Coping with Multicollinearity: An Example on Application of Principal Components Regression in Dendroecology

Abstract

The theory and application of principal components regression, a method for coping with multicollinearity among independent variables in analyzing ecological data, is exhibited in detail. A concrete example of the complex procedures that must be carried out in developing a diagnostic growth-climate model is provided. We use tree radial increment data taken from breast height as the dependent variable and climatic data from the area as the independent data. Thirty-four monthly temperature and precipitation measurements are used as potential predictors of annual growth. Included are monthly average temperatures and total monthly precipitation for the current and past growing season. The underlying theory and detail illustration of the computational procedures provide the reader with the ability to apply this methodology to other situations where multicollinearity exists. Comparison of the principal component selection rules is shown to significantly influence the regression results. A complete derivation of the method used to estimate standard errors of the principal component estimators is provided. The appropriate test statistic, which does not depend on the selection rule, is discussed. The means to recognize and adjust for autocorrelation in the dependent data is also considered in detail. Appendices and directions to internet-based example data and codes provide the user with the ability to examine the code and example output and produce similar results.