An important feature of the quasi loglikelihood is that the marginal distributions of $\boldsymbol{Y}_{hij}$ are completely known but the components of $\underline{\boldsymbol{Y}}_{hi}$, jointly, might be correlated. This means that distribution of their total, $\widehat{\boldsymbol{Y}}_{hi}$, might be also unknown, but the expectation is obtained as the total of the expectation of $\boldsymbol{Y}_{hij}$, $j=1,...,m_{hi}$. The most common assumption is to consider that $\widehat{\boldsymbol{Y}}_{hi}$ has a multinomial sampling scheme, which means that $\boldsymbol{Y}_{hij}$, $j=1,...,m_{hi}$ are independent random variables and$$\boldsymbol{\Sigma }_{hi}=\boldsymbol{\Sigma }_{hi}\left( \boldsymbol{\beta }\right) =m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) ),  \label{var1}$$where $\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right)) =\mathrm{diag}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )-\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \boldsymbol{\pi }_{hi}^{T}\left( \boldsymbol{\beta }\right)$. Since (\[def0\]) is not an approximation, the term quasi should be dropped. A weaker assumption is to consider that $\widehat{\boldsymbol{Y}}_{hi}$ has a multinomial sampling scheme with a overdispersion parameter $\nu _{hi}=1+\rho _{hi}^{2}(m_{hi}-1)$, which means that $\boldsymbol{Y}_{hij}$, $j=1,...,m_{hi}$ are correlated random variables ($Cor[\boldsymbol{Y}_{hia},\boldsymbol{Y}_{hib}]=\rho _{hi}^{2}$, $a\neq b$, $a,b\in \{1,...,m_{hi}\}$) and$$\boldsymbol{\Sigma }_{hi}=\boldsymbol{\Sigma }_{hi}(\nu _{hi},\boldsymbol{\beta })=\nu _{hi}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) ),  \label{var2}$$but the distribution of $\widehat{\boldsymbol{Y}}_{hi}$ is not in principle used for the estimators. Distributions such as Dirichlet Multinomial, Random Clumped and $m$-inflated belong this family (see Alonso et al. (2017), Morel and Neerchal (2012) and Raim et al. 