Weighted Multicollinearity in Logistic Regression: Diagnostics and Biased Estimation Techniques with an Example from Lake Acidification

Abstract

An historical data set from the Adirondack region of New York is revisited to study the relationship between water chemistry variables associated with acid precipitation and the presence/absence of brook trout (Salvelinus fontinalis) and lake trout (Salvelinus namaycush). For the trout species data sets, water chemistry variables associated with acid precipitation, for example pH and alkalinity, are highly correlated. Regression models to assess their effects on the probability of the presence of fish species are therefore affected by multicollinearity. Because the appropriate regressions are logistic, correction techniques based on least squares do not work. Maximum likelihood parameter estimation is highly unstable for the trout presence/absence data. Developments in weighted multicollinearity diagnostics are used to evaluate maximum likelihood logistic regression parameter estimates. Further, an application of biased parameter estimation is presented as an option to the traditional maximum likelihood logistic regression. Biased estimation methods, like ridge, principal component, or Stein estimation can substantially reduce the variance of the parameter estimates and prediction variance for certain future observations. In many cases, only a slight modification to the converged maximum likelihood estimator is necessary.