Estimation of Effect Size under Nonrandom Sampling: The Effects of Censoring Studies Yielding Statistically Insignificant Mean Differences

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Abstract
Quantitative research synthesis usually involves the combination of estimates of the standardized mean difference (effect size) derived from independent research studies. In some cases, effect size estimates are available only if the difference between experimental and control group means is statistically significant. If the quantitative result of a study is observed only when the mean difference is statistically significant, the observed mean difference, variance, and effect size are biased estimators of the corresponding population parameters. The exact distribution of the sample effect size is derived for the case in which only studies yielding statistically significant results may be observed. The maximum likelihood estimator of effect size also is derived under the model in which only significant results are observed. The exact distribution of the maximum likelihood estimator is obtained numerically and is used to study the bias of the maximum likelihood estimator. An empirical sampling study is used to supplement the analytic results.

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