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Meta-analysis when the assumptions are violated: which method is best?

- 1.A comparison of Random Effects meta-analysis methods when study effects are non-normally distributed Evan Kontopantelis & David Reeves NPCRDC

2. How meta-analysis works A search for papers relevant to the research question is conducted. Unsuitable papers are filtered out In each paper for each outcome measure that is directly relevant to the RQ, or a good enough proxy, we calculate an effect (of intervention vs control) and its variance An overall effect and variance is selected Effects and their variances are combined to calculate an overall effect Chronic disease - Risk factors effect -.4 0 .4 .8 Combined Woolard(B), 1995 Woolard(A), 1995 Eckerlund, 1985 Moher, 2001 Cupples, 1994 Campbell, 1998 Van Ree, 1985 3. Heterogeneity Heterogeneity can be attributed to clinical and/or methodological diversity Clinical heterogeneity: variability that arises from different populations, interventions, outcomes and follow-up times Methodological heterogeneity: relates to differences in trial design and quality Detecting (usually with Cochrans Q test) quantifying and dealing with heterogeneity can be very hard 4. Absence of heterogeneity Assumes that the true effects of the studies are all equal and deviations occur because of imprecision of results Analysed with fixed- effects method i iY e 5. Presence of heterogeneity It is assumed that there exists variation in the size of the true effect among studies (in addition to the imprecision in results) Analysed with random-effects methods i i iY e 6. Random-effect MA methods Estimate the between-study variance and use it in estimating the overall effect Parametric: DerSimonian-Laird (1986) Maximum & Profile likelihood (1996) Non-parametric: Permutations method (1999) Non Parametric Maximum Likelihood (1999) 2 7. Potential problems? Heterogeneity is common & the FE model is under fire Parametric RE models assume that both the effects and errors are normally distributed Almost all RE models (except PL) do not take account of uncertainty in DL is usually the preferred method of analysis because it is easy to implement and is available in all software packages 2 8. So far The number of studies and the amount of heterogeneity have been found to affect method performance Performance comparisons usually focus on coverage and ignore power or have not included some important methods (e.g. PL, PE) Evaluations were based on normal data: method robustness has not been assessed with non-normal data 9. Our bit 10. In a nutshell Simulated various non-normal distributions for the true effects: skew normal, bimodal, beta, uniform, U and others Created datasets of 10000 meta-analyses for various numbers of studies k and different degrees of heterogeneity, for each distributional assumption Compared FE, DL, ML, PL and PE methods (along with a simple t-test) in terms of coverage and power across all datasets 11. Generating the data For a single study we simulated the effect size estimate and the within-study variance estimate of a binary outcome The variance was assumed to be a realisation from a distribution, multiplied by .25 and restricted to the (.009, .6) interval involves two components where Four values were used: .01, .03, .07 & .1 Number of studies (MA size) varied from 2 to 35 iY i 2 2 1 iY i i iY e( ) 2 (0, )i ie 2 k i i 2 ?(0, )i 12. Details on the MA methods Fixed effects (FE) DerSimonian-Laird (DL) Q method (Q) Maximum Likelihood (ML) Profile Likelihood (PL) Permutations method (PE) T-test method (T) 13. Performance For each simulated meta-analysis case we calculated confidence intervals for the overall effect estimate , for all the methods Coverage: % of confidence intervals that contain the true overall effect in a sample of 10000 meta-analyses Power: % of CIs that do not contain the 25th centile of the population distribution of the 10000 effect sizes 14. Results 15. Zero 2 16. Normally distributed i 17. Skew normal i 18. Bimodal i 19. PL performance across various distributions 20. Drawing conclusions 21. Summary Within any given method, the results were consistent across all types of distribution shape This can give researchers confidence that methods are highly robust against even the most severe violation of the assumption of normally distributed effect sizes If it is reasonable to assume that the effect size does not vary between studies, the FE, Q and ML methods all provide accurate coverage coupled with good power 22. In the presence of heterogeneity However, zero between study variance is the exception rather than the norm and the presence of even a moderate amount of alters the picture considerably FE, Q and ML quickly lose coverage as heterogeneity increases DL rapidly goes from providing a coverage that is overly high, to one that is overly low PE, and to a lesser extend PL, now provide the best coverage, even with very small sample sizes 2 23. Which method then? If priority is given to maintaining an accurate Type I error rate then the simple t-test is the best method. But its power is very low, making it a poor choice when control of the Type II error rate is also important PE gives accurate coverage in all situations and has better power than T, but the method is more difficult to implement and cannot be used with less than 6 studies PL has reasonable coverage in most situations, giving it an edge over other methods 24. Current & future work Created a freely available Excel add-in that implements all the described MA methods and various measures of heterogeneity Working on a STATA module that will do the same Investigate performance of heterogeneity measures under non-normally distributed data 25. Main references Brockwell SE, Gordon IR. A comparison of statistical methods for meta-analysis. Stat.Med. 2001; 20(6):825-840 Engels EA, Schmid CH, Terrin N, Olkin I, Lau J. Heterogeneity and statistical significance in meta-analysis: an empirical study of 125 meta-analyses. Stat.Med. 2000; 19(13):1707-1728 Follmann DA, Proschan MA. Valid inference in random effects meta- analysis. Biometrics 1999; 55(3):732-737 Hardy RJ, Thompson SG. A likelihood approach to meta-analysis with random effects. Stat.Med. 1996; 15(6):619-629 Micceri T. The Unicorn, the Normal Curve, and Other Improbable Creatures. Psychological Bulletin 1989; 105(1):156-166 Ramberg JS, Dudewicz EJ, Tadikamalla PR, Mykytka EF. A Probability Distribution and Its Uses in Fitting Data. Technometrics 1979; 21(2):201-214 26. Thank you for listening