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6.2 Case-control studies

The study base

A case-control study is best thought of as an efficient way of sampling from a cohort that is not enumerated. The study base is the population, and its person-time, from which the cases arose. If it can be defined first (the residents of a district over a period, the members of an insurance scheme), it is a primary base: all cases arising in it are sought, and controls are sampled from it. If the cases are defined first (all patients with the disease admitted to one hospital), the base is secondary: the population of people who would have been admitted to that hospital had they developed the disease. It is real but hard to sample, which is why hospital-based studies are prone to bias.

The guiding principle is that controls must be sampled from the study base independently of their exposure. They should represent the exposure distribution of the person-time that gave rise to the cases. Whether the controls resemble the cases in other respects, or are “healthy”, is beside the point.

Sources of controls

  • Population controls, sampled from a register of the primary base, satisfy the principle most directly. Their participation is often lower than that of cases, which can bias the study if participation depends on exposure.
  • Hospital controls, patients admitted for other diseases, share the referral pattern of the cases and cooperate readily. They are valid only if their diseases are unrelated to the exposure: patients with chronic bronchitis are useless controls in a study of smoking.
  • Neighbourhood or friend controls are chosen by the case or by proximity. They tend to share the case’s exposures (friends smoke together), which biases the odds ratio towards \(1\).

Incident and prevalent cases

Cases should be incident, that is newly diagnosed during the study period. Prevalent cases over-represent those who survive long with the disease, and their exposure may be a determinant of survival rather than of occurrence (Problem 3.6.5). Newly diagnosed cases also recall past exposure before the illness has changed their habits.

How many controls per case

When cases are scarce, precision can be bought by taking several controls per case. The return diminishes quickly.

Proposition 6.2.1. With \(m_1\) cases and \(k m_1\) controls, and exposure probabilities \(q_1\) among cases and \(q_0\) among controls, \[ \Var (\log \widehat {\OR }) \approx \frac {1}{m_1}\Bigl \{\frac {1}{q_1(1 - q_1)} + \frac {1}{k\,q_0(1 - q_0)}\Bigr \} . \] When \(q_1\) and \(q_0\) are close, the efficiency relative to an unlimited number of controls is \(k/(k + 1)\).

Proof. From the proof of Theorem 4.3.1, the cases contribute \(1/\{m_1 q_1(1 - q_1)\}\) and the controls \(1/\{k m_1 q_0(1 - q_0)\}\). As \(k \to \infty \) only the first term remains, and with \(q_1 = q_0\) the ratio of the limit to the variance is \(1/(1 + 1/k) = k/(k + 1)\). □

Controls per case \(k\) 1 2 3 4 5 10
Relative efficiency 0.50 0.67 0.75 0.80 0.83 0.91

Going from one control to four gains as much as doubling the number of cases would from one-to-one; beyond four or five the gain rarely justifies the cost.

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