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Meta-Analysis of Correlated Data
Meta-Analysis of Correlated Data
Common Forms of Dependence
• Multiple effects per study– Or per research group!
• Multiple effect sizes using same control
• Phylogenetic non-independence
• Measurements of multiple responses to a common treatment
• Unknown correlations…
Multiple Sample Points per Study!
Study Experiment in Study Hedges D V Hedges D
Ramos & Pinto 2010 1 4.32 7.23
Ramos & Pinto 2010 2 2.34 6.24
Ramos & Pinto 2010 3 3.89 5.54
Ellner & Vadas 2003 1 -0.54 2.66
Ellner & Vadas 2003 2 -4.54 8.34
Moria & Melian 2008 1 3.44 9.23
Hierarchical Models
• Study-level random effect
• Study-level variation in coefficients
• Covariates at experiment and study level
Hierarchical Models• Random variation within study (j)
and between studies (i)
Tijij,ij2)
ijj,j2
j,2
Study Level Clustering
Hierarchical Partitioning of One Study
Grand Mean
Study Mean
Variation due to
Variation due to
Example: Data Set 1
Group Effect Variance
1 A 0.2 0.10
2 A 0.6 0.15
3 A 0.5 0.05
4 A 0.1 0.06
5 B 0.8 0.08
6 B 0.4 0.05
7 B 0.9 0.04
8 C 0.2 0.09
...
A Two-Step SolutionTijij,ij
2)
ijj,j2
j,2library(plyr)
data1_study <- ddply(data1, .(Group), function(adf){
mod <- rma(Effect, Variance, data=adf)
cbind(theta_j = coef(mod),
se_theta_j = coef(summary(mod))[1,2],
omega2 = mod$tau2)
})
A Two-Step Solution
Tijij,ij2)
ijj,j2
j,2
> data1_study
Group theta_j se_theta_j omega2
1 A 0.3312500 0.1369306 0.00000000
2 B 0.7005364 0.1654476 0.02854676
3 C 0.6788453 0.1987595 0.17151248
4 D 0.7836646 0.2677693 0.26470540
5 E 0.8552760 0.1556476 0.14561528j j
A Two-Step Solution
Tijij,ij2)
ijj,j2
j,2
> rma(theta_j, I(se_theta_j^2), data=data1_study)
Random-Effects Model (k = 5; tau^2 estimator: REML)
tau^2 (estimated amount of total heterogeneity): 0.0272 (SE = 0.0414)
...
estimate se zval pval ci.lb ci.ub
0.6472 0.1087 5.9545 <.0001 0.4342 0.8603 ***
2
Multiple Effects per Research Group
Solutions to Multiple Hierarchies
• Multiple-Step Meta-analyses
• Multi-level hierarchical model fits– Better estimator– Accommodates more complex data
structures–May need to go Bayesian (don't be scared!)
• Model correlation…
Common Forms of Dependence
• Multiple effects per study– Or per research group!
• Multiple effect sizes using same control
• Phylogenetic non-independence
• Measurements of multiple responses to a common treatment
• Unknown correlations…
Multiple Effect Sizes with Common Control
Effect of each treatment calculated using same control!
The Control Keeps Showing Up!
• nc and sdc are going to be the same for all treatments
• Effect sizes will covary
Calculating Covariance
Formulae available or derivable for all effect sizes
A Mixed Effect Group Model
• Group means, random study effect, and then everything else is error
Tiim,i2)
where
imm,2
A Mixed Effect Group Model
• Group means, random study effect, and then everything else is error
TiMVNi,i)
where
iMVN Xi, 2
What are i and i?
i =
i=
TiMVNi,i)
What about the treatment effects?
Xi =
i= iMVN Xi, 2
What if treatments are correlated?
i =
TiMVNi,i)
Why does covariance matter?
x-y =
x + y + 2
x,y
• In asking if two treatments differ, cov helps tighten confidence intervals
• High cov more weight for a study as treatments share information
Multiple Treatments
study trt m1i m2i sdpi n1i n2i
1 1 1 7.87 -1.36 4.2593 25 25
2 1 2 4.35 -1.36 4.2593 22 25
3 2 1 9.32 0.98 2.8831 38 40
4 3 1 8.08 1.17 3.1764 50 50
5 4 1 7.44 0.45 2.9344 30 30
6 4 2 5.34 0.45 2.9344 30 30CommonControl!
http://www.metafor-project.org/doku.php/analyses:gleser2009
Calculating the Variance/Covariance Matrix
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.113 0.060 0.000 0.000 0.000 0.000
[2,] 0.060 0.098 0.000 0.000 0.000 0.000
[3,] 0.000 0.000 0.105 0.000 0.000 0.000
[4,] 0.000 0.000 0.000 0.064 0.000 0.000
[5,] 0.000 0.000 0.000 0.000 0.098 0.055
[6,] 0.000 0.000 0.000 0.000 0.055 0.082
http://www.metafor-project.org/doku.php/analyses:gleser2009
Fitting a Model with a VCOV Matrix
> rma.mv(yi ~ factor(trt)-1,
V,
random =~ 1|study,
data=dat)
Comparison to No Correlation Model
With correlation estimate se zval pval ci.lb ci.ub
factor(trt)1 2.3796 0.1641 14.4984 <.0001 2.0579 2.7013
factor(trt)2 1.5784 0.2007 7.8662 <.0001 1.1851 1.9716
Without correlation estimate se zval pval ci.lb ci.ub
factor(trt)1 2.3759 0.1511 15.7196 <.0001 2.0797 2.6722
factor(trt)2 1.5177 0.2125 7.1405 <.0001 1.1011 1.9343
Common Forms of Dependence
• Multiple effects per study– Or per research group!
• Multiple effect sizes using same control
• Phylogenetic non-independence
• Measurements of multiple responses to a common treatment
• Unknown correlations…
Effect Size on Related Organisms Not Independent
Warming onLitterfall
Pine TreesRedwoodsFir Trees
Oak Trees{
Phylogenetic Distances Determines Covariances for
Weights
What about Multiple Studies of Some Species?
Common Forms of Dependence
• Multiple effects per study– Or per research group!
• Multiple effect sizes using same control
• Phylogenetic non-independence
• Measurements of multiple responses to a common treatment
• Unknown correlations…
Common Treatments
Treatment
Response 1 Response 2 Response 3
Common Treatments
CO2
CO2 Assimilation
GS
Stomatal Conductance
PN
Correlation Between Responses
What does Correlation between effects mean?
Xi =
i= iMVN Xi, 2
What Do We Do?1. Create a 'composite' measure
– Average
– Weighted Average
2. Estimate different coefficients directly
3. Robust Variance Estimation (RVE)
The CO2 Effect Data
experiment Paper Measurement Hedges Var
1 1 121 GS -0.4862 0.3432
2 1 121 PN 0.9817 0.3735
3 2 121 GS 0.1535 0.3343
4 2 121 PN 2.0668 0.5113
5 3 121 GS 0.0965 0.3337
6 3 121 PN 2.6101 0.6172
7 4 121 GS 0.0000 0.2857
8 4 121 PN 3.6586 0.7638
9 5 168 GS -1.5271 0.4305
10 5 168 PN 1.8355 0.4737
Direct Estimation
rma.mv(Hedges ~ Measurement,
Var,
random =~ Measurement|Paper,
data=co2data,
struct="HCS")
and Different Correlation Structures
• Different structures for different data
• We do not always know which one is correct!
Estimates of Variance, Covariance
Multivariate Meta-Analysis Model (k = 68; method: REML)
Variance Components:
outer factor: Paper (nlvls = 18)
inner factor: Measurement (nlvls = 2)
estim sqrt k.lvl fixed level
tau^2.1 4.5098 2.1236 34 no GS
tau^2.2 3.5799 1.8921 34 no PN
rho 0.4751 no
Disadvantages to Multivariate Meta-Analysis
1. Difficult to estimate with few studies
2. Additional assumptions of covariance structure
3. Often little improvement over univariate meta-analysis
4. Publication bias exacerbated if data not missing at random
Jackson et al. 2011 Satist. Med.
Robust Variance Estimation
• Essentially, bound weights within a group j to 1/mean varj and assume a value of
– Test sensitivity to choice of
– Correct DF for small sample sizes
• Methods developed by Hedges, Tipton, and others
• robumeta package in R
robumeta & RVE
library(robumeta)
robu(Hedges ~ Measurement, data=co2data,
studynum=Paper,
var.eff.size=Var)
RVE: Correlated Effects Model with Small-Sample Corrections
Model: Hedges ~ Measurement
Number of studies = 18
Number of outcomes = 68 (min = 2 , mean = 3.78 , median = 4 , max = 10 )
Rho = 0.8
I2 = 85.59992
Tau.Sq = 2.561661
Struct="CS" only so far
Often, Choice of Matters Little
> sensitivity(co2modRVE)
Type Variable rho=0 rho=0.2 rho=0.4 rho=0.6 rho=0.8 rho=1
1 Estimate intercept 0.00454 0.00457 0.00459 0.00462 0.00464 0.00467
2 - MeasurementPN 1.03149 1.03139 1.03128 1.03118 1.03107 1.03097
3 Std. Err. intercept 0.51173 0.51179 0.51185 0.51192 0.51198 0.51204
4 - MeasurementPN 0.61984 0.61990 0.61996 0.62003 0.62009 0.62015
5 Tau.Sq - 2.55334 2.55542 2.55750 2.55958 2.56166 2.56374
Results May Differ…
Multivariate Meta-AnalysisModel Results:
estimate se zval pval ci.lb ci.ub
intrcpt -0.0503 0.5221 -0.0963 0.9233 -1.0735 0.9730
MeasurementPN 1.0579 0.5359 1.9742 0.0484 0.0076 2.1082 *
Robust Variance EstimationModel Results:
Estimate StdErr t-value df P(|t|>) 95% CI.L 95% CI.U Sig
1 intercept 0.00464 0.512 0.00907 16.7 0.993 -1.077 1.09
2 MeasurementPN 1.03107 0.620 1.66278 16.7 0.115 -0.279 2.34
Other Sources of Unknown Correlations
• Shared system types
• Shared environmental events
• Labs or investigators
• Re-sampling experiments
• Experiments repeated in a region
• More…
Why Model Correlation instead of Hierarchy?
• Depends on question
• Analytical difficulty
• Leveraging correlation to aid with missing data