Goodness of Fit Test and Tests for Contingency Tables

Chapter 6 presents the theory related to the asymptotic null distribution in goodness of fit test procedures and in tests for contingency tables. All tests related to contingency table and all goodness of fit tests are the likelihood ratio tests when the underlying probability model is a multinomial distribution. Section 6.2 is devoted to a study of multinomial distribution, where we discuss the maximum likelihood estimation of cell probabilities and study the asymptotic properties of these estimators. Some tests associated with multinomial distribution are also developed. Section 6.3 presents the role of multinomial distribution in goodness of fit tests, which are essentially the tests for validity of the model. In goodness of test procedures the most frequently used test statistics is Karl Pearson’s test statistic. We prove that the likelihood ratio test statistic and Pearson’s test statistic for testing some hypotheses in a multinomial distribution are equivalent, in the sense that their asymptotic null distributions are the same. In Sect. 6.4, we study Wald’s test procedure and score test procedure. It is proved that asymptotic null distribution of likelihood ratio test statistic, test statistics in Wald’s test procedure and score test procedure is the same. An important finding of this section is the link between a score test statistic and Karl Pearson’s chi-square test statistic. It is proved that while testing any hypothesis about the parameters of a multinomial distribution, these two statistics are identical. A contingency table provides a technique for investigating the suspected relationships. Section 6.5 is devoted to various test procedures for two way and three way contingency tables, which are again likelihood ratio test procedures when underlying probability model is a multinomial distribution. Consistency of a test procedure is an optimality criterion for a test procedure. It is briefly discussed in Sect. 6.6. Section 6.7 elaborates on the application of \(\texttt {R}\) software to validate various results proved in earlier sections, to perform the goodness of test procedures and tests for contingency tables.