Méta régression (MR) Marc Sznajder, Patricia Samb (décors: Roger Hart Costumes: Donald Cardwell ) Staff DIHSP janvier 2009

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1 Méta régression (MR) Marc Sznajder, Patricia Samb (décors: Roger Hart Costumes: Donald Cardwell ) Staff DIHSP janvier

2 Références de Koning L, Merchant AT, Pogue J, Anand SS. Eur Heart J. 2007;28: Waist circumference and waist-to-hip ratio as predictors of cardiovascular events: meta-regression analysis of prospective studies. Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, Ontario, Canada L8L 2X2. Thompson SG, Higgins JP. Stat Med. 2002;21: How should meta-regression analyses be undertaken and interpreted? MRC Biostatistics Unit, Institute of Public Health, Robinson Way, Cambridge CB2 2SR, UK. 2

3 Description of MR extension to meta-analysis, (and a generalization of subgroup analyses) examines the relationship between one or more study-level characteristics and the sizes of effect observed in the studies (continuous covariates). can be used to investigate heterogeneity of effects across studies. 3

4 Points majeurs des MR Prendre en compte l hétérogénéité statistique liée à la diversité clinique et méthodologique des études incluses (idem méta-analyses) Lier la taille de l effet observé à une ou plusieurs caractéristiques des études incluses 4

5 MR Use of trial-level covariates ( regression analyses with individual data) (unité d analyse: l essai, donc si peu d essais: pas valide) MR should be weighted to take account of both: within-trial variances of treatment effects and the residual between-trial heterogeneity (that is, heterogeneity not explained by the covariates in the regression) This corresponds to random effects meta- regression. 5

6 Points majeurs des MR (suite) Diagramme: visualise la précision de l effet estimé de chaque traitement l unité d analyse est l essai 6

7 Log relative risk of stroke in 13 trials of aspirin versus placebo, according to aspirin dose, together with a summary random effects meta-regression. The area of each circle is inversely proportional to the variance of the log relative risk estimate. 7

8 Points majeurs des MR (suite) residual heterogeneity must be acknowledged in the statistical analysis. random effects rather than fixed effect meta-regression (If residual heterogeneity exists, a random effects analysis appropriately yields wider confidence intervals for the regression coefficients than a fixed effect analysis) 8

9 Points majeurs des MR (suite) The MR should clearly be weighted (so that the more precise studies have more influence in th analysis) weight for each trial: inverse of the sum of the within-trial variance and the residual between-trial variance (= random effects analysis). 9

10 Points majeurs des MR (suite) It is appropriate to use meta-regression to explore sources of heterogeneity even if an initial overall test for heterogeneity is non-signicant (test with low power). 10

11 Points majeurs des MR (suite) Estimating the residual between-trial variance is somewhat problematic. The estimate is usually imprecise because it is based on rather a limited number of trials. One way to allow for the imprecision is to adopt a Bayesian approach, using, for example, non-informative priors (help!!) 11

12 Points majeurs des MR (suite) Logiciel permettant de réaliser des méta- régressions à effet aléatoire: Stata 12

13 Point «mineur» des MR (suite) relationship described by a MR: observational association across trials. Although the original studies may be randomized trials, the meta-regression is across trials and does not have the benefit of randomization to underpin a causal interpretation risque de biais de confusion 13

14 Autres points «mineurs» des MR (suite) Si faible nombre d essais dans la MR et nombre élevé de covariables analysées (analyses multiples et post hoc): risque élevé de conclusions faussement positives D où l importance de pré-spécifier dans le protocole les variables à analyser et d en limiter le nombre; Cependant pas tjs facile d identifier à l avance les variables pertinentes (CDSR) 14

15 Weighted Least Squares (WLS) Regression Homoscedasticity: : the variance of residual error should be constant for all values of the independent(s). Violation of homoscedasticity occurs when the magnitude of the dependent is correlated with the variance of the independent (cf figure suivante). One possible cause of this might be a skewed rather than normally distributed dependent variable Wolberg, John (2006). Data analysis using the method of least squares: Extracting the most information from experiments. NY: Springer. 15

16 EX: The variance in Preference increases for higher values of Age. 16

17 Weighted Least Squares (WLS) Regression (suite) WLS regression compensates for violation of the homoscedasticity assumption by weighting cases differentially: cases whose value on the dependent variable corresponds to large variances on the independent variable(s) count less (and inversely) in estimating the regression coefficients. 17

18 Weighted Least Squares (WLS) Regression (suite) Cases are weighted by the reciprocal of their estimated point variance That is, cases with greater weights contribute more to the fit of the regression line. 18

19 Weighted Least Squares (WLS) Regression (suite) Weighted predicted/residual plots can be used to assess the goodness of fit of the weighted model. Weighted predicted is plotted on the x axis and weighted residual on the y axis. When there is good fit, the residuals will no longer form a funnel shape but instead be uniformly distributed around the 0.0 line of the y axis. 19

20 The funnel shape has disappeared. 20

21 Because of the reduction in heteroscedasticity, standard errors will be small but estimates will be very similar 21

22 Waist circumference and waist-to-hip ratio as predictors of cardiovascular events: meta-regression analysis of prospective studies (2007) AIMS: The objectives of this study were to determine the association of waist circumference (WC) and waist-to-hip ratio (WHR) with the risk of incident cardiovascular disease (CVD) events and to determine whether the strength of association of WC and WHR with CVD risk is different. METHODS AND RESULTS: This meta-regression analysis used a search strategy of keywords and MeSH terms to identify prospective cohort studies and randomized clinical trials of CVD risk and abdominal obesity from the Medline, Embase, and Cochrane databases. Fifteen articles (n = participants, 4355 CVD events) reporting CVD risk by categorical and continuous measures of WC and WHR were included. For a 1 cm increase in WC, the relative risk (RR) of a CVD event increased by 2% (95% CI: 1-3%) overall after adjusting for age, cohort year, or treatment. For a 0.01 U increase in WHR, the RR increased by 5% (95% CI: 4-7%). These results were consistent in men and women. Overall risk estimates comparing the extreme quantiles of each measure suggested that WHR was more strongly associated with CVD than that for WC (WHR: RR = 1.95, 95% CI: ; WC: RR = 1.63, 95% CI: ), although this difference was not significant. The strength of association for each measure was similar in men and women. CONCLUSION: WHR and WC are significantly associated with the risk of incident CVD events. These simple measures of abdominal obesity should be incorporated into CVD risk assessments. 22

23 Waist circumference and waist-to-hip ratio as predictors of cardiovascular events: meta-regression analysis of prospective studies (2007) Méthodo: Weighted-least-squares (WLS) regression in studies that reported risk estimates by quantiles of WC or WHR. Outcome: : natural logarithm of CVD risk in each quantile Beta-coefficients represented the change in log CVD risk for a 1 U increase in WC or WHR Inverse (quasi)-variance of risk estimates as regression weights in order to include the reference category in the regression 23

24 Waist circumference and waist-to-hip ratio as predictors of cardiovascular events: meta-regression analysis of prospective studies (2007) (suite) Méthodo (suite): Heterogeneity in beta-coefficients was explored using a random effects meta-regression model ( metareg module, Stata ver 8.2). We included predictors for mean age, mean follow-up, and the type of data (categorical or continuous) used to derive beta-coefficients, with beta-coefficients as the outcome Beta-coefficients were weighted by their inverse variances and pooled using the DerSimonian and Laird random effects model to allow for differences between studies ( meta module, Stata ver 8.2). Cochrane's Q was used to assess heterogeneity among the beta- coefficients. Pooled beta-coefficients with 95% confidence intervals were exponentiated and plotted to assess the statistical significance of the estimates. Risk estimates for WC were evaluated for a 1 cm increase, and estimates for WHR were evaluated for a 0.01 U increase. We calculated the predicted changes in WC and WHR for an equivalent increase in CVD risk to give WHR a meaningful interpretation. 24

25 Funnel plots of moderately and maximally adjusted beta-coefficients. 25

26 Pooled exponentiated beta-coefficients and 95% confidence intervals plotted by sex and level of adjustment. 26