# Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test
## 3 Results
### Models with Random Intercept  
Figures 3 and 4 show the results for type I errors and the power of the statistical Wald test when testing \(\mathrm{H_{0}}:\beta_{2}=0\) vs. \(\mathrm{H_{1}}:\beta_{2}\
eq 0\) in generalized linear mixed models with Poisson and NB response variables, respectively.  
In the first column of Figure 3, we can observe the rejection rates for the null hypothesis \(\mathrm{H_{0}}:\beta_{2}=0\) when in fact, \(\beta_{2}=0\). In this case, we expect rejection rates around the significance level of 5%. For the random intercept with lognormal distribution (dash-dot line), we note that rejection rates are the greatest, thus indicating an impact of the incorrect specification of the distribution of the random intercept.  
Columns 2, 3 and 4 from Figure 3 correspond to the rejection rates of the hypothesis \(\mathrm{H_{0}}:\beta_{2}=0\) for \(\beta_{2}=0.5,\;1.0,\;1.5\), respectively. Here, it is expected that the rejection rates go up to 100% as both the value of \(\beta_{2}\) and the value of \(\mathrm{n_{i}}\) increase. The values of 0.5 and 1.5 for the parameter \(\beta_{2}\) in the simulation study were selected intentionally because with them we obtained rejection curves that show a gradual increase. From Figure 3, it can be seen that for all cases, except when the true distribution is lognormal, the highest rejection rates  
\begin{table}
\begin{tabular}{c c c c} \hline
**Cases** & **Hypothesis** & **Poisson** & **BN** \\ \hline Case 1 & \(\mathrm{H_{0}}:\beta_{1}=0\) vs. \(\mathrm{H_{1}}:\beta_{1}\
eq 0\) & \(\beta_{2}\) fixed at 1.0 & \(\beta_{2}\) fixed at 1.0 \\ Case 2 & \(\mathrm{H_{0}}:\beta_{2}=0\) vs. \(\mathrm{H_{1}}:\beta_{2}\
eq 0\) & \(\beta_{1}=0,\;0.3,\;0.6,\;0.9\) & \(\beta_{1}=0,\;0.7,\;1.4,\;2.1\) \\ Case 2 & \(\mathrm{H_{0}}:\beta_{2}=0\) vs. \(\mathrm{H_{1}}:\beta_{2}\
eq 0\) & \(\beta_{2}=0,\;0.04,\;0.08,\;0.12\) & \(\beta_{2}=0,\;0.2,\;0.4,\;0.6\) \\ \hline \end{tabular} Source: authors.  
\end{table}
Table 2: Hypothesis of interest and values of the parameters used in the simulations of a GLMM with random intercept and slope.  
correspond to the normal distribution. Moreover, for the uniform distribution, represented by the dotted line, there are rates below the normal distribution, which also indicates an impact of the incorrect specification of the distribution of the random intercept.  
Figure 4 exhibits a behaviour similar to that found in Figure 3, as it displays an impact of incorrectly specifying the distribution of the random intercept by assuming it is normal when it actually comes from a lognormal distribution. Similar to the results for a Poisson GLMM, the impact is greater as the variance values increase (higher rejection rates appear, i.e., greater probability of type I error).  
Figure 3: Type I errors and power (H\({}_{0}:\beta_{2}=0\) vs. H\({}_{1}:\beta_{2}\
eq 0\)) in a Poisson GLMM with a random intercept. Type I errors and power for the Wald test with H\({}_{0}:\beta_{2}=0\) vs. H\({}_{1}:\beta_{2}\
eq 0\) in a Poisson GLMM with random intercept, with \(\sigma_{\rm b}^{2}=1,2,4,8\) and n\({}_{\rm i}=5,10,15,20\) and four distributions for random intercept: normal (solid line), mixture of two normal distributions (dashed line), uniform (dotted line) and lognormal (dash–dot line).