# Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test
### Generalized Linear Mixed Models  
When dealing with non-Gaussian data with multiple sources of variation, a commonly used subject-specific model is the generalized linear mixed model [10]. Let \(y_{ij}\) be the response of the \(i\)th subject at the point of time \(j\). Conditional to a vector of random individual effects \(b_{i}\), all the response variables \(y_{ij}\) are independent and have density functions belonging to the exponential family:  
\[f\Big{(}y_{ij},\theta_{ij},\phi\Big{)}=\exp\Big{[}\phi^{-1}\Big{\{}y_{ij}\theta _{ij}-\psi\big{(}\theta_{ij}\big{)}\Big{\}}+c\Big{(}y_{ij},\phi\Big{)}\Big{]} \tag{1}\]  
where \(\theta_{ij}=\eta\big{(}x^{\prime}_{ij}\beta+x^{\prime}_{ij}b_{i}\big{)}\), \(\eta(\cdot)\) is a known inverse link function, \(\beta_{0}\) is an intercept, \(x_{ij}\) and \(x_{ij}\) are covariate vectors, \(\beta\) is a vector of fixed unknown regression coefficients, \(\phi\) is a scale parameter, and \(c(\cdot)\) is a function that depends only on \(y_{ij}\) and \(\phi\). In addition, \(\psi(\cdot)\) is a function that satisfies \(E(y_{ij})=\psi^{\prime}\big{(}\theta_{ij}\big{)}\) and \(Var(y_{ij})=\phi\psi^{\prime\prime}\big{(}\theta_{ij}\big{)}\). Subject-specific effects \(b_{i}\) are generally assumed to be normally distributed with mean \(0\) and a variance-covariance matrix \(D\)[4].  
The probability distributions that belong to the exponential family include normal distributions, gamma, binomial, Poisson and negative binomial, among others. The last two are used for count data and are the two distributions that are used in this article to conduct the simulation study. The Poisson distribution (2) is the cornerstone for modelling these data [27]. This is a parameter model and its conditional mean is supposed to be equal to the conditional variance [28].  
\[f(y|\lambda)=\frac{e^{-\lambda}\lambda y}{y!},\ y=0,1,2,\ldots \tag{2}\]  
where \(E(Y)=\lambda\) and \(Var(Y)=\lambda\), with \(\lambda>0\).  
In many real-life applications, the simple Poisson model cannot describe the data, because, generally speaking, the variance of the sample is higher than the mean obtained by the simple Poisson model [23]. For count data with extra Poisson variation (count data with overdispersion and extra zeros), several modifications or extensions to the Poisson model have been proposed, such as the negative binomial distribution (NB) (3).  
\[\mathrm{f}(y|\mu,\alpha)=\begin{pmatrix}y+\frac{1}{\alpha}-1\\ \frac{1}{\alpha}-1\end{pmatrix}\biggl{(}\frac{1}{1+\alpha\mu}\biggr{)}^{\frac{1 }{\alpha}}\biggl{(}\frac{\alpha\mu}{1+\alpha\mu}\biggr{)}^{y}y=0,1,2,\ldots \tag{3}\]  
where \(\mathrm{E}(\mathrm{Y})=\mu\) and \(\mathrm{Var}(\mathrm{Y})=\mu+\alpha\mu^{2}\), with \(\mu>0\) and \(\alpha>0\).  
NB generalizes the Poisson model through the relaxation of the Poisson assumption of equidispersion (allowing the variance to exceed the mean). This is achieved through the addition of another source of variability, the dispersion parameter, k, beyond which the Poisson process is derived. The added dispersion parameter allows the variance to exceed the mean, and therefore, it allows the NB distribution to account for the overdispersion. The dispersion is expressed as parameter k in some formulations. When this dispersion parameter becomes zero, the NB model is reduced to the Poisson model [24].