# Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test  
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1 Departamento de Ciencias Administrativas, Instituto Tecnologico Metropolitano, Medellin 050036, Colombia 1  
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2 Escuela de Estadistica, Universidad Nacional de Colombia, Medellin 050036, Colombia 2  
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1 Departamento de Ciencias Administrativas, Instituto Tecnologico Metropolitano, Medellin 050036, Colombia 13 Escuela de Ingenieria Industrial, Universidad Serlor de Sipan, Chiclayo 14000, Peru 3  
[PERSON]  
1 Departamento de Ciencias Administrativas, Instituto Tecnologico Metropolitano, Medellin 050036, Colombia 13 Escuela de Ingenieria Industrial, Universidad Serlor de Sipan, Chiclayo 14000, Peru 3  
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authors have addressed the impact of incorrect specification of random-effects distribution in GLMMs [4; 8; 9; 10; 11]. Other types of incorrect specification are also possible, including incorrect specification of the link function, or not considering the presence of overdispersion, among others. Many authors have voiced concern about tests to detect incorrect specifications. For example, ref. [6] proposed a two-stage diagnostic method to detect incorrect specifications of the random-effects model in a GLMM; ref. [12] recommended two diagnostic tests based on the information matrix of the model; ref. [13] stated that, in general, if the model selection for random effects in a GLMM is incorrectly specified, then parameter estimation and inference procedures such as the Wald test may also be affected. Therefore, assessing the results is always recommended. Wald Z, \(\chi^{2}\) and t and F tests for GLMMs evaluate a null non-effect hypothesis by adjusting the parameter estimates or parameter combinations by their estimated standard errors and comparing the resulting test statistic with the zero [14]. Several studies refer to the use of the Wald test for the assessment of statistical significance. For example, [15; 16; 17; 18] used such a test in the context of modelling non-linear mixed effects and evaluating their type I errors and statistical power. In basic research, analysts often emphasize avoiding type I errors more than avoiding type II errors [19]. On the other hand, the statistical power analysis (i.e., the probability that a test will reject the null hypothesis when the null hypothesis is false, or alternatively, \(1-\mathrm{P}\)(type II error)), has gained wide acceptance among scientists over the last thirty years. The number of searches on the Thomson Reuters science website for'sampl * and power analysis' increased from 115 (1996-2000) to 214 (2001-2005 period) and 265 (2006-2010) [20]. Several studies have assessed type I errors and statistical power related to the inference of the fixed parameters. For example, ref. [21] extensively studied how the incorrect specification of the size of the model's cluster affects inference in joint modelling, using the Wald test and type I error, as well as the power associated with it. Refs. [4; 22] addressed the impact of incorrect specification of random-effects distribution on type I errors and the power of the Wald test for the mean structure in GLMMs. Wald Z and \(\chi^{2}\) tests are only suitable for GLMMs without overdispersion, while Wald t and F tests account for uncertainty in overdispersion estimations. This uncertainty depends on the number of degrees of freedom of the residual, which can be very difficult to calculate because the actual number of parameters used by a random effect ranges between 1 (i.e., a single standard deviation parameter) and \(\mathrm{N}-1\) (i.e., a parameter for each additional level of random effect) [14]. Although a comprehensive performance evaluation for small samples of the asymptotic Wald-type test aimed at evaluating fixed effects on the mixed model has not been reported, there is evidence indicating that a normal or chi-square approximation is unreliable [23].