Null Distribution of the Test Statistic for Model Selection via Marginal Screening: Implications for Multivariate Regression Analysis
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 k predictors out of m primary 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 k top variables out of m independent random variables having a 2 1 χ distribution. 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.
