# 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

YES, there is a metafor solution for such a problem.

However, since you are interested in network meta-analysis techniques, I want to point out some other references, since NMA is very active topic nowadays. Firstly Lu and Ades (2004) is outdated, Lu and Ades (2006) is their more recent work (which is also a Bayesian hierarchical model). Then, Higgins et al (2012) have shown that estimates of Lu and Ades (2006) model does depend on treatment ordering, so they propose an improved version of that model (by the way Lu and Ades are themselves coauthors of Higgins et al (2012) paper).

This new model is one of the recent models on network meta-analysis which is called design-by-treatment interaction model. It is implemented in mvmeta a STATA macro (a frequentist approach), or you can use BUGS-variants programs (a Bayesian approach). By the way, there is a distinction between contrast-based and arm-based meta-analysis models. Very basically, a contrast-based uses contrasts of arm-specific parameters (for example log-odds ratio), but an arm-based models directly the arm-specific parameters (for example directly uses the binomial structure of data). And mvmeta fits a contrast-based NMA model and also an arm-based NMA model but using data augmentation.

Lastly, very recently published paper Law et al (2016) introduces a likelihood approach of a contrast based design-by-treatment interaction model and it is implemented in metafor R package (actually metafor developer is one of the coauthor of Law et al (2016)). Furthermore, there is online supplementary material of that paper, and more specifically "Additional file 5 (Fitting Jackson’s model using metafor)" includes R code to fit that model using metafor.

• Thank you very much for your enlightening answer, Burak. If I undestood well, the more traditional Lu-Ades approach is arm-based one. I am currently trying to use the framework Dias et, which is arm-based, too (as far as what i understand). At this moment, arm-based sounds more natural to me. I will certainly give a look to new approach. Thank you. – jabbba Nov 18 '16 at 9:47