The ridge prediction error sum of squares statistic in linear mixed models

Abstract

In case of multicollinearity, PRESS statistics has been proposed to be used in the selection of the ridge biasing parameter of the ridge estimator which is introduced as an alternative to BLUE. This newly proposed PRESS statistic for the ridge estimator, $\textit{CPRESS}_{k}$, depends on the conditional ridge residual and can be computed once at a time by fitting the linear mixed model with all the observations. We also define $R^2_{RidPred}$ statistic to evaluate the predictive ability of the ridge fit. Since the PRESS statistic for the BLUE is a special $\textit{CPRESS}_{k}$ statistic, we indirectly also give closed form solution of the PRESS statistic for the BLUE. Then, we compared the predictive performance of the linear mixed model via the statistics $\textit{CPRESS}_{k}$, $GCV_{k}$ and $C_{p}$ by considering a real data analysis and a simulation study where the optimal ridge biaisng parameter is obtained by minimizing each statistic. The study shows that the ridge predictors improve the predictive performance of a linear mixed model over BLUE in the presence of multicollinearity and each statistic gives a different optimum ridge biasing value and they show the best predictive performance at their optimum ridge biasing values. In addition, the simulation study has shown that the intensity of variance and multicollinearity is effective in determining the optimum ridge biasing value and this optimum ridge biasing value is effective on the superiority of the predictive performance of ridge estimator over BLUE.