Null Distribution of the Test Statistic for Model Selection via Marginal Screening: Implications for Multivariate Regression Analysis

A V Rubanovich,V A Saenko

doi:10.34257/gjsfrgvol21is1pg23

Abstract

Marginal screening (MS) is the computationally simple and commonly used for the dimension reduction procedures. In it, a linear model is constructed for several top predictors, chosen according to the absolute value of marginal correlations with the dependent variable. Importantly, when kpredictors out of mprimary covariates are selected, the standard regression analysis may yield false-positive results if m>> k(Freedman's paradox). In this work, we provide analytical expressions describing null distribution of the test statistics for model selection via MS. Using the theory of order statistics, we show that under MS, the common F-statistic is distributed as a mean of ktop variables out of mindependent random variables having a 21χdistribution. Based on this finding, we estimated critical p-values for multiple regression models after MS, comparisons with which of those obtained in real studies will help researchers to avoid false-positive result. Analytical solutions obtained in the work are implemented in a free Excel spreadsheet program.

Highlights

Marginal screening (MS) is the simplest and most commonly used method of variable selection (Hastie, Tibshirani, 2003; Genovese et al, 2009, 2012; Leek, 2012)
A typical situation is when the number of objects is several orders of magnitude less than the number of covariates from which a statistically significant combination of predictors is derived. This is another side of an old problem of multiple comparisons, Author α: Vavilov Institute of General Genetics, Russian Academy of Sciences, Moscow, Russia. e-mail: rubanovich@vigg.ru Author σ: Department of Radiation Molecular Epidemiology, Atomic Bomb Disease Institute, Nagasaki University, Nagasaki, Japan
The purpose of this work was to explicitly address null-distribution of the F-statistic, which is used for testing the significance of a model (1), when model selection is performed with MS

Summary

Introduction

Marginal screening (MS) is the simplest and most commonly used method of variable selection (Hastie, Tibshirani, 2003; Genovese et al, 2009, 2012; Leek, 2012). MS is intuitively preferred by the researchers, when the number of objects (e.g. participants of a study, samples or outcomes) is much smaller than the number of explanatory variables (the so-called “n

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