Goodness-of-Fit Tests for Autoregressive Logistic Regression Models and Generalized Linear Mixed Models
Goodness-of-fit is a very important concept in data analysis, as most statistical models make some underlying assumptions. When these assumptions are violated, any model inference can be suspect. Thus, a goodness-of-fit check is necessary in order to trust any conclusions drawn from the model. Herein we propose two goodness-of-fit tests, one that addresses autoregressive logistic regression (ALR) models and another that is appropriate for generalized linear mixed models (GLMMs).
Both GLMMs and ALR models are extensions of generalized linear models, a broad class of models that includes logistic regression and Poisson regression. ALR models go a step beyond typical generalized linear models by regressing upon past observations. In contrast, GLMMs go beyond the scope of generalized linear models by incorporating random effects.
For the ALR model, a chi-square test is proposed and the asymptotic distribution of the statistic is derived. General guidelines for a two-dimensional, dynamic binning strategy are provided, which make use of two types of maximum likelihood parameter estimates. For smaller sample sizes, a bootstrap p-value procedure is discussed. Simulation studies indicate that the procedure has the correct size and is sensitive to model misspecification. In particular, the test is very good at detecting the need for an additional lag. An application to a dataset relating to late-onset Alzheimer's disease is provided.
For GLMMs, we propose a Cramer-von-Mises omnibus test statistic, which extends upon a procedure applied to Poisson regression. Here, predictors of the random effects are plugged into the model to approximate a simpler, generalized linear model. The statistic is then calculated by making use of a probability integral transformation. Simulation studies indicate that the test has good size and power for a Poisson GLMM. Some ideas for future research are also proposed.