Multilevel Statistical Models and Ecological Scaling

A useful way to conceptualize ecological processes operating at different spatial scales is through what Wu [1999] calls hierarchical patch dynamics. A key notion is that a few parts of a large hierarchical structure can be studied in isolation insofar as these parts are distinguished from the rest by “near-decomposability.” In effect, a segment of special interest within the hierarchical structure interacts weakly with the rest and then only asymmetrically. In this chapter, we focus on a particular kind of segment comprised of nested elements; higher levels are composed of the components of the level below. We consider multilevel statistical models that can be used to describe how variables characterizing higher levels affect processes operating at lower levels. For simplicity, consider a subset of a hierarchy with two levels The basic idea is to have a regression equation characterizing relationships at the lower, or micro, level and then have one or more of the regression coeffi- cients at the micro level a function of predictors at the macro level. At the micro level, for instance, taxa richness may be a function of stream velocity (and other things). Then at the macro level, the regression coefficient linking stream velocity to taxa richness may be a function of proximity of the stream to land used for agriculture. Thus, one can address how the relationship between stream velocity and taxa richness varies (or not) in different locations, here with locale characterized by proximity to land used for agriculture. That is, one can learn when to generalize over sites and when not to generalize over sites. One can also learn how different temporal and/or spatial scales are related. These sorts of relationships can easily be formulated as interaction effects within a conventional regression analysis. However, the usual estimation procedures will not properly characterize the uncertainty in the output, so that the confidence intervals and hypothesis tests will not perform properly. A key problem is that the model’s errors (or disturbances) are not likely to behave as if drawn independently from a single distribution. Special estimation procedures are required. Such procedures, often constructed within a multilevel framework, are well known and widely available in existing software [Raudenbush and Bryk, 2002]. Our goal, therefore, is to summarize some recent extensions of multilevel models to more complicated and realistic situations common in ecological research