Hierarchical generalized linear model

In statistics, hierarchical generalized linear models (HGLM) extend generalized linear models by relaxing the assumption that error components are independent. This allows models to be built in situations where more than one error term is necessary and also allows for dependencies between error terms. The error components can be correlated and not necessarily follow a normal distribution. When there are different clusters, that is, groups of observations, the observations in the same cluster are correlated. In fact, they are positively correlated because observations in the same cluster share some common features. In this situation, using generalized linear models and ignoring the correlations may cause problems. In statistics, hierarchical generalized linear models (HGLM) extend generalized linear models by relaxing the assumption that error components are independent. This allows models to be built in situations where more than one error term is necessary and also allows for dependencies between error terms. The error components can be correlated and not necessarily follow a normal distribution. When there are different clusters, that is, groups of observations, the observations in the same cluster are correlated. In fact, they are positively correlated because observations in the same cluster share some common features. In this situation, using generalized linear models and ignoring the correlations may cause problems. In a hierarchical model, observations are grouped into clusters, and the distribution of an observation is determined not only by common structure among all clusters but also by the specific structure of the cluster where this observation belongs. So a random effect component, different for different clusters, is introduced into the model. Let y {displaystyle y} be the response, u {displaystyle u} be the random effect, g {displaystyle g} be the link function, η = X β {displaystyle eta =Xeta } , and v = v ( u ) {displaystyle v=v(u)} is some strictly monotone function of u {displaystyle u} . In a hierarchical generalized linear model, the assumption on y | u {displaystyle y|u} and u {displaystyle u} need to be made: y ∣ u ∼   f ( θ , ϕ ) {displaystyle ymid usim f( heta ,,phi )} and u ∼   f u ( α ) . {displaystyle usim f_{u}(alpha ).}