Introduction to Robust Standard Errors

The Campbell Collaboration www.campbellcollaboration.org

Introduction to Robust Standard Errors

Emily E. Tanner-SmithAssociate Editor, Methods Coordinating Group

Research Assistant Professor, Vanderbilt University

Campbell Collaboration ColloquiumCopenhagen, Denmark

May 30th, 2012


Outline• Types of dependencies• Dealing with dependencies• Robust variance estimation• Practical considerations

– Choosing weights– Handling covariates

• Robust variance estimation in Stata


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


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


Correlated Effects Model (εi correlated)

kj correlated estimates of the study specific ES

τ2 = between study variation

in ESMeta-

regression

Dependent effect size

meta-regression

Study 1, θ1

Estimate 1.1 of θ1

Estimate 1.2 of θ1

Estimate 1.3 of θ1Study 2, θ2

Estimate 2.1 of θ2

Study 3, θ3Estimate 3.1 of θ3

Estimate 3.2 of θ3


Hierarchical Model (θi correlated)

ω2= between-study, within-cluster variation in ES

τ2 = between cluster variation

in average ESMeta-

regression

Dependent effect size

meta-regression

Cluster 1, θ1

Study 1.1 estimate of θ1.1



Cluster 2, θ2 Study 2.1 estimate of θ2.1

Cluster 3, θ3Study 3.1 estimate of θ3.1



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)

http://onlinelibrary.wiley.com/doi/10.1002/jrsm.5/pdf


Variance Estimation• Assume T = Xβ + ε, where

T = (T1,…,Tm)’ Tj is a kj x 1 vector of effect sizes for study jX = (X1,…,Xm)’ Xj is a kj x p design matrix for study jβ = (β1,…,βp)’ β is a 1 x p vector of unknown regression coefficientsε = (ε1,…,εm)’ εj is a kj x 1 vector of residuals for study j

E(εj) = 0, V(εj) = Σj


Variance Estimation• We can estimate β by:

• And the covariance matrix for this estimate is

• The problem is that although the variances in Σj are known, the covariances are UNKNOWN


Robust Variance Estimator (RVE)• The RVE of b is

where ej = Tj – Xjb is the kj x 1 estimated residual vector for study j


Robust Variance Estimator (RVE)• A robust test of H0: βa = 0 uses the statistic where vR

aa is the ath diagonal of the VR matrix

Note: the t-distribution with df = m - p should be used for critical values


Robust Variance Estimator (RVE)• Under regularity conditions and as m -> ∞, Vm

R 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


Practical Issues: Choosing Weights• Although the RVE works for any weights, the most efficient

weights are inverse-variance weights

– In the hierarchical model:

W 𝒊𝒋=1/ (𝑉 j+𝜏2+ω2)


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 𝒊𝒋=1

{(𝑉 j+𝜏2 ) [1+(k 𝑗−1 ) 𝜌 ] }


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:

• Conservative approach, because studies do not receive additional weight for contributing multiple effect sizes

W 𝒊𝒋=1

{𝑘 𝑗 (𝑉 j+𝜏2 )}


Practical Issues: Choosing Weights• In the correlated effects model, ρ also occurs in the

estimator of τ2:

• 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)


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


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


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


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


Practical Issues: Handling Covariates• When using the RVE, these two types of regression occur in

one analysis:Within Group Tij = β0j + Xijβ2 + … Between Group

β0j = β0 + Xjβ1 + …

• These two regressions are combined into one analysis and model:

Tij = β0 + Xijβ2 + Xjβ1 + …


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


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 X ij

– 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


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


• 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

Calculating Robust Variance Estimates

http://peabody.vanderbilt.edu/peabody_research_institute/methods_resources.xml

http://www.northwestern.edu/ipr/qcenter/RVE-meta-analysis.html


Robust Variance Estimation in Stata• Install robumeta.ado file from the Statistical Software

Components (SSC) archive


Robust Variance Estimation in Stata• Access example datasets and syntax/output files here:https://my.vanderbilt.edu/emilytannersmith/training-materials/

https://my.vanderbilt.edu/emilytannersmith/training-materials/

https://my.vanderbilt.edu/emilytannersmith/training-materials/


Robust Variance Estimation in Stata• robumeta.ado command structure

robumeta depvar [indepvars] [if] [in], study(studyid) variance(variancevar) weighttype(weightingscheme) rho(rhoval) [options]


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


Robust Variance Estimation in Stata


Intercept only model to estimate random-effects mean effect size with robust standard error, assuming ρ = .80







Robust Variance Estimation in Stata• 4 moderators of interest: 2 vary within and between studies,

2 vary between studies only


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




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



τ2 – between lab variance component; ω2 between-study within-lab variance component𝑑=.25 , p=.001 ;95% CI (.12 ,.38)


Robust Variance Estimation in Stata• 5 moderators of interest: all vary within and between clusters

(research labs)


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



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)


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.,

• 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)

W 𝒊𝒋=1

{𝑘 𝑗 (𝑉 j+𝜏2 )}


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.


P.O. Box 7004 St. Olavs plass0130 Oslo, Norway

E-mail: [email protected]://www.campbellcollaboration.org

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Introduction to Robust Standard Errors