Cases and controls are selected without regard to exposure. For
example, it is possible to randomly sample cases from all those
in the population who develop disease during the study period and
randomly select controls from all those without disease. Sampling
is “blind” to exposure, so that the proportion
of cases in the study that was exposed should, on average, equal
the proportion of cases in the full cohort. Similarly, the proportion
of controls in the study that was exposed should, on average, equal
the proportion of those without disease in the full cohort. Thus,
the exposure odds among cases, a/b, is
an estimate of A/B, the corresponding
odds among new cases arising from the full cohort. In a parallel
manner, the exposure odds among controls, c/d, is
an estimate of P/Q, the corresponding
odds among persons without disease in the full cohort. The cross-product
or odds ratio (OR)
is the odds that a case is exposed, a/b, divided
by the odds that a control is exposed, c/d: