linguistic:voice_change
Group effects

Group effects provide a means to study the collective impact of strongly correlated predictor variables in linear regression models. Individual effects of such variables are not well-defined as their parameters do not have good interpretations. Furthermore, when the sample size is not large, none of their parameters can be accurately estimated by the least squares regression due to the multicollinearity problem. Nevertheless, there are meaningful group effects that have good interpretations and can be accurately estimated by the least squares regression. A simple way to identify these meaningful group effects is to use an all positive correlations (APC) arrangement of the strongly correlated variables under which pairwise correlations among these variables are all positive, and standardize all p {\displaystyle p} predictor variables in the model so that they all have mean zero and length one. To illustrate this, suppose that { x 1, x 2, …, x q } {\displaystyle \{x_{1},x_{2},\dots,x_{q}\}} is a group of strongly correlated variables in an APC arrangement and that they are not strongly correlated with predictor variables outside the group. Let y ′ {\displaystyle y'} be the centred y {\displaystyle y} and x j ′ {\displaystyle x_{j}'} be the standardized x j {\displaystyle x_{j}}. Then, the standardized linear regression model is