Multiple regression for physiological data analysis: the problem of multicollinearity

Abstract

Multiple linear regression, in which several predictor variables are related to a response variable, is a powerful statistical tool for gaining quantitative insight into complex in vivo physiological systems. For these insights to be correct, all predictor variables must be uncorrelated. However, in many physiological experiments the predictor variables cannot be precisely controlled and thus change in parallel (i.e., they are highly correlated). There is a redundancy of information about the response, a situation called multicollinearity, that leads to numerical problems in estimating the parameters in regression equations; the parameters are often of incorrect magnitude or sign or have large standard errors. Although multicollinearity can be avoided with good experimental design, not all interesting physiological questions can be studied without encountering multicollinearity. In these cases various ad hoc procedures have been proposed to mitigate multicollinearity. Although many of these procedures are controversial, they can be helpful in applying multiple linear regression to some physiological problems.