Dealing with collinearity in behavioural and ecological data: model averaging and the problems of measurement error

Abstract

There has been a great deal of recent discussion of the practice of regression analysis (or more generally, linear modelling) in behaviour and ecology. In this paper, I wish to highlight two factors that have been under-considered, collinearity and measurement error in predictors, as well as to consider what happens when both exist at the same time. I examine what the consequences are for conventional regression analysis (ordinary least squares, OLS) as well as model averaging methods, typified by information theoretic approaches based around Akaike’s information criterion. Collinearity causes variance inflation of estimated slopes in OLS analysis, as is well known. In the presence of collinearity, model averaging reduces this variance for predictors with weak effects, but also can lead to parameter bias. When collinearity is strong or when all predictors have strong effects, model averaging relies heavily on the full model including all predictors and hence the results from this and OLS are essentially the same. I highlight that it is not safe to simply eliminate collinear variables without due consideration of their likely independent effects as this can lead to biases. Measurement error is also considered and I show that when collinearity exists, this can lead to extreme biases when predictors are collinear, have strong effects but differ in their degree of measurement error. I highlight techniques for dealing with and diagnosing these problems. These results reinforce that automated model selection techniques should not be relied on in the analysis of complex multivariable datasets.