrast with the two-sided test proposed by Breslow and Day [1980]. Section 3 contains two data sets which serve as examples for the use of the proposed test. A third artificial example is also offered to illustrate the steps in carrying out the directed chi-square test as well as to illustrate its power advantages. * Corresponding author: e-mail: artcohen@rci.rutgers.edu ** Research supprted by NSF Grant DMS 0092659, R01 Grant CA 063050 *** Research supprted by NSA Grant MDA 904-02-1-0039 Biometrical Journal 46 (2004) 6, 633 – 641 DOI: 10.1002/bimj.200410065 # 2004 WILEY-VCH Verlag GmbH &Co. KGaA, Weinheim