# A Modified Pearson’s $\chi^{2}$ Test with Application to Generalized Linear Mixed Model Diagnostics

Session:

### Accepted presentations cancelled by authors

Authors:
Cecilia Dao (University of California, Davis) (Speaker)
Jiming Jiang (University of California, Davis)
Abstract:

We propose a modified version of Pearson’s $\chi^{2}$ test for goodness-of-fit that is applicable to generalized linear mixed models
(GLMMs) diagnostics. The proposed test is based on cell frequencies, which is natural for many cases of GLMM. The procedure is simple and does not involve generalized inverse of a matrix, as was used in a previous study. Furthermore, the unknown parameters are estimated by solving a system of optimal estimating equations, which is computationally more efficient than the maximum likelihood estimators that were used in the previous study. Finally, the asymptotic null distribution of the proposed test is $\chi^{2}_{M-r-1}$, where $M$ is the number of cells and $r$ is the number of unknown parameters that are estimated. A simulation study is carried out to demonstrate the asymptotic theory as well as finite-sample performance of the proposed test, including comparison with the previous method. An example of real-data application is considered. Application to small area estimation will be discussed.

Keywords: asymptotic distribution, cell frequencies, chi-square, generalized linear mixed models, goodness-of-fit, model diagnostics