In statistics, hierarchical generalized linear models 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.
Overview and model.
Model.
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 formula_1 be the response, formula_2 be the random effect, formula_3 be the link function, formula_4, and formula_5 is some strictly monotone function of formula_2. In a hierarchical generalized linear model, the assumption on formula_7 and formula_2 need to be made: formula_9 and formula_10
The linear predictor is in the form:
 formula_11
where formula_3 is the link function, formula_13, formula_14, and formula_5 is a monotone function of formula_2. In this hierarchical generalized linear model, the fixed effect is described by formula_17, which is the same for all observations. The random component formula_2 is unobserved and varies among clusters randomly. So formula_19 takes the same value for observations in the same cluster and different values for observations in different clusters.
Identifiability.
Identifiability is a concept in statistics. In order to perform parameter inference, it is necessary to make sure that the identifiability property holds. In the model stated above, the location of v is not identifiable, since
 formula_20
for constant formula_21. In order to make the model identifiable, we need to impose constraints on parameters. The constraint is usually imposed on random effects, such as formula_22.
Models with different distributions and link functions.
By assuming different distributions of formula_23 and formula_2, and using different functions of formula_3 and 'formula_19, we will be able to obtain different models. 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 formula_2 do not necessarily follow normal distribution. If the distribution of formula_2 is normal and the link function of formula_19 is the identity function, then hierarchical generalized linear model is the same as GLMM.
Distributions of formula_23 and formula_2 can also be chosen to be conjugate, since nice properties hold and it is easier for computation and interpretation. For example, if the distribution of formula_23 is Poisson with certain mean, the distribution of formula_2 is Gamma, and canonical log link is used, then we call the model Poisson conjugate hierarchical generalized linear models. If formula_23 follows binomial distribution with certain mean, formula_2 has the conjugate beta distribution, and canonical logit link is used, then we call the model Beta conjugate model. Moreover, the mixed linear model is the normal conjugate hierarchical generalized linear models.
A summary of commonly used models are: 
Fitting the hierarchical generalized linear models.
Hierarchical generalized linear models are used when observations come from different clusters. There are two types of estimators: fixed effect estimators and random effect estimators, corresponding to parameters in : formula_36 
and in formula_37 , respectively. 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. There are different techniques to fit a hierarchical generalized linear model.
Examples and applications.
Hierarchical generalized linear model have been used to solve different real-life problems.
Engineering.
For example, this method was used to analyze semiconductor manufacturing, because interrelated processes form a complex hierarchy. Semiconductor fabrication is a complex process which requires different interrelated processes. 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.
Business.
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.
References.
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