On Correctly Adjusting the Squared Multiple Correlation Coefficient in Linear Regression: Effect Size Estimation and Significance Testing with Application to Substance Abuse Research

Linear regression analysis is ubiquitous in many areas of scholarly inquiry, including substance abuse research. In linear regression it is common practice to test whether the squared multiple correlation coefficient, R2, differs significantly from zero. However, this test is misleading because the expected value of R2 is not zero under the null hypothesis. In this brief methodological note I discuss the implications of this realization for calculating and interpreting the squared multiple correlation coefficient, R2. In addition, I discuss and offer freely available software that calculates the expected value of R2 under the null hypothesis that ρ-the population value of the multiple correlation coefficient-equals zero, an adjusted R2 value and effect size measure that both take into account the expected value of R2, and an F statistic that tests the significance of difference between the obtained R2 and the expected value of R2 under the null hypothesis that ρ=0.