Abstract
The chi-squared test for independence in two-way categorical tables depends on the assumptions that the data follow the multinomial distribution. Thus, we suggest alternatives when the assumptions of multi nomial distribution do not hold. First, we consider the Bayes factor which is used for hypothesis testing in Bayesian statistics. Unfortunately, this has the problem that it is sensitive to the choice of prior distributions. We note here that the intrinsic Bayes factor is not appropriate because the prior distribu tions under consideration are all proper. Thus, we propose using Bayesian estimation which is generally not as sensitive to prior specifications as the Bayes factor. Our approach is to construct a 95% simultaneous credible re gion (i.e., a hyper-rectangle) for the interactions. A test that all interactions are zero is equivalent to a test of independence in two-way categorical tables. Thus, a 95% simultaneous credible region of the interactions provides a test of independence by inversion.
Highlights
There are many occasions when we need to understand the extent of the association of two attributes
This paper reviews shortcomings of methods based on adjusted chi-squared statistic and the Bayes factor, an alternative in Bayesian hypothesis testing to the chi-squared statistic, and we propose a simple method based on estimation rather than hypothesis testing to “test” for independence in two-way categorical tables
We have demonstrated the difficulties associated with the standard chi-squared test in two-way categorical tables when the multinomial assumptions are violated
Summary
There are many occasions when we need to understand the extent of the association of two attributes. When a simple random sample of individuals is taken from the population, the sampled individuals can be allocated to the cells of the r ×c categorical table (multinomial sampling) to obtain a chi-squared test of association between the two categories based on the data collected. In multinomial sampling the individuals fall in the cells independently, and in this case the standard chi-squared test for association between the two categories of r × c table is correct asymptotically. Rao and Scott (1981, 1984) investigate the effects of stratification and clustering on the asymptotic distribution of Pearson’s chi-squared statistic for goodness of fit and independence in multiway categorical tables. A Bayesian approach to the problem is desirable especially when the assumption of multinomial distribution does not hold In this case, one can use the Bayes factor for testing association versus no association.
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