PSYCHOMETRIKA—VOL. 76, NO. 4, 670–690 OCTOBER 2011 DOI : 10.1007/ S11336-011-9224-6 BIASES AND STANDARD ERRORS OF STANDARDIZED REGRESSION COEFFICIENTS KE-HAI YUAN UNIVERSITY OF NOTRE DAME WAI CHAN THE CHINESE UNIVERSITY OF HONG KONG The paper obtains consistent standard errors (SE) and biases of order O(1/n) for the sample stan- dardized regression coefficients with both random and given predictors. Analytical results indicate that the formulas for SEs given in popular text books are consistent only when the population value of the regression coefficient is zero. The sample standardized regression coefficients are also biased in general, although it should not be a concern in practice when the sample size is not too small. Monte Carlo results imply that, for both standardized and unstandardized sample regression coefficients, SE estimates based on asymptotics tend to under-predict the empirical ones at smaller sample sizes. Key words: asymptotics, bias, consistency, Monte Carlo. 1. Introduction Linear regression is typically the first model introduced in statistical text books. It is also one of the most widely used statistical methods across all disciplines. Statistical inference for regression models includes obtaining a confidence interval for each regression coefficient and performing an F -test for the overall significance of all the explanatory variables. There also exist various diagnostic tools for properly applying regression models. In addition to the regression coefficients, the standardized regression coefficients are also of substantial interest in practice (Kelley and Maxwell, 2003). This is especially true in social sci- ences where variables are typically measured in different and arbitrary units. In the psychometric literature, standardized regression coefficients are called Beta-coefficients while the conventional regression coefficients are called B-coefficients. The importance of the standardized regression coefficients can be testified by its coverage in popular statistical textbooks (e.g., Cohen, Cohen, West, & Aiken, 2003; Harris, 2001; Hays, 1994) as well as most widely used statistical software (e.g., SAS, SPSS). However, our knowledge of standardized regression coefficients is very lim- ited. For example, there does not exist a formula to properly estimate the standard errors (SEs) of the sample standardized regression coefficients. We also do not know whether the sample stan- dardized coefficients are biased or not. The purpose of this paper is to study SEs and biases of the sample standardized regression coefficients. In regression analysis, the most widely used statistic is probably the t -statistic for testing the null hypothesis that a regression coefficient is zero. Under the assumptions of given predictors and normally distributed errors, the t -statistic follows a Student t -distribution. However, data in social sciences are seldom normally distributed (see Micceri, 1989) and the predictors are typi- cally not controllable. Nevertheless, linear regression is frequently used in social science research when studying the relationship of a response with potential predictors. Thus, random predictors This research was supported by Grants DA00017 and DA01070 from the National Institute on Drug Abuse. Requests for reprints should be sent to Ke-Hai Yuan, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: kyuan@nd.edu © 2011 The Psychometric Society 670