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.

[1]  D. Hinkley.  Inference about the intersection in two-phase regression , 1969 .

[2]  D. Hinkley.  Inference in Two-Phase Regression , 1971 .

[3]  David W. Bacon,et al.  Estimating the transition between two intersecting straight lines , 1971 .

[4]  Bruce A. Wales.  Vegetation Analysis of North and South Edges in a Mature Oak‐Hickory Forest , 1972 .

[5]  David W. Bacon,et al.  Using An Hyperbola as a Transition Model to Fit Two-Regime Straight-Line Data , 1974 .

[6]  P. Feder.  On Asymptotic Distribution Theory in Segmented Regression Problems-- Identified Case , 1975 .

[7]  P. Feder.  The Log Likelihood Ratio in Segmented Regression , 1975 .

[8]  Hugh G. Gauch,et al.  A COMPARATIVE STUDY OF RECIPROCAL AVERAGING AND OTHER ORDINATION TECHNIQUES , 1977 .

[9]  W. Cleveland.  Robust Locally Weighted Regression and Smoothing Scatterplots , 1979 .

[10]  A. Tishler,et al.  A New Maximum Likelihood Algorithm for Piecewise Regression , 1981 .

[11]  A Functional Approach to Estimating Habitat Edge Width for Birds , 1981 .

[12]  E. C. Pielou.  The Interpretation of Ecological Data: A Primer on Classification and Ordination , 1984 .

[13]  G. Williams‐Linera.  Vegetation structure and environmental conditions of forest edges in Panama. , 1990 .

[14]  P. G. Murphy,et al.  Disturbance versus edge effects in sugar-maple/beech forest fragments , 1990 .

[15]  T. Spies,et al.  Vegetation Responses to Edge Environments in Old-Growth Douglas-Fir Forests. , 1992, Ecological applications : a publication of the Ecological Society of America.

[16]  Glenn R. Matlack,et al.  MICROENVIRONMENT VARIATION WITHIN AND AMONG FOREST EDGE SITES IN THE EASTERN UNITED STATES , 1993 .

[17]  S. Fraver.  Vegetation Responses along Edge‐to‐Interior Gradients in the Mixed Hardwood Forests of the Roanoke River Basin, North Carolina , 1994 .

[18]  H. Andrén,et al.  Effects of habitat fragmentation on birds and mammals in landscapes with different proportions of suitable habitat: a review , 1994 .

[19]  J. Shao,et al.  The jackknife and bootstrap , 1996 .

[20]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[21]  Ransom A. Myers,et al.  Still more spawner-recruitment curves: the hockey stick and its generalizations , 2000 .

[22]  David R. Anderson,et al.  Null Hypothesis Testing: Problems, Prevalence, and an Alternative , 2000 .

[23]  L. Fahrig.  How much habitat is enough , 2001 .