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.
 A: This would typically refer to something like logistic regression that allows you to model a binary outcome (or ordinal regression for more than 2 ordered categories) adjusted for some covariates.
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).
A: There are two primary methods used to get adjusted odds ratios.  If the variables one wants to adjust for purely categorical one may be able to cross-classify on them to compute stratified odds ratios, and do some sort of averaging on the log odds ratio scale.  Second, and more common because of its flexibility in handling continuous variables and more variables in general, is logistic regression.  Assuming no interactions between the exposure and the adjustment variables, the regression coefficient for exposure is the adjust log odds ratio, and one anti-logs that to estimate the adjusted odds ratio.
