A modified efficient empirical Bayes regression model for predicting phenomena with a large number of independent variables and fewer observations; examples of its application in human disease, protein bioinformatics, and microarray gene expression profiling

Abstract

For a long time, it thought impossible to find a precise predictor model with a large number of independent variables (predictors) and a fewer number of observations (replications), since there was a large prediction error in estimating the model. Recently, empirical Bayes, the third main strategy in statistics, has provided a new approach in solving this problem. In 2007, Srivastava and Kubokawa introduced an efficient empirical Bayes estimator, named Crude estimator, and demonstrated its superiority over the previously used estimator (least square estimator). Crude estimator is based on λ coefficient in this equation: ()()1ˆ1EBttpλ−=βXX+IXy. According to Srivastava and Kubokawa (2007) when the λ is negative, it should be replaced with zero. We showed that when λ is zero, applying the Crude estimator is not possible. In this paper, the Crude estimator has been modified introducing more efficient and precise regression model, applicable in all cases with lower amounts of prediction error. Then, the model performance was evaluated using datasets of important biological phenomena including human disease, protein bioinformatics, and microarray gene expression profiling. Cross validation confirmed the accuracy of the model based on estimating the prediction error. Precise modeling, presented here, can open a new vista in applied statistics when the number of observations is smaller than the number of variables. In addition, this model can clarify the main structure underlying high-dimensional predictors by reducing the dimensions of variables via eigenvalues.