In the example below we are concerned about the confounding effect of smoking on the relationship between matches and lung cancer, we stratify on the two values of smoking:
After we create these strata, we observe that each of the smoking specific-stratum odds ratios have become one.
Crude OR = 2x2 table - no adjustment for confounder
Stratified OR =A 2x2 table for each of the stratified variables - adjusted for confounder
In the smoking, matches, lung cancer example (below), the measures of association from the different strata were identical; they are both equal to 1.
Summary Adjusted Estimate = Summary of all measurements from the stratified variables
But, in real life it is seldom the case that the estimates from the various strata are identical.