I am doing a meta-analysis of studies which reported their estimates sometimes as OR and sometimes as RR. I wanted to convert the OR to RR, so that I can pool them together. I know that I can use this formulas for unadjusted odds

$$\text{Relative Risk} = {\text{Odds Ratio} \over (1–p_0)+(p_0 *\text{Odds Ratio}) }$$

in which $p_0$ is the incidence of the outcome of interest in the non-exposed group.

Can the same formula can be used to convert adjusted ORs to their RR equivalent?


You can do this calculation for an adjusted OR (I presume from a logistic regression) to a RR, but the end result may not be useful for your goal of meta-analysis. The essential problem is that the adjusted OR $exp(\beta_1)$ from a logistic regression is not an "average" over the population. And so there's no way to calculate a population average relative risk from a logistic regression OR. Simply using the population baseline risk to convert $exp(\beta_1)$ to an RR will be incorrect.

Instead, you only can calculate relative risks for fixed sets of covariates. Say you have: $$g(Y) = \beta_0 + \beta_1 Treatment + \beta_2 Age + \beta_3 Gender$$ Then $exp(\beta_1)$ represents the multiplicative change in odds given fixed values for $Age$ and $Gender$. You essentially have different $p_0$ for different sets of covariates, so you end up with different relative risks for say, a (40, Female) vs a (30, Male).

Thus unless you're concerned with comparing a very specific set of fixed covariates, this likely isn't useful for meta-analysis. Separating the analysis into those that report RR and those that report OR is probably the best bet, as suggested here.


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