PIECEWISE REGRESSION: A TOOL FOR IDENTIFYING ECOLOGICAL THRESHOLDS

We demonstrate the use of piecewise regression as a statistical technique to model ecological thresholds. Recommended procedures for analysis are illustrated with a case study examining the width of edge effects in two understory plant communities. Piece-wise regression models are “broken-stick” models, where two or more lines are joined at unknown points, called “breakpoints.” Breakpoints can be used as estimates of thresholds and are used here to determine the width of edge effects. We compare a sharp-transition model with three models incorporating smooth transitions: the hyperbolic-tangent, bent-hyperbola, and bent-cable models. We also calculate three types of confidence intervals for the breakpoint estimate: an interval based on the computed standard error of the estimate from the fitting procedure, an empirical bootstrap confidence interval, and a confidence interval derived from an inverted F test. We recommend use of the inverted F test confidence interval when sample sizes are large, and cautious use of bootstrapped confidence intervals when sample sizes are smaller. Our analysis demonstrates the need for a careful study of the likelihood surface when fitting and interpreting the results from piecewise-regression models.