# Misspecification in Generalized Linear Mixed Models and Its Impact on the Statistical Wald Test
One strategy that has been suggested for the improvement of Wald-type tests involves the replacement of the asymptotic approximation based on normal and chi-square distributions with approximations based on t and F distributions. Several effective methods have been proposed to define the degrees of freedom in the denominator used in the t and F approximations [23]. The degrees of freedom for random effects, required for Wald t or F tests, must range between 1 and \(\mathrm{N}-1\) (where N is the number of levels of random effects). Software packages vary significantly in their approach to calculating degrees of freedom. The most straightforward approach (the default value in SAS) uses the minimum number of degrees of freedom provided by the random effects that affect the term that is being tested. Satterthwaite and Kenward-Roger (KR) approximations use more complicated rules to approach the degrees of freedom and adjust standard errors. KR, only available in SAS, generally works best (at least for mixed linear models). The Satterthwaite approximation is available in the PROC MIXED of SAS [14]. Another approach is to use the test based on the Wald-type test with the sandwich method. The sandwich procedure for estimating the covariance matrix is valid even if the model is specified incorrectly, provided that the structure of the mean of \(y_{i}\) has been specified correctly. This estimation procedure is therefore referred to as robust estimation [23]. An alternative to the Wald test for fixed effects is the likelihood ratio test (LRT). The likelihood ratio test compares the logarithm of the likelihood of two models where one is nested into the other [24]. The Wald test is less computationally intensive since it can be performed on any analysis without the need to repeat the analysis for the null model. It may bring advantages for some strategies of covariance model building [15]. Ref. [23] proposed a Bartlett-type correction for the likelihood ratio test. The essence of the Bartlett correction is to increase the LRT statistic by a scale factor, resulting in a statistic that has moments closest to a \(\chi^{2}\). Additionally, ref. [25] developed modified versions of the likelihood ratio test to infer fixed effects in linear mixed models. Namely, they derived a Bartlett correction into a test of this kind and a test based on a modified profile likelihood function. According to [26], Wald tests and likelihood ratio (LR) tests, along with Score (another test for evaluating hypotheses), are known for having incorrect type I errors; for this reason, they considered an extension of the results that [PERSON] (1984) achieved for general consistent estimators and tried to derive their own Bartlett-type corrections. In their research, they studied the behaviour of type I errors and the power of the three classical tests mentioned above, as well as that of the adjusted tests with Bartlett-type corrections through a Monte Carlo simulation. The simulation study presented here allowed us to focus on the possibilities and limitations of Wald tests applied to GLMMs. This manuscript is part of ongoing research on the strengths and limitations of Wald statistical tests taking into account the incorrect specification of distribution of random effects in these models.