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
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Outline
• Types of dependencies
• Dealing with dependencies
• Robust variance estimation
• Practical considerations
– Choosing weights
– Handling covariates
• Robust variance estimation in Stata
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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Practical Issues: Choosing Weights
• Although the RVE works for any weights, the most efficient
weights are inverse-variance weights
– In the hierarchical model:
W𝒊𝒋 = 1
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𝑉j + 𝜏 2 + ω2
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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 Vj
W𝒊𝒋 =
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1
{ 𝑉j + 𝜏 2 1 + k𝑗 − 1 𝜌 }
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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:
W𝒊𝒋 =
1
𝑘𝑗 𝑉j + 𝜏 2
• Conservative approach, because studies do not receive
additional weight for contributing multiple effect sizes
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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)
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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
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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
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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
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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
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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 + Xjβ1 + …
• These two regressions are combined into one analysis and
model:
Tij = β0 + Xijβ2 + Xjβ1 + …
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Practical Issues: Handling Covariates
• To properly separate estimation of within- and between-
group effects of covariates, use group mean centering:
Xcij = Xij – Xj
where Xj is the mean value of Xij in group j (and where
group is either study or cluster). So now,
Tij = β0 + Xcijβ2 + Xjβ1 + …
• 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
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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
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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
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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
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Robust Variance Estimation in Stata
• Install robumeta.ado file from the Statistical Software
Components (SSC) archive
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Robust Variance Estimation in Stata
• Access example datasets and syntax/output files here:
https://my.vanderbilt.edu/emilytannersmith/training-materials/
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Robust Variance Estimation in Stata
• robumeta.ado command structure
robumeta depvar [indepvars] [if] [in], study(studyid)
variance(variancevar) weighttype(weightingscheme) rho(rhoval)
[options]
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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
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Robust Variance Estimation in Stata
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Robust Variance Estimation in Stata
Intercept only model to estimate random-effects mean effect size with robust standard error,
assuming ρ = .80
𝑑 = .23, p = .001; 95% CI (.11, .36)
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Robust Variance Estimation in Stata
The Campbell Collaboration
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Robust Variance Estimation in Stata
The Campbell Collaboration
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Robust Variance Estimation in Stata
• 4 moderators of interest: 2 vary within and between studies,
2 vary between studies only
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Robust Variance Estimation in Stata
• To model both the within- (Xcij) and between- effects (Xj) of
the type of alcohol outcome and follow-up time frame, create
group mean and group mean centered variables
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Robust Variance Estimation in Stata
The Campbell Collaboration
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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
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Robust Variance Estimation in Stata
τ2 – between lab variance component; ω2 between-study within-lab variance component
𝑑 = .25, p = .001; 95% CI (.12, .38)
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Robust Variance Estimation in Stata
• 5 moderators of interest: all vary within and between clusters
(research labs)
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Robust Variance Estimation in Stata
• To model both the within- (Xcij) and between- effects (Xj) of
the covariates of interest, create group mean and group
mean centered variables
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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)
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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.,
W𝒊𝒋 =
1
𝑘𝑗 𝑉j + 𝜏 2
• For each covariate Xij in your model, remember that you can
estimate:
– Between-group effect: group mean (Xj)
– Within-group effect: group mean centered variable (Xcij = Xij – Xj)
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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.
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