Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test

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Abstract

Generalized linear mixed models are commonly used in repeated measurement studies and account for the dependence between observations obtained from the same experimental unit. The designs of repeated measurements in which each experimental unit (e.g., subject) is proven in more than one experimental condition are widespread in psychology, neuroscience, medicine, social sciences and agricultural research. Estimation in generalized linear mixed models is often based on the maximum likelihood theory, which assumes that the assumptions about the underlying probability model are correct. These assumptions include the specification of the distribution of random effects. This research study aimed to identify the impact of the incorrect specification of this distribution on the probability of a type I error and the statistical power of the Wald test. This was achieved through a simulation study where different distributions were considered for random effects in generalized linear mixed models with Poisson and negative binomial responses. Evidence of the impact of the incorrect specification was presented in distributions for random effects different from the normal ones. Lognormal was used for random intercepts and bivariate exponential and Tukey for random intercepts and slopes. Lognormal has positive asymmetry and high kurtosis. Exponential has moderate asymmetry and kurtosis, and Tukey has moderate asymmetry and high kurtosis.

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Generalized linear mixed models: a practical guide for ecology and evolution.

Benjamin M Bolker, Mollie E. Brooks, Connie Clark … (2009)

How should ecologists and evolutionary biologists analyze nonnormal data that involve random effects? Nonnormal data such as counts or proportions often defy classical statistical procedures. Generalized linear mixed models (GLMMs) provide a more flexible approach for analyzing nonnormal data when random effects are present. The explosion of research on GLMMs in the last decade has generated considerable uncertainty for practitioners in ecology and evolution. Despite the availability of accurate techniques for estimating GLMM parameters in simple cases, complex GLMMs are challenging to fit and statistical inference such as hypothesis testing remains difficult. We review the use (and misuse) of GLMMs in ecology and evolution, discuss estimation and inference and summarize 'best-practice' data analysis procedures for scientists facing this challenge.

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Evaluating the robustness of repeated measures analyses: the case of small sample sizes and nonnormal data.

Daniel Oberfeld, Thomas Franke (2013)

Repeated measures analyses of variance are the method of choice in many studies from experimental psychology and the neurosciences. Data from these fields are often characterized by small sample sizes, high numbers of factor levels of the within-subjects factor(s), and nonnormally distributed response variables such as response times. For a design with a single within-subjects factor, we investigated Type I error control in univariate tests with corrected degrees of freedom, the multivariate approach, and a mixed-model (multilevel) approach (SAS PROC MIXED) with Kenward-Roger's adjusted degrees of freedom. We simulated multivariate normal and nonnormal distributions with varied population variance-covariance structures (spherical and nonspherical), sample sizes (N), and numbers of factor levels (K). For normally distributed data, as expected, the univariate approach with Huynh-Feldt correction controlled the Type I error rate with only very few exceptions, even if samples sizes as low as three were combined with high numbers of factor levels. The multivariate approach also controlled the Type I error rate, but it requires N ≥ K. PROC MIXED often showed acceptable control of the Type I error rate for normal data, but it also produced several liberal or conservative results. For nonnormal data, all of the procedures showed clear deviations from the nominal Type I error rate in many conditions, even for sample sizes greater than 50. Thus, none of these approaches can be considered robust if the response variable is nonnormally distributed. The results indicate that both the variance heterogeneity and covariance heterogeneity of the population covariance matrices affect the error rates.