Fisher's exact approach for post hoc analysis of a chi-squared test

This research is motivated by one of our survey studies to assess the potential influence of introducing zebra mussels to the Lake Mead National Recreation Area, Nevada. One research question in this study is to investigate the association between the boating activity type and the awareness of zebra mussels. A chi-squared test is often used for testing independence between two factors with nominal levels. When the null hypothesis of independence between two factors is rejected, we are often left wondering where does the significance come from. Cell residuals, including standardized residuals and adjusted residuals, are traditionally used in testing for cell significance, which is often known as a post hoc test after a statistically significant chi-squared test. In practice, the limiting distributions of these residuals are utilized for statistical inference. However, they may lead to different conclusions based on the calculated p-values, and their p-values could be over- o6r under-estimated due to the unsatisfactory performance of asymptotic approaches with regards to type I error control. In this article, we propose new exact p-values by using Fisher's approach based on three commonly used test statistics to order the sample space. We theoretically prove that the proposed new exact p-values based on these test statistics are the same. Based on our extensive simulation studies, we show that the existing asymptotic approach based on adjusted residual is often more likely to reject the null hypothesis as compared to the exact approach due to the inflated family-wise error rates as observed. We would recommend the proposed exact p-value for use in practice as a valuable post hoc analysis technique for chi-squared analysis.