metafor: estimate correlation between response variables (meta-analysis)

I have a hard time making sense of that toy example.

But if I understand you correctly, you have data like this:

study trt respvar yi vi
-----------------------
1     1   1       .  .
1     1   2       .  .
1     2   1       .  .
1     2   2       .  .
2     1   1       .  .
2     1   2       .  .
2     2   1       .  .
2     2   2       .  .
...

where 'yi' and 'vi' are the observed outcomes and corresponding variances.

Within studies, can we assume that 'yi' is independent for different treatments? Then V (the var-cov matrix of the 'yi' vector) will be block-diagonal, each block being a 2x2 matrix (since you only need to consider the covariance between 'yi' for respvar 1 and 'yi' for respvar 2). And the problem is that the covariances are unknown. Correct so far?

If so, the 'R' argument has nothing to do with this. If you want to approach this by means of a sensitivity analysis, then just impute the unknown covariances directly into 'V' and analyze by means of an appropriate multilevel/multivariate model. Something like this:

dat$study.trt <- interaction(dat$study, dat$trt)
rma.mv(yi, V, mods = ~ factor(trt) - 1, random = list(~ 1 | study, ~ respvar | study.trt), struct="UN", data=dat)

To impute the covariances, you may be able to use cor*sqrt(v1*v2), where 'cor' is some kind of assumed correlation (maybe constant across studies, maybe not). However, whether this is appropriate depends on your outcome measure. For example, this would be fine for means or mean differences, but the covariance between standardized mean differences cannot be computed that way (see Gleser & Olkin, 2009, for the correct equation).

Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 357-376). New York: Russell Sage Foundation.

Best,
Wolfgang

-----Original Message-----
From: R-sig-mixed-models [mailto:r-sig-mixed-models-bounces at r-
project.org] On Behalf Of Gustaf Granath
Sent: Tuesday, December 13, 2016 11:42
To: r-sig-mixed-models at r-project.org
Subject: [R-sig-ME] metafor: estimate correlation between response
variables (meta-analysis)

Hi,
Im trying the estimate the correlation between two response variable in
a meta-analysis. Basically, are effects on y1 associated with effects on
y2 across studies. Unfortunately, I dont have information on y1-y2
correlations within each study. In addition, each study contain multiple
treatments, adding within-study dependence for each response variable.

Because I dont have the y1-y2 covariance in each study, my idea is to
run analyses with different covariance/correlation values to explore how
this covariance affect the result. Reading the rma.mv() documentation,
this seems possible through the R= argument, but I cant figure out how.
Alternatively, I can add covariances in the matrix describing the known
var-covariance matrix of the within-study dependence, but it seemed like
the R argument is a easier solution (I may be wrong though). Below
follow code illustrating my problem using the dat.berkey1998 example.

Cheers

Gustaf

# Estimate correlation between two response variables
# with within-study dependence
library(metafor)

# Make up an example based on the berkey1998 data.
# Multiple outcomes (multiple treatments)
# within each study, for each response variable, are created.
# Covariances between response variables (within studies) are unknown.
dat <- get(data(dat.berkey1998))
st.vcov <- dat[, c("v1i", "v2i")] # save vcov matrices to add later
dat <- rbind(dat, dat)
dat <- dat[order(dat$trial, dat$outcome),] # fix order
dat[,c("v1i", "v2i")] <- rbind(st.vcov,st.vcov) # put back matrices
dat$trial.treat <-paste(rep(1:2, nrow(dat)/2), dat$trial, sep="_") # id
for treatments within trials

# Covariances within studies, for each response variable, is known.
Hence, a
# varcov-matrix for each study and response, can be made and added as
blocks
# in a large varcov-matrix for the data set.
# First a within-study dependence dummy must be added
dat$stud.unit <-  interaction(dat$trial, dat$outcome) # within-trial
dependence dummy
# Put together known varcov matrix
V <- bldiag(lapply(split(dat[,c("v1i", "v2i")], dat$stud.unit),
as.matrix))

# plot relationship between response variables
dat.wide <- reshape(dat, direction="wide",  v.names = "yi", timevar =
"outcome", idvar = "trial.treat")
plot(yi.AL ~yi.PD, dat.wide)
cor(dat.wide$yi.AL, dat.wide$yi.PD)
# r = 0.39, if using the study outcomes ignoring all
dependence/uncertainty.

# Run analysis
res <- rma.mv(yi, V, mods = ~ outcome - 1,
                random = ~ outcome | trial,
                struct="UN", data=dat, method="REML")
print(res, digits=3)
# correlation = 0.51, when accounting for dependence etc. But, y1 and y2
are assumed to be independent
# within each study. How to perform sensitivity analyses by adding
different values
# on this within-study correlation between the two response variables?

--
Gustaf Granath (PhD)
Post doc
Swedish University of Agricultural Sciences
Department of Ecology