Odds are already the ratio of two probabilities, so the odds ratio involves four probabilities:
So in the above example the p of the event in the exposed is 2/5, 1 - p of the event in the exposed is 3/5.
For probability (# events/ # exposed ) / (# non-events/ # not exposed)
Notice that the odds ratio is larger than the probability ratio (either prevalence or risk ratio) in the above example.
Unless both ratios = 1.0,
We begin with our 2 x 2 table below:
We then do a little algebra to show how these four fractions reduce down to what is called the cross-product of the 2x2 table,
We find that the odds ratio of disease in the exposed and unexposed equals the odds ratio of exposure in the diseased and not diseased.
In the equations below we are calculating the probabilities of exposure and non-exposure in those with disease and similarly for those without disease and then forming an odds ratio.
So the probabilities, with our arrangement of disease across the top, are calculated within the columns rather than within the rows.
In a cross-sectional study, we would normally not be interested in looking at exposure by disease status because we are always interested in how exposure leads to disease, i.e., how disease is distributed among the exposed. We are looking at this way in order to show that the odds ratio for exposure equals the odds ratio for disease.
This is an important property of the odds ratio and its use in case-control studies. It was originally pointed out by Cornfield, a pioneer of modern observational epidemiology, in the 1950's and drew attention to the usefulness of the case-control design.
An important property of the odds ratio is that the:
Language like "X times as likely to" implies a comparison of probabilities, not odds -
Regardless of the actual measure of association that was calculated by the study or what measure it validly estimates,
The language of probability can be justified if the rare disease assumption is met
Language like "7% more likely" suggests an absolute risk difference, not a ratio (also a misuse of risk ratios) -
Language like 7% more likely sounds as if an absolute risk difference is being reported