مطالعه شبیه سازی آزمون ارتباط یکنواخت بر اساس واگرایی قدرت

In this paper, a simulation study is presented to analyze the behavior of the family of test statistics proposed by Conde and Salicrú [J. Conde, M. Salicrú, Uniform association in contingency tables associated to Csiszár divergence, Statistics and Probability Letters, 37 (1998) 149–154] using the ϕ-divergence measures, that include as special case the power-divergence [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, Journal of the Royal Statistic Society, Series B, 46 (1984) 440–464] for the analysis of uniform association between two classification processes, based on the local odd ratios. For the above test statistics the significance level and its power are evaluated for different sample sizes when we consider a 3 × 2 contingency table.

Let X and Y denote two categorical response variables, X and Y having I and J levels, respectively. When we classify subjects on both variables, there are I × J possible combinations of classifications. The responses (X, Y) of a subject randomly chosen from some population have a probability distribution. Let pij=P(X=i,Y=j), with pij>0,i=1,…,I,j=1,…,J and we denote by p=(p11,…,pIJ)T the joint distribution of X and Y. We display this distribution in a contingency table having I rows for the categories of X and J columns for the categories of Y. We consider the I × J contingency table with ordered rows and adjacent columns. If the responses are independent we have pij=pi.×p.j for all i and j, where View the MathML source and View the MathML source. When the responses are not independent, there is an association between them, and the question arises of how the association can be measured. It is possible to consider single summary numbers that describe relationships between the categorical response variables, X and Y. Examples include Yule’s coefficient, Pearson’s coefficients, Tschuprow’s coefficient and Cramer’s coefficient among others. These numbers, are generally called “measures of association” , and summarize the deviations of the pij from the hypotheses of independence. Bartolucci and Forcina [1], Menéndez et al. [11], Tomizawa [17] and references therein used the odd ratios to establish any measure of association of rows and columns for an I × J contingency table. In this contingency table the association may be viewed in terms of the association in each of the 2 × 2 subtables formed from the full I × J table. It is well-known that the (I − 1)(J − 1) local odd ratios formed from adjacent rows i and i + 1, and adjacent columns j and j + 1 determine all odd ratios that can be formed from the full I × J contingence table. The value θij is a measure of the row–column interaction in the 2 × 2 subtable formed from adjacent rows i and i + 1 and adjacent columns j and j + 1. Interesting topics in categorical data analysis can be seen, for example, in [2], [20], [19] and [21] and references therein. We denote π=(π11,…,π(I-1)(J-1))T and View the MathML source, where View the MathML source and View the MathML source. Goodman [7] considered the uniform association model, defined by View the MathML source Turn MathJax on Tomizawa [17] proposed a measure to represent the degree of departure from uniform association based on the information in the vector π=(π11,…,π(I-1)(J-1))T and using the Kullback–Leibler divergence. This measure of divergence is a particular case of the power-divergence measure introduced by Cressie and Read [4]. Tomizawa and Hashimoto [18] use the power-divergence to represent the degree of departure from uniform association, which includes Tomizawa’s [17] as a special case, but they do not present any test for uniformity association. This family of measures of divergence is defined for two different contingence tables, characterized by the probability distributions πu and π, by View the MathML source Turn MathJax on I0(πu,π)=limλ→0Iλ(πu,π) and I-1(πu,π)=limλ→-1Iλ(πu,π). For more details of the power-divergence, see [4] and [15]. Note that I0(πu,π) is the Kullback–Leibler divergence measure between the probability distributions πu and π. We can see that Iλ(πu,π)=0 if and only if the uniform association model holds. In Section 2, we consider a measure based on the power-divergence which represents the degree of departure from uniform association in the I × J contingency table. Finally, in Section 3 a simulation study is carried out obtaining new test statistics that are good alternatives to the classical test for this problem.

In this paper, the power-divergence family of statistics is considered for the problem of uniform association in a two way contingency table. It is obtained, by means of a wide study of simulation, that in this family of test statistics there are some elements that present a good behavior. In fact, in this study it emerges the test statistic associated with λ=2/3 which presents an attractive alternative to the likelihood ratio test. The test statistic associated with λ=2/3 has appeared in other contexts as a good alternative to the classical test statistics. See for instance [13], [4] and [5] and references therein.