  Extrapolation is defined as making predictions beyond the range of the data
used to estimate a statistical model. In ecological studies, it is not always
obvious when and where extrapolation occurs because of the multivariate nature
of the data. Previous work on identifying extrapolation has focused on
univariate response data, but these methods are not directly applicable to
multivariate response data, which are more and more common in ecological
investigations. In this paper, we extend previous work that identified
extrapolation by applying the predictive variance from the univariate setting
to the multivariate case. We illustrate our approach through an analysis of
jointly modeled lake nutrients and indicators of algal biomass and water
clarity in over 7000 inland lakes from across the Northeast and Mid-west US. In
addition, we illustrate novel exploratory approaches for identifying regions of
covariate space where extrapolation is more likely to occur using
classification and regression trees.
