Given that there is a single effect size estimate from each study (see comments above), the analysis can be carried out with regular meta-regression methods. You can carry out such an analysis with the metafor package. The "trick" is to code variables that indicate what treatments have been compared within a particular study:
library(metafor)
my_data$A1 <- ifelse(treat1 == "A1", 1, 0)
my_data$A2 <- ifelse(treat1 == "A2", 1, 0)
my_data$B1 <- ifelse(treat2 == "B1", -1, 0)
my_data$B2 <- ifelse(treat2 == "B2", -1, 0)
res <- rma(TE, sei=seTE, mods = ~ A1 + A2 + B1 - 1, data=my_data)
res
yields:
Mixed-Effects Model (k = 38; tau^2 estimator: REML)
tau^2 (estimated amount of residual heterogeneity): 0.2898 (SE = 0.1578)
tau (square root of estimated tau^2 value): 0.5384
I^2 (residual heterogeneity / unaccounted variability): 59.02%
H^2 (unaccounted variability / sampling variability): 2.44
Test for Residual Heterogeneity:
QE(df = 35) = 93.5215, p-val < .0001
Test of Moderators (coefficient(s) 1,2,3):
QM(df = 3) = 435.5223, p-val < .0001
Model Results:
estimate se zval pval ci.lb ci.ub
A1 2.2446 0.2837 7.9123 <.0001 1.6886 2.8006 ***
A2 0.9060 0.3387 2.6751 0.0075 0.2422 1.5699 **
B1 -2.2983 0.3467 -6.6294 <.0001 -2.9778 -1.6188 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Since variable B2 has been left out, this becomes the "reference" treatment. So, the coefficient for A1 is the estimated average effect when comparing treatment A1 against B2. The coefficient for A2 is the estimated average effect when comparing treatment A2 against B2. And the coefficient for B1 is the estimated average effect when comparing treatment B1 against B2.
The network that is analyzed here looks like this:
A1 A2
|\ /|
| \ / |
| X |
| / \ |
|/ \|
B1 B2
So, the comparison between B1 and B2 is based purely on indirect evidence.
There are 3 more comparisons that can be obtained here besides the ones above (i.e., A1 vs A2, A1 vs B1, and A2 vs B1). You can obtain those by changing the "reference" treatment.
An assumption made here is that the amount of heterogeneity is the same regardless of the comparison. This may or may not be true.
An article that describes this type of analysis is:
Salanti et al. (2008). Evaluation of networks of randomized trials. Statistical Methods in Medical Research, 17, 279-301.
Edit: To test whether the effect of the first factor (A & B) depends on the second factor (1 & 2), that is, whether (A1 vs B1) = (A2 vs B2) or not, first note that:
(A1-B2) - (A2-B2) - (B1-B2) = (A1-A2) - (B1-B2)
= (A1-B1) - (A2-B2).
So, you just have to test whether b1 - b2 - b3 = 0. You can do this with:
predict(res, newmods=c(1,-1,-1))
or install/load the multcomp package and use:
summary(glht(res1, linfct=rbind(c(1,-1,-1))), test=adjusted("none"))
which yields:
Simultaneous Tests for General Linear Hypotheses
Linear Hypotheses:
Estimate Std. Error z value Pr(>|z|)
1 == 0 3.6369 0.5779 6.293 3.12e-10 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Adjusted p values reported -- none method)
Best Answer
I think, the modeling approaches and estimation techniques should be viewed seperately. From modeling point of view, Lumley model only works for two-arm trials only. So it is not preferable. To my understanding, node-splitting approach, which you listed as Dias et al, is very intuitive. Also, I think you should add the design-by-treatment interaction approach (http://www.ncbi.nlm.nih.gov/pubmed/24777711). From estimation point of view, I dont know much about frequentist techniques, but one can use MCMC for almost all models for NMA. Lastly, there is a different technique (which is not widely known unfortunately) called INLA. You can use INLA from within R and fit NMA models, it is faster and no need to check convergence diagnostics. Here is the paper http://www.ncbi.nlm.nih.gov/pubmed/26360927. So, at the end I would prefer node-splitting and the design-by-treatment interaction approach using INLA.