Using regression models for prediction: shrinkage and regression to the mean

The use of a fitted regression model in predicting future cases, either as a diagnostic tool or as an instrument for risk assessment is discussed. The regression to the mean effect implies that the future values of the response variable tend to be closer to the overall mean than might be expected from the predicted values. The extent of this shrinkage is studied for multiple and logistic regression models, and is found to be related to simple goodness-of-fit statistics of the original regression. Shrinkage is a particularly serious problem if the sample size is small and/or the number of covariates is large. Shrinkage of predictors is illustrated by two examples. A more general formulation is suggested.