Graphical spatial models: a new view on interpreting spatial pattern

Graphical models provide an important tool for facilitating communication between scientists, decision-makers, and statisticians—many complicated ecological processes can be described in terms of “box-and-arrow” conceptual diagrams (e.g., Shipley in Cause and correlation in biology: a user’s guide to path analysis, structural equations and causal inferences, Cambridge Universtiy Press, Cambridge, 2000; Clark and Gelfand TRENDS in Ecology and Evolution 21:375–380, 2006). In particular, problems in landscape ecology often involve modeling relationships among multiple physical and/or biological variables that may operate on differing spatial scales (e.g., Rossi et al. in Ecol Monographs 62:277–314, 1992; Legendre et al. in Ecography 25:601–615, 2002; Overmars et al. in Ecol Model 164:257–270, 2003; Brown and Spector in J Appl Ecol 45:1639–1648, 2008; Koniak and Noy-Meir in Ecol Model 220:1148–1158, 2008). These problems are inherently multivariate, though researchers commonly rely on univariate methods, such as spatial regression models, to address them. In this paper, we introduce a multivariate method—graphical spatial models—that extends path analysis to incorporate spatial autocorrelation in one or more variables in a directed graph. We show how both exogenous and endogenous ecological processes as defined by Legendre et al. (Ecography 25:601–615, 2002) and Lichstein et al. (Ecol Monographs 72:445–463, 2002) can be represented in a graph. Most importantly, we show how to translate graphs representing these ecological processes into statistically estimable models. We motivate our theoretical results using an example of stream health data from the Willamette Valley, Oregon. For these data we are interested in the spatial pattern within both riparian land use and an index of stream health, and whether there is an association between land use and stream health, after accounting for these spatial patterns. We use a graphical spatial model to address these ecological questions simultaneously. We find that the health of a stream decreases as the percent of developed land within a 120-m riparian buffer increases; interestingly, there is only evidence of spatial pattern within land use.