# Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test
## 4 Conclusions  
In this article, the aim was to identify whether there is actually an impact of the incorrect specification of the distribution of random effects on the inferential procedures of Wald-type hypothesis tests. Data sets were simulated with Poisson or NB response variables, which had generalized linear mixed models adjusted, taking into consideration, in some cases, a random intercept, and in others, a random intercept and slope.  
The simulation study presented here allows us to conclude that the impact of the incorrect specification appears to depend on the complexity of the structure of the random effects, the variance of the distribution of the underlying random effects and the parameters of interest. Additionally, the results indicate an impact of incorrect specification in the assumed distributions for random effects, which are quite different from the well-known characteristics of normal distribution: lognormal (positive asymmetry and high kurtosis) for random intercepts and bivariate exponential (moderate asymmetry and kurtosis) and bivariate Tukey (moderate asymmetry and high kurtosis) for random intercepts and slopes.  
For both cases--Poisson and BN with a random intercept--an impact was found from incorrectly specifying the distribution of said intercept, as greater rejection rates were obtained for hypothesis \(\mathrm{H}_{0}:\beta_{2}=0\) vs. \(\mathrm{H}_{1}:\beta_{2}\
eq 0\), where \(\beta_{2}=0\), i.e., higher probabilities of making a type I error when the true distribution of the intercept is lognormal, and also, the impact was greatest as the variance values increased for the random intercept. Moreover, evidence was found of said impact with the lowest rejection rates when the true distribution was uniform when \(\beta_{2}\
eq 0\); i.e., with lower rates for the test power.  
In the cases where generalized linear mixed models were adjusted with a Poisson or NB response and a random intercept and slope, and where the interest was focused on the Wald-type hypothesis tests for determining an effect of the parameter \(\beta_{1}\) or the parameter \(\beta_{2}\), an impact was indeed found arising from the incorrect specification of the random effects when proving hypothesis \(\mathrm{H}_{0}:\beta_{1}=0\) vs. \(\mathrm{H}_{1}:\beta_{1}\
eq 0\), for the adjustment of both the Poisson and the NB models. This impact was observed with the highest rejection rates when the null hypothesis was true; i.e., the highest probability of making a type I error was found when the real distribution of the random effects was bivariate Tukey, and with the lowest rejection rates when the null hypothesis was false, being related to the statistical power when the true distribution was a bivariate exponential. When testing hypothesis \(\mathrm{H}_{0}:\beta_{2}=0\) vs. \(\mathrm{H}_{1}:\beta_{2}\
eq 0\), no evidence was found for the incorrect specification of the distribution of the random effects.  
The results presented herein contribute to understanding the implications of an incorrect specification of the distribution of random effects in GLMMs. However, this contribution is limited by the computational behaviour of considering only two functions in the simulation study (glmer, glmer.nb), which are found in the lme4 package of software R. Considerations and comparisons with other packages and software would be ideal; but this would imply a higher computational cost, which is beyond the scope and resources of this research.