Odds Ratio (Cross-Product)

The odds ratio divides the odds of exposure among cases by the odds of exposure among controls, and the algebra collapses that to the cross-product of the 2×2 table: $ad/bc$. It exists because of a design constraint. A case-control study recruits a fixed number of cases and a fixed number of controls, so its denominators are set by the investigator rather than by nature, and no risk can be computed from it. The odds ratio can, and it comes out the same whether the table is read by rows or by columns — which is exactly the property the design requires.

The attribution here needs care. The cross-product arithmetic is older than any modern paper and is essentially definitional: it falls straight out of writing down two odds and dividing. What Jerome Cornfield earned with his 1951 paper is the result that actually made case-control studies interpretable — the proof that the odds ratio approximates the relative risk when the disease is rare. That rare-disease assumption is the whole content of the contribution, and citing Cornfield for the cross-product rather than for the approximation gets both the history and the caveat backwards.

The caveat matters because the approximation fails in the direction that misleads. As the outcome becomes common the odds ratio moves further from the relative risk, always further from 1. With a control-group risk of 10 % an odds ratio of 3 corresponds to a relative risk near 2.3; at 30 % it corresponds to about 1.9. Reading an odds ratio from a common outcome as though it were a risk ratio overstates the effect, sometimes substantially, and this is a routine error in reporting cross-sectional studies where the outcome is anything but rare.

Two footnotes. The odds ratio survives outside case-control work because logistic regression estimates it natively and because it is symmetric — the odds ratio for having the outcome is exactly the reciprocal of the odds ratio for avoiding it, which no risk ratio manages. And a zero in any cell sends the cross-product to zero or infinity; the usual repair is a continuity correction, adding 0.5 to every cell, which is a reporting decision that belongs to the analyst rather than to the arithmetic.