The magnitude of the difference of crude and adjusted log odds ratios

Crude and adjusted odds ratios, calculated from a collapsed 2×2 table or a stratified 2×2×K table, can be very similar or quite different when significant associations are found between each dichotomous variable and the K-level stratifying variable. It is demonstrated here that the magnitude of the difference between the logs of the two estimators can be approximated by 4 times the covariance between log linear interactions describing the associations of each of the binary variables with the stratifying variable. Two data examples illustrate the usefulness of the variability and covariability of the interactions in providing a statistical accounting for the magnitude of the difference between the logs of the crude and adjusted odds ratios. Other interpretations and applications of the variances and covariances of the log linear interactions are discussed.