The Conditional Random-Effects Variance Component in Meta-regression Michael T. Brannick Guy Cafri University of South Florida

Background What is the random-effects variance component (REVC)? What is the conditional random-effects variance component (CREVC)? Who cares? (Tells whether we are done!)

Fixed and Mixed Regression CREVC = 0 CREVC > 0

Items of Interest Point Estimators of the CREVC Significance tests Method of Moments (WLS) Maximum Likelihood (iterated WLS) Significance tests Fixed chi-square Random chi-square (2 of these) Lower bound > 0 (3 of these) Confidence Intervals (3 types) ML, bootstrap, bootstrap adjusted Bias RMSE Type I error Power Coverage probability Width

Monte Carlo Method Effect size: d Conditions (based on literature) REVC: 0, .04, .10, .19, .35, .52 Proportion A/C: 0, .02, .18, .50 K studies: 13, 22, 30, 69, 112, 234 Average N (skewed): 53, 231, 730 Reps: 10k times each for 378 cells

Results – Point Estimates Note: results are averages over cells Method of moments is less biased than max like until k > 100

Results – Point Estimates Meta-analysis results for one cell (10k trials for each method)

Results – Significance Tests CREVC ML ran Chi-sq CI ml CI bs CI bc .045 .002 .004 .044 .02 .732 .507 .537 .710 .04 .859 .674 .696 .845 .10 .960 .837 .849 .952

Results – Confidence Intervals Bias corrected bootstrap has best coverage; similar width Coverage Width

Implications Slight preference for method of moments WLS when k is small Use the fixed-effects chi-square for testing the CREVC Use the bias-corrected bootstrap for constructing confidence intervals

Conclusions Please indicate the uncertainty of the estimates when reporting a meta-analysis (confidence intervals and/or standard errors of parameter estimates) Free software: http://luna.cas.usf.edu/~mbrannic/files/meta/MetaRegsMB1.sas