Report

Introduction to Robust Standard Errors Emily E. Tanner-Smith Associate Editor, Methods Coordinating Group Research Assistant Professor, Vanderbilt University Campbell Collaboration Colloquium Copenhagen, Denmark May 30th, 2012 The Campbell Collaboration www.campbellcollaboration.org Outline • Types of dependencies • Dealing with dependencies • Robust variance estimation • Practical considerations – Choosing weights – Handling covariates • Robust variance estimation in Stata The Campbell Collaboration www.campbellcollaboration.org Types of Dependencies • Most meta-analysis techniques assume effect sizes are statistically independent • But there are many instances when you might have dependent effect sizes – Multiple measures of the same underlying outcome construct – Multiple measures across different follow-up periods – Multiple treatment groups with a common control group – Multiple studies from the same research laboratory The Campbell Collaboration www.campbellcollaboration.org Types of Dependencies • Assume Ti = θi + εi, where – Ti is the effect size estimate – θi is the effect size parameter – εi is the estimation error • Statistical dependence can arise because – εi are correlated – θi are correlated – or both The Campbell Collaboration www.campbellcollaboration.org Correlated Effects Model (εi correlated) Meta-regression Dependent effect size metaregression τ2 = between study variation in ES Study 1, θ1 Study 2, θ2 kj correlated estimates of the study specific ES Estimate 1.1 of θ1 Estimate 1.2 of θ1 Estimate 1.3 of θ1 Estimate 2.1 of θ2 Study 3, θ3 Estimate 3.1 of θ3 Estimate 3.2 of θ3 The Campbell Collaboration www.campbellcollaboration.org Hierarchical Model (θi correlated) Metaregression τ2 = between cluster variation in average ES ω2= between-study, within-cluster variation in ES Study 1.1 estimate of θ1.1 Dependent effect size metaregression Cluster 1, θ1 Study 1.2 estimate of θ1.2 Study 1.3 estimate of θ1.3 Cluster 2, θ2 Cluster 3, θ3 Study 2.1 estimate of θ2.1 Study 3.1 estimate of θ3.1 Study 3.2 estimate of θ3.2 The Campbell Collaboration www.campbellcollaboration.org Dealing with Dependencies • Ignore it and analyze the effect sizes as if they are independent (not recommended) • Select a set of independent effect sizes – Create a synthetic mean effect size – Randomly select one effect size – Choose the “best” effect size • Model the dependence with full multivariate analysis – This requires information on the covariance structure • Use robust variance estimation (Hedges, Tipton, & Johnson, 2010) The Campbell Collaboration www.campbellcollaboration.org Variance Estimation • Assume T = Xβ + ε, where T = (T1,…,Tm)’ X = (X1,…,Xm)’ β = (β1,…,βp)’ ε = (ε1,…,εm)’ Tj is a kj x 1 vector of effect sizes for study j Xj is a kj x p design matrix for study j β is a 1 x p vector of unknown regression coefficients εj is a kj x 1 vector of residuals for study j E(εj) = 0, V(εj) = Σj The Campbell Collaboration www.campbellcollaboration.org Variance Estimation • We can estimate β by: æ ö b = ç å X j'Wj X j ÷ è ø m j=1 -1 m å X 'W T j j j j=1 • And the covariance matrix for this estimate is -1 æ ö æ öæ ö V (b) = ç å X j'Wj X j ÷ ç å X j 'Wj S j Wj X j ÷ ç å X j'Wj X j ÷ è j=1 ø è j=1 ø è j=1 ø m m m -1 • The problem is that although the variances in Σj are known, the covariances are UNKNOWN The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimator (RVE) • The RVE of b is -1 æ ö æ öæ ö V = ç å X j'Wj X j ÷ ç å X j 'Wj e j e j 'Wj X j ÷ ç å X j'Wj X j ÷ è j=1 ø è j=1 ø è j=1 ø R m m m -1 where ej = Tj – Xjb is the kj x 1 estimated residual vector for study j The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimator (RVE) • A robust test of H0: βa = 0 uses the statistic where vRaa is the ath diagonal of the VR matrix t = R a ba ( )v m m-p R aa Note: the t-distribution with df = m - p should be used for critical values The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimator (RVE) • Under regularity conditions and as m -> ∞, VmR is a • • • • • consistent estimator of the true covariance matrix RVE theorem is asymptotic in the number of studies m, not the number of effect sizes Results apply to any type of dependency No distributional assumptions needed for the effect sizes Correlations do not need to be known or specified, though may impact the standard errors RVE theorem applies for any set of weights The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Choosing Weights • Although the RVE works for any weights, the most efficient weights are inverse-variance weights – In the hierarchical model: W = 1 The Campbell Collaboration j + 2 + ω2 www.campbellcollaboration.org Practical Issues: Choosing Weights • In the correlated effects model, we can estimate approximately efficient weights by assuming a simplified correlation structure: – Within each study j, the correlation between all pairs of effect sizes is a constant ρ – ρ is the same in all studies – kj sampling variances within the study are approximately equal with average Vj W = The Campbell Collaboration 1 { j + 2 1 + k − 1 } www.campbellcollaboration.org Practical Issues: Choosing Weights • In the correlated effects model, we can take a conservative approach when calculating weights by also assuming ρ = 1, and weights become: W = 1 j + 2 • Conservative approach, because studies do not receive additional weight for contributing multiple effect sizes The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Choosing Weights • In the correlated effects model, ρ also occurs in the estimator of τ2: æ m wj ö æ m wj ö QE - m + tr ç V å X j'X j ÷ + r tr ç V å éë X j'J j X j - X j'X j ùû÷ è j=1 k j ø è j=1 k j ø 2 ˆ t = m æ m 2 ö å k jw j - tr çè V åw j X j'J j X j ÷ø j=1 j=1 • Use external information about ρ if available (test reliabilities, correlations reported in studies, etc.) • Take a sensitivity approach when estimating τ2 by estimating the model with various values of ρ in (0, 1) The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Choosing Weights • For the correlated effects model, RVE software is currently programmed to default to (per Hedges, Tipton, & Johnson recommendation): – Conservative approach to estimate weights (assume ρ = 1) – User must specify ρ for estimation of τ2 ; sensitivity tests recommended The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • Some covariates may vary within groups (i.e., studies or clusters) and between groups, e.g., – Length of follow-up after intervention – Time frame of outcome measure – Outcome reporter (self-report vs. parent-report) – Type of outcome construct (frequency vs. quantity of alcohol use) • When modeling the effects of a covariate, ask if the effect of interest is between- or within-groups The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • In a standard meta-regression with independent effect sizes, Tj = β0 + Xjβ1 + … where Xj is length to follow-up, β0 and β1 can be interpreted as: – β0 = the average effect size when Xj = 0 • e.g. the average effect size in studies in which the intervention just occurred – β1 = the effect of a 1-unit increase in Xj on Tj • e.g. the effect size change associated with moving from a study in which the intervention just occurred to a study in which the effect size was measured at a 1 month posttest follow-up The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • In the correlated effects model, for a fixed study (j = 1), now assume there are multiple outcomes. This study has its own regression equation: Ti1 = β01 + Xi1β11 +… • The coefficients β01 and β11 can be interpreted as: – β01 = the average effect size when Xi1 = 0 • e.g. the average effect size for units in the study (j = 1) when the intervention just occurred – β11 = the effect of a 1-unit increase in Xi1 on Ti1 • e.g. the effect size change for units in the study (j = 1) at the time of intervention and at follow-up 1 month later The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • When using the RVE, these two types of regression occur in one analysis: Within Group Between Group Tij = β0j + Xijβ2 + … β0j = β0 + Xjβ1 + … • These two regressions are combined into one analysis and model: Tij = β0 + Xijβ2 + Xjβ1 + … The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • To properly separate estimation of within- and between- group effects of covariates, use group mean centering: Xcij = Xij – Xj where Xj is the mean value of Xij in group j (and where group is either study or cluster). So now, Tij = β0 + Xcijβ2 + Xjβ1 + … • If you don’t center Xij you are actually modeling a weighted combination of the within- and between-study effect, which is difficult to interpret The Campbell Collaboration www.campbellcollaboration.org Practical Issues: Handling Covariates • When using a covariate, ask if the effect of interest is between- or within-groups • Make sure to group-center your within-group variables • Acknowledge that if only a few groups have variability in Xij – Within-group estimate of β2 (associated with Xcij) will be imprecise (i.e. have a large standard error) – The types of groups in which Xij varies may be different than (i.e. not representative of) groups in which Xij does not vary The Campbell Collaboration www.campbellcollaboration.org Calculating Robust Variance Estimates • Variables you will need in your dataset – Group identifier (e.g., study/cluster identification number) – Effect size estimate – Variance estimate of the effect size – Any moderator variables or covariates of interest • Additional pieces of information you will need to specify – In a correlated effects model: assumed correlation between all pairs of effect sizes (ρ) – Fixed, random, hierarchical, or user-specified weights The Campbell Collaboration www.campbellcollaboration.org Calculating Robust Variance Estimates • Stata ado file available at SSC archive: type ssc install robumeta • SPSS macro available at: http://peabody.vanderbilt.edu/peabody_research_institute/methods_resources.xml • R functions available at: http://www.northwestern.edu/ipr/qcenter/RVE-meta-analysis.html The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • Install robumeta.ado file from the Statistical Software Components (SSC) archive The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • Access example datasets and syntax/output files here: https://my.vanderbilt.edu/emilytannersmith/training-materials/ The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • robumeta.ado command structure robumeta depvar [indepvars] [if] [in], study(studyid) variance(variancevar) weighttype(weightingscheme) rho(rhoval) [options] The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • Example of a correlated effects model (correlated ε) • Fictional meta-analysis on the effectiveness of alcohol abuse treatment for adolescents • Effect sizes represent post-treatment differences between treatment and comparison groups on some measure of alcohol use (positive effect sizes represent beneficial treatment effects) – Number of effect sizes k = 172 – Number of studies m = 39 – Average number of effect sizes per study = 4.41 The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata Intercept only model to estimate random-effects mean effect size with robust standard error, assuming ρ = .80 = .23, p = .001; 95% CI (.11, .36) The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • 4 moderators of interest: 2 vary within and between studies, 2 vary between studies only The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • To model both the within- (Xcij) and between- effects (Xj) of the type of alcohol outcome and follow-up time frame, create group mean and group mean centered variables The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • Let’s say we have a similar meta-analysis, but now need to estimate a hierarchical model (correlated θ) • Effect sizes represent post-treatment differences between treatment and comparison groups on some measure of alcohol use (positive effect sizes represent beneficial treatment effects) – Number of effect sizes k = 68 – Number of research labs m = 15 – Average number of effect sizes per research lab = 4.5 The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata τ2 – between lab variance component; ω2 between-study within-lab variance component = .25, p = .001; 95% CI (.12, .38) The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • 5 moderators of interest: all vary within and between clusters (research labs) The Campbell Collaboration www.campbellcollaboration.org Robust Variance Estimation in Stata • To model both the within- (Xcij) and between- effects (Xj) of the covariates of interest, create group mean and group mean centered variables The Campbell Collaboration www.campbellcollaboration.org The Campbell Collaboration www.campbellcollaboration.org Conclusions & Recommendations • Robust variance estimation is one way to handle dependencies in effect size estimates, and allows estimation of within- and between-study effects of covariates – Method performs well when there are 20 or more studies with an average of 2 or more effect size estimates per study • Choose the proper model for the type of dependencies in your data (correlated ε or correlated θ) The Campbell Collaboration www.campbellcollaboration.org Conclusions & Recommendations • When using the correlated effects model (correlated ε), with efficient weights, if you have no information on ρ: – Use a sensitivity approach for estimating τ2 – Assume ρ = 1 in your weights, i.e., W = 1 j + 2 • For each covariate Xij in your model, remember that you can estimate: – Between-group effect: group mean (Xj) – Within-group effect: group mean centered variable (Xcij = Xij – Xj) The Campbell Collaboration www.campbellcollaboration.org Recommended Reading Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1, 39-65. The Campbell Collaboration www.campbellcollaboration.org P.O. Box 7004 St. Olavs plass 0130 Oslo, Norway E-mail: [email protected] http://www.campbellcollaboration.org The Campbell Collaboration www.campbellcollaboration.org