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?
Best Answer
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.