A: Some logistic regression routines -- e.g., Stata's logistic command or SAS PROC LOGISTIC -- give the option of reporting odds ratios OR instead of logistic regression coefficients b. In any case, it's easy to convert one to the other, since OR = exp (b). (A shortcut: If b is not too far from zero, then OR is close to 1+b.)
Often, however, the odds ratios are almost as hard to interpret as the regression coefficients. For example, suppose that a one-unit change in X increases the probability of Y from 50% to 66%. This means that the odds of Y increase from 1:1 to 2:1 -- i.e., the odds double; the odds ratio is 2.
But the probability of Y hasn't come close to doubling, so it's wrong to give the typical interpretation and say that "a one-unit increase doubles the chances of Y." If the word "chances" is replaced with "odds", the interpretation is technically correct but still misleading in the sense that it doesn't square with intuition. Most people have very poor intuitions about odds; I certainly do.
Rather than odds ratios, it may be clearer to interpret effects in terms of probabilities. For example, "holding other variables at their average values, increasing X from 0 to 1 results in increasing the probability of Y from 50% to 66%." Most logistic regression software will give you predicted probabilites for each case in your data set; often you can get predicted probabilities for hypothetical cases as well.
If the probability of Y is generally low -- say less than 10% -- then the odds ratio does come close to describing changes in the probability of Y. But if the probability is higher, then the odds ratio has the problems described above.
If p is the probability, then the odds are o = p / (1-p). Conversely, p = o / (o+1).
The logistic regression model is linear in the log odds
ln (o) = b0 + b1 X1 + b2 X2 + ...
So to get the probability of Y for given values of X, calculate ln(o) from the logistic regression equation, then use
o = exp (ln (o))to get p. (A spreadsheet can be helpful.)
and
p = o / (o+1)
Long, J.S. (1997). Regression Models for Categorical and Limited Dependent Variables. (Sage), especially the discussion around Table 3.10.