# Reporting the difference of two probabilities: as “times” OR as odds-ratios

I work with probabilities in my article and I need to report the difference between two probabilities.

Group A has a happening probability of 0.7

Group B has a happening probability of 0.35

Solution 1

0.7/0.35


2

REPORTING: Group A has two times higher probability for ...

Solution 2

OR between two groups

(0.7/(1-0.7))/(0.35/(1-0.35))


4.333333

REPORTING: Group A has 4.3 times higher odds for ...

Question

Are both of these approaches correct?

Yes. First metric is called risk ratio (RR), while the second is called odds ratio (OR).

First one is considered to be more intuitive so it should probably be preferred. Second one is (very) often used when RR can't be estimated (case-control studies or logistic regression models).

• Am I correct that some logistic regression alternatives - bayesian Bernoulli models for example - also give output in probabilities. Thus, there I can also report RR-s? – st4co4 Oct 3 at 14:21
• point is, with logistic models you assume constant odds ratios (proportional odds), assuming constant RRs would be tricky because it can result in predicted probabilities outside [0, 1]. This, however, doesn't mean that it is impossible to train such a model (in both bayesian or frequentist setting) or that it has never been tried. It is never used in common practice though, and it is never reccomended. – carlo Oct 3 at 15:19

Ideally, I would like to see reported the Odds-Ratio (OR) and the cell counts underlying the formation of the constituting probabilities.

Also, displaying the natural log transform of the OR would be informative as to quote a source:

The distribution of the log odds ratio is approximately normal...

and the source further notes, that a standard error estimate could be computed based on an asymptotic approximation (namely, the square root of the sum of the respective reciprocals of the constituting cell counts).

This would allow the construction of Normal distribution based confidence intervals (CIs) for the log(OR) (and also, median-centered CIs upon reversing the log transform), which would provide important information on precision (or lack there-of).

Further, if there are likely explanatory variables to account for variation across Log(OR), a generalized least-squares (as the individual logit points, per the standard error discussion above, have a known approximate relative error structure) model could be facilely explored to advance the information content.