# Is there a frequentist approach to Lu & Ades network meta-analysis model?

I'm currently reading the seminal Lu & Ades network meta-analysis paper. It proposes a bayesian hierachical model for dichotomous outcomes which treats each study's arm as an (conditionally) indepenent binomial sample $r_{ik} \sim Bin(p_{ik}, n_{ik})$, $i$ being of the index for the study ($i = 1, \ldots, N$) and $k$ the index for the treatment ($k = 1, \ldots, K$). Hyperparameters are a (nuisance) random scalar "location" parameter $\mu_i$ and a random log-odds-ratios ($K - 1$)-vector $\mathbf{\delta_i}$, so that

$$logit(p_{i1}) = \mu_i - \sum_{k = 2}^K \delta_{i1} / K \\ logit(p_{ik}) = \mu_i + \delta_{ik} - \sum_{k = 2}^K \delta_{ik} / K\\ \mu_i = \sum_{k = 1}^K \delta_{ik} / K \\ \delta_{ik} = logit(p_{ik}) - logit(p_{i1})$$ So the first treatment is the baseline for comparisons. The $\delta$ vector is modeled a Nomal $N_{K -1} (\mathbf{d, \Sigma})$. Putting things together and defining the appropriate matrix and notation, the likelihood easily turns out to be: $$p(r | \theta) = \prod_i \prod_k \frac{\exp(r_{ik}X_k^T\theta_i)}{(1+\exp(X_k^T\theta_i))^{n_{ik}}}$$ This model resembles a multivariate version of the random effect binomial-normal model, which I usually use to meta-analyze log-odds-ratios with metafor. Since I'm not very familiar with priors and simulations, I wonder if there is a relatively straightforward [R-|metafor-] way to estimate this model with a multivariate GLMM, using a likelihood-based approach. Thank you for reading and for any advice in advance.

• I am not sure whether you want a frequentist model that exactly resembles the Lu and Ades model. Possibly you can find some additional information in this similar post: stats.stackexchange.com/questions/202090/… – Joe_74 Oct 26 '16 at 12:54
• Thank you for the comment, Giuseppe. I already read that thread and also bookmarked it for future reference (and let me say thank you, I found it really useful, since reading it gave me a coordinate system in this complex field). The answer to your question is sometimes in the middle. I don't need exactly the same parametrization of Lu and Ades, but I have a similar problem to solve. So not "exactly", but neither as general as that comparison between different paradigms. In the end I hope to I'll have time to explore some different options. – jabbba Oct 26 '16 at 14:19
• @jabba That sounds great. If you can come up thanks to CV to a novel approach then it would prove an important adjunct for evidence synthesis, possibly also suitable for a scholarly publication – Joe_74 Oct 27 '16 at 8:06
• The netmeta package to which @GiuseppeBiondi-Zoccai refers in that post does seem to be very fully featured and is being updated fairly regularly. – mdewey Oct 27 '16 at 14:32