Bootstrap Confidence Intervals for Multilevel Standardized Effect Size.

Although many methodologists and professional organizations have urged applied researchers to compute and report effect size measures accompanying tests of statistical significance, discussions on obtaining confidence intervals (CIs) for effect size with clustered/multilevel data have been scarce. In this paper, I explore the bootstrap as a viable and accessible alternative for obtaining CIs for multilevel standardized mean difference effect size for cluster-randomized trials. A simulation was carried out to compare 17 analytic and bootstrap procedures for constructing CIs for multilevel effect size, in terms of empirical coverage rate and width, for both normal and nonnormal data. Results showed that, overall, the residual bootstrap with studentized CI had the best coverage rates (94.75% on average), whereas the residual bootstrap with basic CI had better coverage in small samples. These two procedures for constructing CIs showed better coverage than using analytic methods for both normal and nonnormal data. In addition, I provide an illustrative example showing how bootstrap CIs for multilevel effect size can be easily obtained using the statistical software R and the R package bootmlm. I strongly encourage applied researchers to report CIs to adequately convey the uncertainty of their effect size estimates.

On the Computation of the RMSEA and CFI from the Mean-And-Variance Corrected Test Statistic with Nonnormal Data in SEM.

A new type of nonnormality correction to the RMSEA has recently been developed, which has several advantages over existing corrections. In particular, the new correction adjusts the sample estimate of the RMSEA for the inflation due to nonnormality, while leaving its population value unchanged, so that established cutoff criteria can still be used to judge the degree of approximate fit. A confidence interval (CI) for the new robust RMSEA based on the mean-corrected ("Satorra-Bentler") test statistic has also been proposed. Follow up work has provided the same type of nonnormality correction for the CFI (Brosseau-Liard & Savalei, 2014). These developments have recently been implemented in lavaan. This note has three goals: a) to show how to compute the new robust RMSEA and CFI from the mean-and-variance corrected test statistic; b) to offer a new CI for the robust RMSEA based on the mean-and-variance corrected test statistic; and c) to caution that the logic of the new nonnormality corrections to RMSEA and CFI is most appropriate for the maximum likelihood (ML) estimator, and cannot easily be generalized to the most commonly used categorical data estimators.

Effect size measures in a two-independent-samples case with nonnormal and nonhomogeneous data.

In psychological science, the "new statistics" refer to the new statistical practices that focus on effect size (ES) evaluation instead of conventional null-hypothesis significance testing (Cumming, Psychological Science, 25, 7-29, 2014). In a two-independent-samples scenario, Cohen's (1988) standardized mean difference (d) is the most popular ES, but its accuracy relies on two assumptions: normality and homogeneity of variances. Five other ESs-the unscaled robust d (d r* ; Hogarty & Kromrey, 2001), scaled robust d (d r ; Algina, Keselman, & Penfield, Psychological Methods, 10, 317-328, 2005), point-biserial correlation (r pb ; McGrath & Meyer, Psychological Methods, 11, 386-401, 2006), common-language ES (CL; Cliff, Psychological Bulletin, 114, 494-509, 1993), and nonparametric estimator for CL (A w ; Ruscio, Psychological Methods, 13, 19-30, 2008)-may be robust to violations of these assumptions, but no study has systematically evaluated their performance. Thus, in this simulation study the performance of these six ESs was examined across five factors: data distribution, sample, base rate, variance ratio, and sample size. The results showed that A w and d r were generally robust to these violations, and A w slightly outperformed d r . Implications for the use of A w and d r in real-world research are discussed.