4.6 What a case-control odds ratio estimates
Section 1.4 showed that the case-control odds ratio estimates the disease odds ratio, which approximates the risk ratio for a rare disease. How the controls are chosen decides what it estimates, and two ways of choosing them remove the need for the rare disease assumption altogether.
Consider a cohort followed over a period, in which \(A_1\) of \(N_1\) exposed and \(A_0\) of \(N_0\) unexposed people become cases, with \(T_1\), \(T_0\) person-years at risk. All cases are taken; controls are sampled in one of three ways.
- 1.
- Cumulative sampling: controls from those still free of disease at the end of the period. Their exposure odds estimate \((N_1 - A_1)/(N_0 - A_0)\), and the odds ratio estimates \[ \frac {A_1/(N_1 - A_1)}{A_0/(N_0 - A_0)} , \] the risk odds ratio.
- 2.
- Case-cohort (case-base) sampling: controls from the whole cohort as it was at the start, regardless of later disease. Their exposure odds estimate \(N_1/N_0\), and the odds ratio estimates \((A_1/A_0)/(N_1/N_0) = (A_1/N_1)/(A_0/N_0)\), the risk ratio.
- 3.
- Density (risk-set) sampling: whenever a case occurs, controls are drawn from those at risk at that moment. A person is then eligible as a control for as long as they are at risk, so the chance that a control is exposed is proportional to exposed person-time.
Theorem 4.6.1 (Density sampling). Under density sampling, if the exposure odds among controls estimate \(T_1/T_0\), the odds ratio estimates the incidence rate ratio \(\lambda _1/\lambda _0\), with no rare disease assumption.
Proof. The expected numbers of cases are \(A_1 = \lambda _1 T_1\) and \(A_0 = \lambda _0 T_0\), so the exposure odds among cases are \(\lambda _1 T_1/(\lambda _0 T_0)\). Each control is a person drawn from those at risk at a time point, so it is exposed with probability equal to the exposed share of the person-time at risk, and the control exposure odds are \(T_1/T_0\). The ratio is \(\{\lambda _1 T_1/(\lambda _0 T_0)\}/(T_1/T_0) = \lambda _1/\lambda _0\). □
A person sampled as a control under density sampling may later become a case, and should then appear in both roles; a person may also be sampled as a control more than once. Excluding such people reintroduces the rare disease assumption. Nested case-control studies within large cohorts, in which exposure is measured (say, from stored blood) only for the cases and a few matched controls per case, use density sampling and estimate rate ratios directly.
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