Estimating the coefficient of cross-validity in multiple regression: A comparison of analytical and empirical methods

In predictive applications of multiple regression, interest centers on the estimation of the population coefficient of cross-validation rather than the population multiple correlation. The accuracy of 3 analytical formulas for shrinkage estimation (Ezekiel, Browne, & Darlington) and 4 empirical techniques (simple cross-validation, multicross-validation, jackknife, and bootstrap) were investigated in a Monte Carlo study. Random samples of size 20 to 200 were drawn from a pseudopopulation of actual field data. Regression models were investigated with population coefficients of determination ranging from .04 to .50 and with numbers of regressors ranging from 2 to 10. For all techniques except the Browne formula and multicross-validation, substantial statistical bias was evident when the shrunken R(2) values were used to estimate the coefficient of cross-validation. In addition, none of the techniques examined provided unbiased estimates with sample sizes smaller than 100, regardless of the number of regressors.