Some empirical comparisons of exact, modified exact, and higher-order asymptotic tests of independence for ordered categorical variables

Abstract

We present results of a simulation study investigating the behavior of some exact, modified exact,and asymptotic tests of independence for two-way contingency tables having ordinal row and column classifications. The exact analysis utilizes the sufficient statistic for the association parameter in a loglinear model of linear-by-linear association, conditional on marginal totals. Approximate methods include a modified exact test using the mid-P value, a first-order asymptotic test usingthe efficient score statistic, and a higher-order asymptotic test using a double saddlepoint approximation. For most cases considered, the saddlepoint approximation to the exact P-value is outstanding. The extra discreteness induced by the conditioning in the exact test results in that test being conservative. For fixed sample size, the conservativeness diminishes as the numbers of rows and columns increase or as row scores and column scores for the statistic depart from equal-interval.The test based on the mid-P value gives a good compromise between ensuring exactness and eliminating conservativeness due to discreteness

Keywords:

• conditional inference
• contingency tables
• Fisher's exact test
• linear-by-linear association
• mid P-values
• ordinal data
• saddlepoint approximations