# Performance assessment of shrinkage estimator for prediction in multiple regression with future random X

Regression models in medical research are widely used for prediction. When predicting the response at a future randomly chosen covariate vector x, problems can arise. The fit of a regression model to new data is nearly always worse than its fit to the original data, a deterioration called shrinkage. The Stein-type predictors give a uniformly lower expected mean squared error for prediction (MSEP) than least squares estimators under certain assumptions Different forms of the Stein-type and nonparametric shrinkage predictors were computed and compared using re-sampling and sample re-using. Data were generated from models with normally and nonnormally distributed error terms. Normally distributed data with residual variances following particular patterns of heteroscedasticity were investigated as well. The number of predictors, sample size, beta and the correlation between covariates were varied. Both the mean and median of the mean squared error for prediction (MSEP) and predicted residual sum of squares (PRESS) were computed from 10,000 simulations (5,000 for PRESS) for each shrinkage predictor and the results compared The results showed that the shrinkage predictors gave a uniformly lower MSEP/PRESS than the least squares predictors. The percentage of loss saving went up to 35% under certain conditions. The Stein-type predictors usually gave a lower MSEP/PRESS than the nonparametric form. One of the modified Stein-form/class estimators. This conclusion held even when moderate correlation existed in the covariates. The positive part estimator also dominated under different error structures The percentage of loss saving decreased as the sample size increased or the number of covariates decreased. The percentages of loss saving were fairly comparable when the covariates were either independent or had low (pairwise r = 0.1) correlation. When there were moderate correlations among covariates, the loss saving decreased significantly. Such findings were the same whether means or medians of MSEP and PRESS were considered