Annals of Mathematical Sciences and Applications

Volume 1 (2016)

Number 1

A modified Pearson’s $\chi^2$ test with application to generalized linear mixed model diagnostics

Pages: 195 – 215



Cecilia Dao (Department of Statistics, University of California at Davis)

Jiming Jiang (Department of Statistics, University of California at Davis)


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.


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

Published 12 April 2016