[next] [prev] [prev-tail] [tail] [up]

7.2 Selection bias

Selection bias arises when the probability of being included in the study, or in the analysis, depends on both exposure and disease. A simple model shows exactly when it biases the odds ratio.

Theorem 7.2.1. Suppose that people in the source population with disease status \(i\) and exposure status \(j\) enter the study with probability \(s_{ij}\) (\(i = 1\) for cases, \(0\) for non-cases; \(j = 1\) for exposed, \(0\) for unexposed). Then the odds ratio in the study population is \[ \OR _{\text {obs}} = \OR \times \frac {s_{11}\,s_{00}}{s_{10}\,s_{01}} , \] where \(\OR \) is the odds ratio in the source population.

Proof. If \(N_{ij}\) is the number of people in cell \((i, j)\) of the source population, the expected number in the study is \(s_{ij}N_{ij}\). The odds ratio of the expected study table is \[ \frac {(s_{11}N_{11})(s_{00}N_{00})}{(s_{10}N_{10})(s_{01}N_{01})} = \frac {N_{11}N_{00}}{N_{10}N_{01}} \cdot \frac {s_{11}s_{00}}{s_{10}s_{01}} . \] □

The selection bias factor \(s_{11}s_{00}/(s_{10}s_{01})\) equals \(1\) whenever selection depends on disease only (\(s_{1j} = s_1\), \(s_{0j} = s_0\)), which is exactly what case-control sampling does, and whenever it depends on exposure only, which is what cohort sampling does. Bias arises only when the selection probabilities depend on exposure and disease jointly, in the sense that the ratio of the case’s selection probabilities for exposed and unexposed differs from the non-case’s.

Example 7.2.2. In a case-control study \(90\%\) of eligible cases agree to take part whatever their exposure. Among population controls, people with the exposure (say, heavy drinkers) are less willing: \(50\%\) of exposed and \(70\%\) of unexposed controls take part. The bias factor is \((0.9 \times 0.7)/(0.9 \times 0.5) = 1.4\), so a true odds ratio of \(1\) would appear as \(1.4\). Response rates of \(90\%\) and \(60\%\) are not by themselves a problem; their dependence on exposure among the controls is.

Common forms of selection bias

  • Non-response and loss to follow-up related to both exposure and outcome.
  • Hospital controls, when the conditions for which they were admitted are associated with the exposure (Problem 7.7.1).
  • Berkson’s bias: when admission to hospital depends on having either of two conditions, the conditions become associated among hospital patients even if they are independent in the population (Problem 7.7.2).
  • The healthy worker effect: employment selects healthier people, so comparing workers with the general population is biased against finding harm.
  • Survival bias: prevalent cases are survivors (Section 3.4).

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.