Explained Residual Variation, Explained Risk, and Goodness of Fit

A loss function approach is used to define the concepts of explained residual variation and explained risk for general regression models.Explained risk measures the ability of the covariates in a correctly specified model to distinguish differing outcomes.Explained residual variation, which is R 2 for a linear model, estimates the explained risk with a penalty for poorly fitting models.Application of the general definitions to linear regression, logistic regression, and survival analysis is given.The importance of distinguishing the concepts of explained residual variation, explained risk, and goodness of fit is discussed.