# How to combine multiple dose levels in a study for meta-analysis

I am conducting a meta-analysis of RCTs for treatment A vs. control. Some of the studies have multiple dose levels of A. Should I combine these groups with different doses of A and compute one effect size for each study? Or may I choose one dose, for example, the highest dose of A used in the study?

Apart from the statistical considerations, you have to also consider the clinical implications of the decision to combine all doses vs. a singular comparator vs. adding each dose against the comparator in the meta-analysis. Let's assume that you are including all FDA approved doses (or any dose used by the trial authors) then there is an underlying assumption that the lowest dose will have a similar (even if not exact) effect as the highest dose (and vice versa). That is often a fair assumption used in meta-analyses. In this case, you would want to combine all the intervention arms into a single 'treatment' arm versus the comparator. This is pretty simple for binary outcomes and a little more tricky for continuous outcomes.

If, on the other hand, you believe there is heterogeneity in true benefit or harm from the different doses then you may rather compare each dose against the comparator. What you will want to make sure is that you split the n and N of the comparator equally (as much as possible) across all the doses so that you are not counting the same individuals multiple times in your analysis.

One thing to note (that I've observed) is that when you combine the intervention arms first, you often get a slightly higher weight in the meta-analysis for that trial as compared to the cumulative weight after splitting the arms. That shouldn't be a reason to split or not, but just an observation I have made.

If you want a worked out example, please let me know and I'll try to post one.

The best approach in most cases is to conduct an analysis where you lump together all dosages, and another in which different dosages are analyzed separately (the latter at least for sensitivity purposes).

In addition, you can explore the effect of dosage using meta-regression (see for instance https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.1187), specific methods for dose-response analysis (see for instance https://academic.oup.com/aje/article-abstract/135/11/1301/331121), or even consider applying network meta-analysis methods (see for instance these two pertinent books: https://books.google.it/books/about/Network_Meta_Analysis_for_Decision_Makin.html, https://books.google.it/books/about/Network_Meta_Analysis.html).

Apart from statistical considerations, it is important from a pathophysiologic perspective to recognize whether a threshold effect for drug dose is plausible (an uncommon scenario). If this is not the case, any claim for differential effects of different doses of the same drug should be considered hypothesis generating only.

This sounds like a case where network meta-analysis (NMA), also known as multiple treatment comparison, would be helpful. Although NMA was originally used for cases where studies compared different treatments there is no reason to restrict it and in this case you could assume that each dose and control were separate treatments. This might be especially useful if different studies had used different sub-sets of the possible doses as then you get indirect comparison evidence even for dose levels which have never appeared together in any primary study. The only disadvantage I can see at the moment is that it would treat the different doses as categorical rather than as values from a continuous variable and so you could not fit linear or non-linear effects of dose. That may not matter too much as you would be able to see where doses level off in their effect.

• One possible approach is to model the outcomes for each treatment arm. E.g. for 3 doses and something that is normally distributed (e.g. a continuous outcome, or a log-rate, or log-odds given sufficiently sample size), you could assume a $$Y_{ij} \sim N(\mu_i + \beta_1 * x_{ij1}+ \beta_2 * x_{ij2}+ \beta_3 * x_{ij3}, \text{standard error})$$ sampling distribution for arm $$j$$ in study $$i$$ , where $$\mu_i$$ is the study main effect, $$x_{ijk}$$ is the indicator of arm $$j$$ in trial $$i$$ being on dose $$k$$ and where the $$\beta_k$$ are the effects of each dose over the control. As needed you can turn $$\mu_i$$ into a fixed effect or a random trial effect and similarly, you could turn the $$\beta_k$$ into $$\beta_{ik}$$s (random treatment effects for each study).
• As needed one can introduce constaints in the above e.g. $$\beta_3\geq \beta_2\geq \beta_1\geq 0$$. Or you could specify some parametric dose response function $$Y_{ij} \sim N(\mu_i + f(\text{dose}_{ij}), \text{standard error})$$, where e.g. $$f(\text{dose}_{ij}) = \text{Emax} \frac{\text{dose}_{ij}^h}{\text{dose}_{ij}^h + \text{ED}_{50}^h}$$ (sigmoid Emax dose response) or with h=1 (Emax dose response).