Precision of Prediction1, 2

ABSTRACT Precision of prediction in multiple linear regression is examined. Two measures of predictive precision, predictive mean square error, d 2 , and the squared weight validity, w 2 , are employed. The use of an existing estimator of ε(d 2 ) as an estimator of d 2 is proposed and the mean squared error of estimation of this estimator about d 2 is obtained. A significance test is given. Estimators of w 2 are derived and an asymptotic approximation for their variances is given. These estimators of w 2 are functions of estimators of p 2 , the squared multiple correlation coefficient, and of p 4 . The bias and mean squared error of estimation of some known estimators of p 2 and of a proposed estimator of p 4 are examined. Monte Carlo experiments are used to compare the proposed estimators of w 2 with an estimator due to Burket 1964. An efficient procedure for generating w 2 and the estimates of w 2 is described. The mean squared errors of estimation of cross‐validation estimators of d 2 and w 2 are obtained and disadvantages of the cross‐validation procedure are discussed. An example is used to illustrate relationships between predictive precision and the number of predictors. The paper is primarily concerned with a random predictor model but results for a fixed predictor model are also given.