Construction of Predictors
Prediction means in general that we have a new subject outside of the original sample that had been used to fit the regression model. We denote the covariate values of such a subject by X∗1 ,X∗2 , . . . , X∗p , and the (usually unknown) outcome by Y∗, just to distinguish these values from the covariate values of subjects in the original sample. A predictor is now a function of the covariate values X∗1 ,X∗2 , . . . , X∗p of the new subject and the estimated regression parameters βˆ0, βˆ1, . . . , βˆp estimated in the original sample, which generates a values on the scale of Y . So the precise notation of a prediction would be something like
Yˆ (βˆ0, βˆ1, . . . , βˆp,X∗1 ,X∗2 , . . . , X∗p ) . However, we will use in the following just the short hand notation Yˆ .