Type I and Type II Error Under Random‐Effects Misspecification in Generalized Linear Mixed Models

Generalized linear mixed models (GLMMs) have become a frequently used tool for the analysis of non-Gaussian longitudinal data. Estimation is based on maximum likelihood theory, which assumes that the underlying probability model is correctly specified. Recent research is showing that the results obtained from these models are not always robust against departures from the assumptions on which these models are based. In the present work we have used simulations with a logistic random-intercept model to study the impact of misspecifying the random-effects distribution on the type I and II errors of the tests for the mean structure in GLMMs. We found that the misspecification can either increase or decrease the power of the tests, depending on the shape of the underlying random-effects distribution, and it can considerably inflate the type I error rate. Additionally, we have found a theoretical result which states that whenever a subset of fixed-effects parameters, not included in the random-effects structure equals zero, the corresponding maximum likelihood estimator will consistently estimate zero. This implies that under certain conditions a significant effect could be considered as a reliable result, even if the random-effects distribution is misspecified.

[1]  P. Diggle,et al.  Analysis of Longitudinal Data. , 1997 .

[2]  G Molenberghs,et al.  The impact of a misspecified random‐effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models , 2008, Statistics in medicine.

[3]  A. Agresti.  Categorical data analysis , 1993 .

[4]  M. Aitkin.  A General Maximum Likelihood Analysis of Variance Components in Generalized Linear Models , 1999, Biometrics.

[5]  J. D. Kalbfleisch,et al.  Conditions for consistent estimation in mixed-effects models for binary matched-pairs data† , 1994 .

[6]  G. Molenberghs,et al.  Models for Discrete Longitudinal Data , 2005 .

[7]  T A Louis,et al.  Random effects models with non-parametric priors. , 1992, Statistics in medicine.

[8]  W. Pan,et al.  Small‐sample adjustments in using the sandwich variance estimator in generalized estimating equations , 2002, Statistics in medicine.

[9]  P. Heagerty,et al.  Misspecified maximum likelihood estimates and generalised linear mixed models , 2001 .

[10]  R. Platt,et al.  A generalized linear mixed models approach for detecting incident clusters of disease in small areas, with an application to biological terrorism. , 2004, American journal of epidemiology.

[11]  J. Kent.  Robust properties of likelihood ratio tests , 1982 .

[12]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[13]  Eric R. Ziegel,et al.  Multivariate Statistical Modelling Based on Generalized Linear Models , 2002, Technometrics.

[14]  J. Kalbfleisch,et al.  The effects of mixture distribution misspecification when fitting mixed-effects logistic models , 1992 .

[15]  G. Verbeke,et al.  The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data , 1997 .

[16]  Alan Agresti,et al.  Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies , 2004, Comput. Stat. Data Anal..

[17]  J. Kalbfleisch,et al.  An Algorithm for Computing the Nonparametric MLE of a Mixing Distribution , 1992 .

[18]  T. Derouen,et al.  A Covariance Estimator for GEE with Improved Small‐Sample Properties , 2001, Biometrics.

[19]  Geert Molenberghs,et al.  Validation of surrogate markers in multiple randomized clinical trials with repeated measurements: canonical correlation approach. , 2004 .

[20]  Marie Davidian,et al.  A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution. , 2002, Biostatistics.

[21]  Alan Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[22]  R J Tempelman,et al.  Generalized linear mixed models in dairy cattle breeding. , 1998, Journal of dairy science.