Antedependence (AD) of order p, also known as the Markov property of order p, is a property of index-ordered random variables in which each variable, given at least p immediately preceding variables, is independent of all further preceding variables. Zimmerman and Nunez-Anton (2010) present statistical methodology for fitting and performing inference for AD models for continuous (primarily normal) longitudinal data. But analogous AD-model methodology for categorical longitudinal data has not yet been well developed. In this thesis, we derive maximum likelihood estimators of transition probabilities under antedependence of any order, and we use these estimators to develop likelihood-based methods for determining the order of antedependence of categorical longitudinal data. Specifically, we develop a penalized likelihood method for determining variable-order antedependence structure, and we derive the likelihood ratio test, score test, Wald test and an adaptation of Fisher's exact test for pth-order antedependence against the unstructured (saturated) multinomial model. Simulation studies show that the score (Pearson's Chi-square) test performs better than all the other methods for complete and monotone missing data, while the likelihood ratio test is applicable for data with arbitrary missing pattern. But since the likelihood ratio test is oversensitive under the null hypothesis, we modify it by equating the expectation of the test statistic to its degrees of freedom so that it has actual size closer to nominal size. Additionally, we modify the likelihood ratio tests for use in testing for pth-order antedependence against qth-order antedependence, where q > p, and for testing nested variable-order antedependence models. We extend the methods to deal with data having a monotone or arbitrary missing pattern. For antedependence models of constant order p, we develop methods for testing transition probability stationarity and strict stationarity and for maximum likelihood estimation of parametric generalized linear models that are transition probability stationary AD(p) models. The methods are illustrated using three data sets.

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