## Hierarchical Models

One way to handle the problem of overparametrization is to fit a model using data collected from multiple settings (e.g., multiple locations, species, or times). However,

It should be clear by now that hierarchical models can be viewed as a special type of latent variable model, in which the common population parameters function as the unobserved, latent variables. There are also obvious relations to the random effects model described for ANOVA. However, while hierarchical models can be analyzed in the classical regression framework, this is usually cumbersome and requires some significant approximating assumptions. Therefore, it is more natural and common to work with hierarchical models in the Bayesian framework, as described in the next section.

Figure 6 Structure of a hierarchical model, assuming a normal population distribution described by parameters ^ and A. Reproduced from Borsuk ME, StowCA, Higdon D, and Reckhow KH (2001) A Bayesian hierarchical model to predict benthic oxygen demand from organic matter loading in estuaries and coastal zones. Ecological modelling 143: 165-181 with permission from Elsevier.

assuming common parameter values across all settings may not always be valid. This assumption can be relaxed by adopting a 'hierarchical modeling approach'. Under the hierarchical structure, each setting has its own set of parameter values, but some commonality in values is assumed across settings. This commonality is structured by an underlying population distribution, thereby avoiding problems of overfitting (Figure 6). The hierarchical approach is, therefore, a practical compromise between entirely site-specific and globally common parameter estimates. It can also be seen as a way to avoid the problems of limited local data by 'borrowing strength' from other settings on the basis of the population distribution.

Once one begins to think in terms of hierarchical models, it becomes clear that they might apply in even the most conventional of settings. For example, in an analysis of data collected on multiple individuals from the same population, we can expect each individual to have a different response to measured predictor variables, depending on other variables that were not measured. These are 'random effects' that can be modeled as hierarchical parameters. When this is done, it is generally the case that the prediction interval for a new value ofy, given the value of x, is substantially different and wider than the interval obtained from a classical, nonhierarchical fit. This is because, by allowing each individual to have its own parameter values, we effectively reduce the amount of information we have to estimate those parameters. However, if the hierarchical model is believed to be a more realistic description, then the higher uncertainty represents a more proper translation of our knowledge into predictions.

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