Type I errors and power of the parametric bootstrap goodness-of-fit test: full and limited information

In sparse tables for categorical data well-known goodness-of-fit statistics are not chi-square distributed. A consequence is that model selection becomes a problem. It has been suggested that a way out of this problem is the use of the parametric bootstrap. In this paper, the parametric bootstrap goodness-of-fit test is studied by means of an extensive simulation study; the Type I error rates and power of this test are studied under several conditions of sparseness. In the presence of sparseness, models were used that were likely to violate the regularity conditions. Besides bootstrapping the goodness-of-fit usually used (full information statistics), corrected versions of these statistics and a limited information statistic are bootstrapped. These bootstrap tests were also compared to an asymptotic test using limited information. Results indicate that bootstrapping the usual statistics fails because these tests are too liberal, and that bootstrapping or asymptotically testing the limited information statistic works better with respect to Type I error and outperforms the other statistics by far in terms of statistical power. The properties of all tests are illustrated using categorical Markov models.