Interaction effects in network meta-analysis (multiple treatment comparison) in R?

Question

I would like to use network meta-analysis (or multiple treatment comparison) to investigate the main-effect and interaction of two factors on a continuous outcome variable. As an example the effect of hair-color (dark/light) and gender (male/female) on a blood pressure (continuous). I have mean, variance and sample size from studies that compared dark-males to dark-females (studies 1 & 2), studies that compared dark-males to light-males (study 5), studies that compared dark-females with light-females, studies that compared dark-males with light-males and light-females (study 7). Most importantly I also have studies that compared males to females irrespective of hair-color (studies 3 and 4).

my_data <- data.frame(study = c("Study_1","Study_1", "Study_2", "Study_2", "Study_3", "Study_3", "Study_4", "Study_4", "Study_5", "Study_5", "Study_6", "Study_6", "Study_7", "Study_7", "Study_7"), treatment = c("dark_male", "dark_female","dark_male", "dark_female","male", "female","male", "female", "dark_male", "light_male", "dark_female", "light_female", "dark_male", "light_male", "light_female"), mean=c(10,12,11,14,11,21,14,25,10,12,14,20,12,14,20), std.dev=c(1,1.2,1,1.2,2,2.1,1,1.3,1.3,1,1,1,1.2,1.2,1.3), sampleSize=c(10,10,20,20,15,15,35,35,18,19,20,22,40,41,40))

my_data

I am not sure if I can include studies 3 and 4 since they include light- and dark-haired subjects.

Is it possible to investigate the main effects of hair-color, gender as well as their interaction using one of the available r packages for network meta-analysis (gemtc, netmeta, metafor)? What would be a suitable way to calculate post-hoc tests?

Answer 1

I think you will have to make some kind of assumption about the proportion of dark/light hair-color individuals in studies 3 and 4. Then you can treat the 'dark' color variable as a proportion that is equal to 0 in samples composed entirely of light hair-color individuals, 1 in samples composed entirely of dark hair-color individuals, and something in between when it is a mix (as in studies 3 and 4). For gender, you just set up a standard dummy variable. Then you can analyze these data with the rma.mv function from the metafor package (which allows fitting network-type meta-analytic models). Here is how this would be done, assuming that 50% of the individuals in studies 3 and 4 were dark hair-color individuals:

# dummy variable for gender and proportion of dark hair-color individuals
my_data$male <- c(1,0,1,0,1,0,1,0,1,1,0,0,1,1,0)
my_data$dark <- c(1,1,1,1,.5,.5,.5,.5,1,0,1,0,1,0,0)

# compute sampling variances of the means
my_data$vi <- my_data$std.dev^2 / my_data$sampleSize

# need a variable to indicate the arm within each study 
my_data$arm <- sequence(table(my_data$study))

# meta-analysis model using correlated random effects within each study
res <- rma.mv(mean, vi, mods = ~ male*dark, random = list(~ arm | study), data=my_data)
res

The output from this is:

Multivariate Meta-Analysis Model (k = 15; method: REML)

Variance Components: 

outer factor: study (nlvls = 7)
inner factor: arm   (nlvls = 3)

            estim    sqrt  fixed
tau^2      6.8909  2.6250     no
rho        0.3316             no

Test for Residual Heterogeneity: 
QE(df = 11) = 1353.4541, p-val < .0001

Test of Moderators (coefficient(s) 2,3,4): 
QM(df = 3) = 38.9377, p-val < .0001

Model Results:

           estimate      se     zval    pval     ci.lb    ci.ub     
intrcpt     21.9548  1.5913  13.7965  <.0001   18.8359  25.0738  ***
male        -9.0630  2.0448  -4.4322  <.0001  -13.0708  -5.0552  ***
dark        -6.8660  2.1176  -3.2424  0.0012  -11.0164  -2.7156   **
male:dark    4.9911  2.7210   1.8343  0.0666   -0.3419  10.3241    .

---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Maybe there is an interaction, but strictly speaking, it is not significant at $\alpha = .05$ (two-sided). Actually, let's get the predicted means for the 'female-light', 'female-dark', 'male-light', and 'male-dark' combinations:

predict(res, newmods=rbind(c(0,0,0), c(0,1,0), c(1,0,0), c(1,1,1)))

     pred     se   ci.lb   ci.ub   cr.lb   cr.ub
1 21.9548 1.5913 18.8359 25.0738 15.9383 27.9714
2 15.0888 1.3667 12.4101 17.7676  9.2882 20.8894
3 12.8918 1.5816  9.7919 15.9917  6.8851 18.8985
4 11.0169 1.2026  8.6599 13.3739  5.3577 16.6761

So, for females, going from 'light' to 'dark' results in a pretty substantial decrease, but not so for males.

Again, I made an assumption about the proportion of dark hair-color individuals in studies 3 and 4. My suggestion now would be to carry out a sensitivity analysis, varying those proportions within a sensible range (e.g., .3 to .7), checking whether the conclusions depend on the specific proportions assumed.