metafor package: study level variation

Thanks for this.  A few things

First, yes, my lmer syntax was indeed bad - I was writing this as an 
example of what my data & code look like.  Apologies. So, 1|studyID 
indeed.  Also 1/variance.

I've also been wondering - I often have more than 2 drugs - so, T1, 
T2, T3, etc.  It depends on what slice of the data I'm looking 
at.  As some of the data sets have, say, T1 and T2, and others have 
T1 and T3, while still others have just T1  - perhaps the way to go 
is to look at each effect size for each drug independently, rather 
than pool them in one analysis.  That does not count for the 
non-independence, though, of different effect sizes being calculated 
sometimes using the same control - which you rightly identify.

I'm curious, what other packages in R would be useful for this?  I 
can also change these from log ratios to log odds ratios - or at 
least obtain # with desirable response v. # with negative response 
for all treatment - perhaps this would be another way to go about it 
using meta.bin in the meta package - again, looking at each metric separately.

On Sep 10, 2012, at 4:52 AM, Viechtbauer Wolfgang (STAT) wrote:

As usual, Michael was faster than I in responding. Let me add a 

few thoughts of my own. See comments below in text.

Best,
Wolfgang

--
Wolfgang Viechtbauer, Ph.D., Statistician
Department of Psychiatry and Psychology
School for Mental Health and Neuroscience
Faculty of Health, Medicine, and Life Sciences
Maastricht University, P.O. Box 616 (VIJV1)
6200 MD Maastricht, The Netherlands
+31 (43) 388-4170 | http://www.wvbauer.com

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Jarrett Byrnes
Sent: Friday, September 07, 2012 16:02
To: R help
Subject: [R] metafor package: study level variation

Hello.  A quick question about incorporating variation due to 

study in the

metafor package.  I'm working with a particular data set for 

meta-analysis

where some studies have multiple measurements.  Others do not.  So, let's
say the effect I'm looking at is response to two different kinds of drug
treatment - let's call their effect sizes T1 and T2.  Some studies have
multiple experiments measuring  T1 and T2.  Some have one of each.  Some
only have T1 or T2.

I assume there is also a control group/condition in each of these 

studies, so in other words, you have a bunch of studies where some 
are two-arm studies comparing Trt1 *or* Trt2 to control and some 
are three-arm studies comparing both Trt1 *and* Trt2 to control.

Now, in metafor, I've been using

rma(yi = logRatio, vi=varLogRatio, mods=~ Drug.Type, data=mydata)

So, drug.type is a dummy variable (either Trt1 or Trt2), so the 

code above will fit the model:

yij = beta0 + beta1 Trt2 + uij + eij,

where yij is the jth observed outcome in the ith study, beta0 

then corresponds to the (average) outcome for Trt1, beta1 indicates 
how much higher or lower the (average) outcome is for Trt2 compared 
to Trt1, uij ~ N(0, tau^2), and eij ~ N(0, varLogRatio). This model 
will treat three-arm studies as if they were two (independent) 
two-arm studies. Probably not ideal.

This works fine.  Out of curiosity, I ran a quickie model in lme4

lmer(logRatio ~ Drug.Type + (1+studyID), data=mydata, 

weights=varLogRatio)

and I noticed that the results are quite different, and this appears due
to some variation due to study (after inspecting ranef - note, I included
Drug.Type as a fixed effect as there were only two levels).

1) Did you use (1+studyID) or (1 | studyID)? The latter is 

probably what you meant/want to use.

2) You need to specify the *inverse* of the variances as weights.
3) This model assumes that the sampling variances are known up to 

a proportionality constant, not exactly known. You will therefore 
get what is sometimes called a multiplicative model for 
heterogeneity, with heterogeneity reflected in a residual variance 
estimate larger than 1. This model is different from the additive 
model (which is typically used), where the sampling variances are 
assumed to be known exactly and we *add* an additional random 
effect to reflect heterogeneity.

So, with (1 | studyID) and inverse sampling variance weights, you 

get the model:

yij = beta0 + beta1 Trt2 + ui + eij,

where ui ~ N(0, tau^2), eij ~ N(0, sigma^2 * varLogRatio). Now 

tau^2 reflects study-level variability and sigma^2 reflects 
multiplicative heterogeneity.

So, I went back to metafor and ran

rma(yi = logRatio, vi=varLogRatio, mods=~ Drug.Type+studyID, data=mydata)

which yielded the error

Error in qr.solve(wX, diag(k)) : singular matrix 'a' in solve
In addition: Warning message:
In rma(yi = logRatio, vi = varLogRatio, data = mydata, mods = ~Drug.Type
:
  Cases with NAs omitted from model fitting.

which appears to be due to the unbalanced nature of the dataset (some
studies having T1 and T2, some having multiple measures of T1 and T2).

I would have to see:

with(mydata, model.matrix(~ Drug.Type + studyID))

to figure out what is going on here. The error indicates that you 

have some linear dependency between the columns of the design 
matrix. That should not happen based on what you describe. For 
example, suppose there are 4 studies, the first and fourth a 
three-arm studies, the second only examines Trt1 and the third only Trt2. Then:

study <- factor(c(1,1,2,3,4,4))
trt <- factor(c(1,2,1,2,1,2))
model.matrix(~ trt + study)

  (Intercept) trt2 study2 study3 study4
1           1    0      0      0      0
2           1    1      0      0      0
3           1    0      1      0      0
4           1    1      0      1      0
5           1    0      0      0      1
6           1    1      0      0      1

which is of full rank. That should be true regardless of how many 

studies (of each type) I add.

So, is there a way to properly incorporate studyID in a metafor 

using rma?

Is there an argument I'm missing, or perhaps should be using a different
function?

At the moment, metafor is really only a set up for univariate 

models (that should change in the near? future). The kind of 
multilevel/multivariate structure you are dealing with will require 
(at the moment) other tools.

Note that there is an additional issue with your data: If 

logRatio reflects the difference between Trt1 or Trt2 and Control, 
then in three-arm studies the two logRatio values are dependent 
since the data from the control group/condition is used twice. Note 
that this is statistical dependence over and beyond what is induced 
by potentially correlated true effects within the three-arm 
studies. See chapter 19 in the Handbook of Research Synthesis and 
Meta-Analysis (i.e., the 2nd ed).

As pointed out by Michael, you are essentially in a network 

meta-analysis type of situation. You may want to take a look at the 
following article for more details:

Salanti, G., Higgins, J. P. T., Ades, A. E., & Ioannidis, J. P. 

A. (2008). Evaluation of networks of randomized trials. Statistical 
Methods in Medical Research, 17(3), 279-301.

Thanks!

-Jarrett

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