Predictive Validity of a Linear Regression Equation

The squared correlation coefficient, w 2 , between an empirically chosen linear function of predictors, B̂ 0 + B̂′x, and a criterion, y , is employed as a measure of predictive precision. This coefficient is defined over the entire population but is conditional on B̂. Assuming a multinormal distribution for x and y, approximations for the expected value and variance of w 2 are derived. If too many predictors are employed, precision of prediction can decrease. This is illustrated by means of an example of a sequence of values of ℰ ( w 2 ). A function of the sample squared multiple correlation coefficient, r 2 , is proposed as an estimator of w 2 . Results of Monte Carlo experiments are employed to give an impression of the precision of the estimates of w 2 , and the accuracy of the approximations for ℰ ( w 2 ) and var ( w 2 ).