Publication bias and other forms of outcome reporting bias are critical threats to the validity of findings from research syntheses. A variety of methods have been proposed for detecting selective outcome reporting in a collection of effect size estimates, including several methods based on assessment of asymmetry of funnel plots, such as the Egger's regression test, the rank correlation test, and the Trim‐and‐Fill test. Previous research has demonstated that the Egger's regression test is miscalibrated when applied to log‐odds ratio effect size estimates, because of artifactual correlation between the effect size estimate and its standard error. This study examines similar problems that occur in meta‐analyses of the standardized mean difference, a ubiquitous effect size measure in educational and psychological research. In a simulation study of standardized mean difference effect sizes, we assess the Type I error rates of conventional tests of funnel plot asymmetry, as well as the likelihood ratio test from a three‐parameter selection model. Results demonstrate that the conventional tests have inflated Type I error due to the correlation between the effect size estimate and its standard error, while tests based on either a simple modification to the conventional standard error formula or a variance‐stabilizing transformation both maintain close‐to‐nominal Type I error.