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On detecting hump‐shaped relationships in ecology: a bootstrap test for monotonicity
Author(s) -
Murtaugh Paul A.
Publication year - 2003
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/env.607
Subject(s) - monotonic function , monotone polygon , isotonic regression , mathematics , linear regression , quadratic equation , statistics , regression analysis , curvature , regression , econometrics , quadratic function , mathematical analysis , geometry , estimator
Ecologists and environmental scientists are often interested in determining whether a response achieves a maximum at an intermediate value of some explanatory variable. Typically, statistical significance of a quadratic term added to a simple linear regression model is interpreted as evidence for an intermediate maximum. Relationships that are monotone but non‐linear can easily be misclassified by this criterion. A bootstrap test of the monotonicity of the relationship between two variables is proposed, based on fitting an isotonic regression and examining the curvature of scatterplots generated by adding resampled residuals to that regression function. The test operates at or below the nominal level when applied to data sets simulated from a variety of monotone functions, and it has reasonable power against several non‐monotone alternatives. By comparison, a test of the significance of the second‐order coefficient in a quadratic regression may have very high rejection rates when applied to monotone functions with appreciable curvature. This argues for caution in the interpretation of a significant quadratic regression as strong evidence against monotonicity. Copyright © 2003 John Wiley & Sons, Ltd.