C2 Training: May 9 – 10, 2011

1. C2 Training: May 9 – 10, 2011 Introduction to meta-analysis

2. Pooled effect sizes • Average across studies • Calculated using inverse variance weights • Studies with more precise estimates (larger N, smaller sd) contribute more to overall average than those with less precise estimates • Choice between fixed and random effects models (and mixed models): 1. a priori expectations, 2. statistical tests • Do we expect studies to estimate a single population parameter? • If Yes, use fixed effect model and test homogeneity assumption • Usually use random effect model or mixed model

3. Combining effect sizes across studies • Compute effect sizes within each study • Create a set of independent effect sizes • Compute weighted mean and variance of effect sizes • Compute 95% confidence interval for weighted mean effect size • Test for the homogeneity of effect sizes

4. Create a set of independent effect sizes • Likely to have multiple effect sizes per study • Effect sizes within a study can be: • Multiple measures of study participants • Measures of independent groups of participants

5. What to do with multiple effect sizes per study? • Use only independent groups in each analysis • Use only one measure of the study participants in each analysis • Use results within studies that are derived from independent groups of study participants

6. Compute weighted mean and variance of effect sizes • Compute individual study effect sizes • Correct effect sizes for any biases, e.g., Hedges correction • Compute study effect size variance

8. p. 62 of MST pdf

9. Use weighted mean for effect sizes • Use weighted means because each effect size has a different variance that depends on the study’s sample size and effect size value • Weight all analyses by the inverse of the effect size variance or

10. Weighted Mean Effect Size k is the number of effect sizes

12. Standard error of weighted mean effect size

13. Confidence Interval for Mean Effect Size α = Significance level, z = Critical value from standard normal distribution

15. Test of Homogeneity • Statistical test that addresses whether the k effect sizes that are averaged into a mean value all estimate the same population effect size • In a homogeneous distribution, effect sizes differ from population mean only by sampling error

16. Assessment and adjustments for bias • Small sample bias – Hedges’ g correction for SMD • Range restriction – Hunter & Schmidt • Missing data – sensitivity analysis, concerns about integrity of intent-to-treat analysis (missing cases), outcome reporting bias (missing data) • Publication bias – missing studies (and missing data for available studies)

17. Assessing risk of publication bias • Funnel plots – plot study effect sizes by their standard errors • “interoccular analysis” of funnel plots is unreliable • Trim and fill analysis (need ~ 10+ studies) • Statistical tests (Egger’s test and others) Do NOT use Failsafe N (Becker, 2005) See Rothstein, Sutton & Bornstein (2005)

21. Homogeneity/heterogeneity tests Statistical tests of homogeneity of results across studies • Are all studies estimating a common population parameter with • Differences between studies due to chance (sampling error) alone? Homogeneity tests performed under the fixed effect model

22. Form of the homogeneity test When we reject the null hypothesis of homogeneity, the variability of the effect sizes is more than would be expected from sampling error

23. Computational form

24. To Test Homogeneity • Compare Q to the (1 – α) critical value of the chi-square distribution with k-1 degrees of freedom • Significant Q = heterogeneity • Non-significant Q = homogeneity

26. Preliminary analysis • Graph effect sizes and 95% CI • Compute overall mean effect size and 95% CI • Compute homogeneity test • Interpret findings

27. Forest plots • Easy to produce from raw data • Cochrane’s RevMan software free, downloadable at: http://www.cc-ims.net/revman

28. p. 59 of MST pdf

29. Moderator models: ANOVA • When we find that a group of studies are heterogeneous, we can explore whether moderator variables explain this variation • When we have continuous moderators, we use regression models • When we have categorical moderators, we use ANOVA • Caution needed: • Moderator analysis is correlational • Moderators may be confounded within studies

30. Example from Sirin paper Sirin, S. R. (2005). Socioeconomic status and academic achievement: A meta-analytic review of research. Review of Educational Research, 75, 417-453.

33. Group exercise • Using one of the studies provided from the MST review, code the elements in Level 4. Choose at least 2 outcomes from the study.