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I have data on a number of studies comparing response rates to various treatments of a disorder. There are six different treatments, and some studies only test a single treatment. For each treatment in each study, I have the response rate and the sample size (plus some other variables).

Normally, if there were only two treatments, and each study compared the two, I might use something like $$D_i=\beta_0+\beta_1\cdot SS_i+u_i+\epsilon_i$$ where $D_i$ is the difference between the logits of the response rates in study $i$, $\epsilon_i$ is the corresponding sampling error, $SS_i$ is the sample size of study $i$, and $u_i$ is the residual error term.

Is it possible to sensibly analyze my data in a similar way? Are there keywords / references I should be aware of?

For example, I'd like to use something like $${\rm logit}(RR_{ij})=\beta_0+\beta_1\cdot SS_{ij}+\beta_2\cdot T_j+u_i+\epsilon_{ij}$$ where $i$ is the study and $j$ is the treatment (1 to 6), $T_j$ is an indicator variable for the treatment (alternatively, could be turned into a second random effect), and $u_i$ and $\epsilon_{ij}$ are as above.

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In the years since this question was asked theory and practice have advanced and the current answer would be that this is an example of network meta-analysis also known as multi-treatment comparison. It is perfectly possible to do this and also incorporate studies which only have one arm (single treatment). The disadvantage is that it merges direct comparisons (which preserve the benefits of random allocation) with indirect ones. The advantage is that it can provide information about comparisons which have never been tested directly and which may be unlikely to be tested. There is a tag on this site for [network-meta-analysis] which is worth following up.

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