The impact of nonnormality on the asymptotic confidence interval for an effect size measure in multiple regression

The increase in the squared multiple correlation coefficient, Delta R2, associated with an individual predictor in a regression analysis is a measure commonly used to evaluate the importance of that variable in a multiple regression analysis. Previous research using multivariate normal data had shown that relatively large sample sizes are necessary for an acceptably accurate confidence interval for this regression effect size measure.; The coverage probability that an asymptotic confidence interval contained the population squared semipartial correlation, Deltarho2, was investigated by simulating data from a range of nonnormal distributions such that (a) the predictors were nonnormal, (b) the error distribution was nonnormal, or (c) both predictors and errors were nonnormal. Additional factors manipulated included (a) the number of predictor variables, (b) the magnitude of the population squared multiple correlation coefficient in the original model, r2r (c) the magnitude of the population squared semipartial correlation, Deltarho 2, and (d) sample size.; This study showed that when nonnormality is introduced, empirical coverage probability was always less than the nominal confidence level, often dramatically so. The degree of nonnormality in the predictors was the most important factor influencing poor coverage probability. Although coverage probability increased as a function of sample size, when nonnormality in the predictors was substantial, the confidence interval is likely to be inaccurate no matter how large a sample size is used. With multivariate normal data, coverage probability improved as both r2r and Deltarho2 increased. When predictors are sampled from a nonnormal distribution, coverage probability tended to decrease as r2r and Deltarho2 increased and became even worse as the degree of nonnormality increased. It was further demonstrated that the asymptotic variance underestimates the sampling variance of Delta R2. This produces standard errors that are too small and results in a confidence interval that is too narrow. Reliance on this confidence interval as a measure of the strength of the effect size will lead us to underestimate the importance of an individual predictor to the regression.