# Combined relative risk in meta study

I am looking at a meta study, and I've never done one (or looked at it much) before. From what I've learned about it so far, the way you calculate the compounded effect is by first calculating this $$\tau^{2}$$, which is a measure of the heterogeneity of the results.

If $$\tau^{2}$$ is in fact 0, as is the case here, we have reason to assume all studies are estimating a fixed, underlying parameter of interest, and the point estimate for the parameter of interest becomes the simple weighted average of the individual studies. Correct?

Applying that to the table below, we would have:

weights = [4.3, 3.6, 4.1, 4.3, 3.7]

y = [0.11, 0.24, 0.14, 0.12, 2.92]

and the weighted average (sum(w*y)/sum(w)) is then 0.66, by my calculation.

However, the authors claim it to be 0.24, see table. I'm sure they're correct and I'm wrong, but could someone tell me where I'm going wrong? Thanks a lot!

• It seems odd that studies without events in both groups are not included in the analysis. Aug 7, 2021 at 14:28

For risk ratios, the averaging is done on the log scale. So, if you take the log of y, then compute the weighted average, and then back-transform via exponentiation, you get the expected result (0.24). For example, in R:

w <- c(4.3, 3.6, 4.1, 4.3, 3.7)
y <- log(c(0.11, 0.24, 0.14, 0.12, 2.92))
exp(sum(w*y)/sum(w))


yields 0.2485498 (which is 0.25 when rounded, but this discrepancy is due to rounding of the weights and risk ratios shown in the figure).

You have a couple questions/ assumptions in your question so let's break it down one by one:

1. Model choice (fixed-effect vs. random-effects): The model choice should be done a priori (aka not driven by the statistical analysis) based assumptions of whether or not you are measuring a single effect or a range of effects (parameters). In other words, using Tau/ Chi/ I-squared to drive your model choice is not good practice. You can also run another model to test the robustness of the analyses and whether the results significantly change based on model choice. [Personal note: In the protocol for this particular meta-analysis I would have suggested the authors use the Peto-modified Odds Ratio (Fixed-effect model) for rare outcomes, but that's another discussion all together].

2. As Wolfgang nicely summarized it you need to pool on the log scale.