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# Hierarchical Model

Statistical Theory and Methods

Published Online: 15 SEP 2006

DOI: 10.1002/9780470057339.vah010

Copyright © 2002 John Wiley & Sons, Ltd

Book Title

## Encyclopedia of Environmetrics

Additional Information

#### How to Cite

Carlin, B. P. 2006. Hierarchical Model. Encyclopedia of Environmetrics. 2.

#### Publication History

- Published Online: 15 SEP 2006

This is not the most recent version of the article. View current version (15 JAN 2013)

A hierarchical model refers to a statistical model specified in stages, often with the purpose of borrowing strength across the various experimental units. For example, suppose our dataset consists of repeated measurements *Y*_{ijkl} of the toxicity detected in measurement *l* taken at dump site *k* in census block group *j* within census tract *i*. It might be natural to assume that all the observations from a particular dump site *ijk* are rather similar, so we might assume , where N denotes the normal distribution. But we might further expect that all the site-specific mean levels θ_{ijk} within a given block group *ij* are also similar, and thus postulate a *second stage* distributional assumption, namely that . Natural third and fourth stages to our model might be and , respectively. This ‘deletion of one subscript’ as we progress up the stages of the hierarchy is typical, and particularly appropriate in many environmental applications.

Estimation and inference for hierarchical models can be performed from either a frequentist or a Bayesian point of view. To clarify the distinction, consider a simple two-stage hierarchy. Here we would have a distributional model *f*(**y**|**θ**) for the observed data **y** = (*y*_{1}, …, *y*_{n}) given a vector of unknown parameters **θ** = (θ_{1}, …, θ_{k}). Denoting the second stage of the model by π(**θ**|**η**), the frequentist would then base inference for **η** on its likelihood function, which is simply the *marginal distribution* of the data:

- (1)

The Bayesian on the other hand would specify a third stage distribution *h*(**η**|**λ**) for **η**, and subsequently base inference on its posterior distribution, obtained by Bayes' rule as

- (2)

(*see* Bayesian Methods and Modeling). Either approach obviously depends on an ability to evaluate complex integrals. In the special case where , and (the so-called *compound sampling* or *conditionally independent hierarchical model*; see [5]), it is easy to show that , where *m*_{i}(*y*_{i}|**η**) = ∫ *f*_{i} (*y*_{i}|θ_{i})π_{i}(θ_{i}|**η**) dθ_{i}. This then reduces the *k*-dimensional integration in 1 to a product of *k* one-dimensional integrals. If in addition *f* and π are chosen as a *conjugate* pair of distributions (i.e. if *p*(θ_{i}|*y*_{i}, **η**) belongs to the same distributional family as π), then *m*_{i}(*y*_{i}|**η**) will be available in closed form, avoiding the integration in 1 altogether.

While such model simplifications were once critical to fitting hierarchical models, modern computational techniques have ameliorated this need to a great extent. For instance, likelihoods of the form 1 may be maximized using the EM algorithm [2] and its variants. Posterior distributions of the form 2 may be estimated using modern Markov chain Monte Carlo (MCMC) techniques, such as the Gibbs sampler [3] and the Metropolis–Hastings algorithm [4]. MCMC methods are especially useful in hierarchical Bayesian modeling since they remain powerful even for models having a large number of stages. For example, if **λ** were unknown in 2 above, than we would simply add another stage to the model with corresponding distribution *g*(**λ**|**μ**), and extend our MCMC algorithm accordingly. Carlin and Louis [1] provide an overview of computational approaches to hierarchical modeling, as well as a review of several modern software packages for fitting hierarchical models. These include SAS Proc MIXED and the BUGS package (http://www.mrcbsu.cam.ac.uk/bugs/); see also the packages described in [6] and [7].

#### Acknowledgments

This work was supported in part by National Institute of Allergy and Infectious Diseases (NIAID) Grant R01-AI41966 and by National Institute of Environmental Health Sciences (NIEHS) Grant 1-R01-ES07750.

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