A comparison of some alternatives to least squares multiple regression
Multiple linear regression techniques are often employed in the statistical analysis of data. The most frequently used regression procedure is ordinary least squares. However, it is accepted that the method of least squares does not necessarily provide either accurate estimates of unknown regression coefficients or accurate predictions at future data points. Several classes of biased estimators have emerged as possible alternatives to ordinary least squares. We review the origins, definitions and properties of existing biased estimation procedures such as ridge, shrinkage, principal components and partial least squares regression. In addition, two new classes of estimator, multistage and multistep, are introduced. Simulation is the obvious means for assessing the relative merits of different estimation procedures. We review the design and results of previous simulation studies in which comparisons have been made between the performances of different estimation procedures. The designs of most previous studies are somewhat limited and unrealistic. Consequently, few clear guidelines have emerged regarding the circumstances in which individual procedures should either be applied or avoided. To provide some clarification, we conducted a series of simulation experiments that were designed to compare the performances of different regression procedures over a broad range of realistic situations.