In statistics, hierarchical generalized linear models extend generalized linear models by relaxing the assumption that error components are independent.
[2] The error components can be correlated and not necessarily follow a normal distribution.
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.
[3] 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.
In a hierarchical generalized linear model, the assumption on
In this hierarchical generalized linear model, the fixed effect is described by
is unobserved and varies among clusters randomly.
In order to perform parameter inference, it is necessary to make sure that the identifiability property holds.
[4] In the model stated above, the location of v is not identifiable, since for constant
[2] In order to make the model identifiable, we need to impose constraints on parameters.
The constraint is usually imposed on random effects, such as
Moreover, the generalized linear mixed model (GLMM) is a special case of the hierarchical generalized linear model.
In hierarchical generalized linear models, the distributions of random effect
do not necessarily follow normal distribution.
is the identity function, then hierarchical generalized linear model is the same as GLMM.
can also be chosen to be conjugate, since nice properties hold and it is easier for computation and interpretation.
is Gamma, and canonical log link is used, then we call the model Poisson conjugate hierarchical generalized linear models.
has the conjugate beta distribution, and canonical logit link is used, then we call the model Beta conjugate model.
There are different ways to obtain parameter estimates for a hierarchical generalized linear model.
If only fixed effect estimators are of interests, the population-averaged model can be used.
If inference is focused on individuals, random effects will have to be predicted.
[3] There are different techniques to fit a hierarchical generalized linear model.
Hierarchical generalized linear model have been used to solve different real-life problems.
For example, this method was used to analyze semiconductor manufacturing, because interrelated processes form a complex hierarchy.
[7] Hierarchical generalized linear model, requiring clustered data, is able to deal with complicated process.
Engineers can use this model to find out and analyze important subprocesses, and at the same time, evaluate the influences of these subprocesses on final performance.
[6] Market research problems can also be analyzed by using hierarchical generalized linear models.
Researchers applied the model to consumers within countries in order to solve problems in nested data structure in international marketing research.