# 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?

The first thing is, what is your question? Are you trying to say something about how effects differ across doses or just whether the drug does something (either assuming that all the doses are the same in terms of the effect of interest or considering some dose response) or are you just interested in the approved dose(s)?

The other question (massively informed by the first one), how you implement this technically. Depending on the question this will differ.

• If you want to simply lump all doses together, then as long as you can do a standard treatment comparison within the study from what you have available, you may be able to proceed with the usual estimate with standard error approach.
• If you want to treat doses separately, either to estimate separate effects for them, or to e.g. look at some monotonic dose response test, the usual estimate and standard error approach will fail. If you naively calculated an effect for each dose in a study and then treated those multiple estimate from a study like separate studies, you would ignore the within study correlation of the outcomes of each arm. That would be a massive error that is sadly very common. Instead:
• One tends to need to use specialized statistical software (e.g. R or the like) instead of some point-and-click interface in some meta-analysis software (unless it specifically covers this setting) to deal with this situation.
• 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).

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.