How to calculate the Adjusted Odds Ratio?

I saw various articles reporting the "adjusted odds ratio" however nowhere it was indicated how exactly it was calculated. I understand the motivation for computing an adjusted odds ratio, however I couldn't find a reference for algorithms to compute that quantity. Most articles I read about this topic just explain the motivation or state that it can be calculated via SPSS for example. I am however interested in the mathematical procedure that is used for the computation.

• Hard to say, but it might be logistic regression. May 10 '21 at 11:48

Example: A randomized clinical trial of a drug versus a matching placebo might analyze whether patients admitted to an intensive care unit die within 30 days of randomization to either treatment. We might have the APACHE IV score before randomization available and might use it (or the logit transformed probability it predicts) as a covariate. We then have a regression model that looks like $$\text{logit}(\pi_i) = \beta_0 + \beta_1 \times \text{treatment}_i + \beta_2 \times \text{APACHE-IV-score}_i$$ and $$Y_i \sim \text{Bernoulli}(\pi_i)$$ for patient $$i$$. The (maximum-likelihood) estimate $$\hat{\beta}_1$$ of the regression coefficient for treatment would then be an adjusted estimate of the log-odds ratio for the effect of treatment (and correspondingly $$\text{logit}^{-1}(\hat{\beta}_1)$$ the adjusted odds-ratio estimate).
• In that case, what is the purpose of reporting adjusted odds ratios? Odds ratios indicate a correlation between a variable and the outcome within the population, but adjusted odds ratios don't seem to be much different. What is the purpose of "adjusting" when the resulting estimates ($\beta_i$) are unreliable in their magnitude? Suppose there is a strong correlation between "treatment" and "APACHE-IV score" (or whatever other predictors there might be) then the resulting coefficients $\beta_i$ won't capture that; the same for multicollinearity. So what's the purpose of reporting these numbers? May 10 '21 at 16:50