1. Younghun Han Department of Epidemiology UT MD Anderson Cancer Center
Meta Analysis
Younghun Han
Department of Epidemiology
UT MD Anderson Cancer Center

2. Introduction
Meta-analysis is the statistical procedure for combining the results of several studies that address a set of related research hypotheses.
When the treatment effect (or effect size) is consistent from one study to the next, meta-analysis can be used to identify this common effect
When the effect is varies from one study to the next, meta-analysis may be used to identify reason for the variation
In epidemiological terms, meta-analysis provide a better estimate of effect size.
A meta-analysis can increase power and provide standards of reporting results in genome-wide association studies (GWAS)

3. Terminology
Effect size : In statistics, effect size is a measure of the strength of the relationship between two variables.
Binary outcomes : odds ratio, relative risk, risk difference
Continuous outcomes: difference in means, standardized difference in means, ∙ ∙ ∙
Summary effect size
(pooled, overall, or combined effect size)
Weighted average = ∑i(effecti × weighti) / ∑weighti
weight : sample size, Inverse of the variance, Quality …..

4. Terminology (continuous)
Fixed effect model (assume that the studies are homogeneous)
Mantel-Haenszel
Peto
Inverse Variance
Random effect model
DerSimonian and Laird
When the studies are found to be homogeneous, random and fixed effects models are indistinguishable.
Test for heterogeneity
Cochrans’s Q statistics I2

5. Terminology (continuous)
Forest plot
left-hand column lists the names of the studies
The right-hand column is a plot of the effect size with confidence interval for each study
The size of each square is proportional to the study's weight in the meta-analysis. The overall meta-analysed measure of effect is represented on the plot as a vertical line. This meta-analysed measure of effect is commonly plotted as a diamond, the lateral points of which indicate confidence intervals for this estimate.

6. Terminology (continuous)
Publication bias
Publication bias is the tendency to publish research with a positive outcome more frequently than research with a negative outcome.
Publication bias can lead to misleading results
Check publication bias by Funnel plots

7. Terminology (continuous)
Funnel plot
Effect size vs Sample size
Effect size vs S.E.
Effect size vs 1/S.E. (precision)
(a) (b)
The asymmetry in the right plot imply that a publication bias might be a problem

8. Mantel-Haenszel methods
Most commonly used method for meta-analysis
Need 2×2 frequency table for Mantel-Haenszel
Exposed
Non-Exposed
Case ai bi
Control ci di

9. Mantel-Haenszel methods (continuous)
summary effectMH
=∑i(effecti ×weighti)/ ∑i weighti
For OR: ORi = (ai×di)/ (bi×ci) , weighti= (bi×ci)/ni
For RR: RRi = [ai/(ai+ci)]/[bi/(bi×di)] , weighti=(ai+ci)bi/ni
For RD: RDi = [ai/(ai+ci)]-[bi/(bi×di)] , weighti=(ai+ci)(bi +di) /ni
where ni = ai + bi + ci + di

10. Meta-Analysis using R
Datasets : Four GWAS for Lung Cancer case-control data (US, Canada, France, and UK)
Example1 : rs

11. Meta-Analysis using R (continuous)
Fixed effect meta-analysis (Mantel-Haenszel)
> LungOR <- meta.MH(data[,1],data[,2],data[,3],data[,4],names=name)
> # data[,1]=Number of subjects in treated/exposed group
> # data[,2]=Number of subjects in control group
> # data[,3]=Number of events in treated/exposed group
> # data[,4]=Number of events in control group
> # names = names or labels for studies
>
> summary(LungOR)
Fixed effects ( Mantel-Haenszel ) meta-analysis
Call: meta.MH(ntrt = data[, 1], nctrl = data[, 2], ptrt = data[, 3],
pctrl = data[, 4], names = name)
OR (lower 95% upper) US
Canada
France UK
Mantel-Haenszel OR = % CI ( 1.27,1.41 )
Test for heterogeneity: X^2( 3 ) = ( p-value 0 )

12. Meta-Analysis using R (continuous)
>plot(LungOR, ylab="") # Forest Plot

13. Meta-Analysis using R (continuous)
tabletext<-cbind(c("Study",NA,LungOR$names,NA,"Summary"),
c("OR", NA, format(exp(LungOR$logOR),digits=2), NA, format(exp(LungOR$logMH),digits=3)),
c( NA,NA,format(exp(LungOR$logOR-LungOR$selogOR*1.96),digits=2), NA, format(exp(LungOR$logMH-LungOR$selogMH*1.96), digits=3 )),
c("(95% CI)", NA, format(exp(LungOR$logOR+LungOR$selogOR*1.96), digits=3), NA, format(exp(LungOR$logMH+LungOR$selogMH*1.96), digits=3 )))
m<- c(NA,NA,LungOR$logOR,NA,LungOR$logMH)
l<- m-c(NA,NA,LungOR$selogOR,NA,LungOR$selogMH)*1.96
u<- m+c(NA,NA,LungOR$selogOR,NA,LungOR$selogMH)*1.96
forestplot(tabletext, m, l, u, zero=0,
is.summary=c( TRUE, rep(FALSE,6),TRUE),
clip=c(log(0.4),log(3.5)), xlab="Odds Ratio", xlog=TRUE,
col=meta.colors(box="royalblue",line="darkblue", summary="royalblue"))
# xlog=TRUE : x-axis tick marks are exponentiated

14. Meta-Analysis using R (continuous)

15. Meta-Analysis using R (continuous)
Random effect meta-analysis ( DerSimonian-Laird )
> LungDSL <- meta.DSL(data[,1],data[,2],data[,3],data[,4],names=name)
> summary(LungDSL)
Random effects ( DerSimonian-Laird ) meta-analysis
Call: meta.DSL(ntrt = data[, 1], nctrl = data[, 2], ptrt = data[, 3],
pctrl = data[, 4], names = name)
OR (lower 95% upper) US
Canada
France UK
SummaryOR= % CI ( 0.77,2.16 )
Test for heterogeneity: X^2( 3 ) = ( p-value 0 )
check publication bias
>funnelplot()

16. Meta-Analysis using Stata
install commands
ssc install metan
ssc install metafunnel
ssc install metabias
metan (the main meta-analysis routine) requires either two, three, four or six variables to be declared.
Two variables : effect estimate and standard error
Three variables: effect estimate and its lower and upper confidence interval
Four variables: the number of events and non-events in the experimental group followed by those of the control group, and analysis of binary data is performed on the 2x2 table.
Six variables : the data are assumed continuous sample size, mean and standard deviation of the experimental group followed by those of the control group.
metafunnel plots funnel plots.
metabias performs the test for publication bias

17. Meta-Analysis using Stata (continuous)
Example1 : rs
. metan case_risk con_risk case_ref con_ref, or label(namevar=study) xlabel(0.5 , 3.5 ) texts(200) classic
Study | OR [95% Conf. Interval] % Weight
US |
Canada |
France |
UK |
M-H pooled OR |
Heterogeneity chi-squared = (d.f. = 3) p = 0.000
I-squared (variation in OR attributable to heterogeneity) = 98.7%
Test of OR=1 : z= p = 0.000

18. Meta-Analysis using Stata (continuous)

19. Meta-Analysis using Stata (continuous)
. metan case_risk con_risk case_ref con_ref, or random label(namevar=study)
xlabel(0.5, 3.5 ) texts(200) classic
Study | OR [95% Conf. Interval] % Weight
US |
Canada |
France |
UK |
D+L pooled OR |
Heterogeneity chi-squared = (d.f. = 3) p = 0.000
I-squared (variation in OR attributable to heterogeneity) = 98.7%
Estimate of between-study variance Tau-squared =
Test of OR=1 : z= p = 0.326

20. Meta-Analysis using Stata (continuous)

21. Meta-Analysis using Stata (continuous)
Example2 : rs

22. Meta-Analysis using Stata (continuous)
. metan case_risk con_risk case_ref con_ref, or label(namevar=study) texts(200) classic
Study | OR [95% Conf. Interval] % Weight
US |
Canada |
France |
UK |
M-H pooled OR |
Heterogeneity chi-squared = (d.f. = 3) p = 0.995
I-squared (variation in OR attributable to heterogeneity) = 0.0%
Test of OR=1 : z= p = 0.000

23. Meta-Analysis using Stata (continuous)

24. Meta-Analysis using Stata (continuous)
. metan case_risk con_risk case_ref con_ref, or random label(namevar=study) texts(200) classic
Study | OR [95% Conf. Interval] % Weight
US |
Canada |
France |
UK |
D+L pooled OR |
Heterogeneity chi-squared = (d.f. = 3) p = 0.995
I-squared (variation in OR attributable to heterogeneity) = 0.0%
Estimate of between-study variance Tau-squared =
Test of OR=1 : z= p = 0.000

25. Meta-Analysis using Stata (continuous)
. gen logor=ln((case_risk*con_ref)/(case_ref*con_risk))
. gen selogor=sqrt(1/case_risk + 1/case_ref + 1/con_risk +1/con_ref)
. metan logor selogor, eform effect(Odds ratio) xlabel(0.5, 3.5 ) texts(200)
classic
Study | ES [95% Conf. Interval] % Weight | | | |
I-V pooled ES |
Heterogeneity chi-squared = (d.f. = 3) p = 0.995
I-squared (variation in ES attributable to heterogeneity) = 0.0%
Test of ES=1 : z= p = 0.000

26. Thank you!!!