Nested Case-Control Sampling

Why Nested Case-Control Sampling Is Event-Specific

A prospective cohort may contain many people whose covariates must be measured repeatedly or expensively. A nested case-control design reduces that burden: when a cohort member experiences the event of interest, controls are sampled from the people still at risk at that event time. The resulting data are therefore organized around event-specific risk sets, not around one permanent division of the cohort into “cases” and “controls.”

This distinction matters for both analysis and interpretation. A person sampled as a control at one event time may be sampled again later, and may subsequently become a case. The relevant probability for design-based weighting is the probability that the person is included as a control in a particular risk set. It is generally not the probability that the person appears anywhere in the study sample.

This convention makes the sampling unit a subject-time pair: subject \(i\) as a potential control at event time \(t_j\). Write \(A_j\) for the eligible control pool and \(R_j=|A_j|\) for its size. Let \(S_j\) be the sampled controls from that pool. If a risk-set definition or sampling protocol instead permits the case to be sampled as a control for another event at the same time, that protocol requires its own explicit eligibility rules and inclusion probabilities.

Inclusion Probabilities Within a Risk Set

Suppose \(m_j\) controls are selected by simple random sampling without replacement from the \(R_j\) eligible controls. Each eligible person then has the same probability of inclusion in that event-specific sample.

To verify the probability, there are \(\binom{R_j}{m_j}\) equally likely samples. Of these, \(\binom{R_j-1}{m_j-1}\) contain a specified subject \(i\). Thus

The formula applies to the eligible control pool, not automatically to a risk set that includes the case. For example, if a risk set contains 20 people including the case and 4 controls are sampled from the other 19 people, the control inclusion probability is \(4/19\), not \(4/20\).

Why Inverse-Probability Weights Recover Risk-Set Totals

Let \(g(X_{ij})\) be a fixed quantity for subject \(i\) at time \(t_j\), such as an exposure indicator or a covariate value. The finite-population total of this quantity over the eligible control pool is \(T_j=\sum_{i\in A_j}g(X_{ij})\). If the inclusion probability \(\pi_{ij}\) is known and positive, the Horvitz–Thompson estimator of that total is \(\widehat T_j=\sum_{i\in S_j}g(X_{ij})/\pi_{ij}\).

For a proof, let \(I_{ij}\) equal 1 if subject \(i\) is sampled as a control for event \(j\), and 0 otherwise. Then \(E[I_{ij}]=\pi_{ij}\), and

By linearity of expectation, with the covariate values fixed under the design,

This proof does not require the inclusion indicators to be independent. In simple random sampling without replacement, for example, they are dependent because the sample size is fixed; the expectation argument still holds.

Repeated Selection and Subject-Time Weights

A control’s weight is attached to a particular risk-set contribution. Suppose a person is eligible at two event times, and at each time a separate simple random sample is drawn. If that person’s inclusion probabilities are \(1/4\) and \(1/2\), the corresponding event-specific weights are 4 and 2. They are not replaced by one weight based on the probability of appearing at least once.

For example, if the two samples are independent, the probability of appearing at least once is \(1-(1-1/4)(1-1/2)=5/8\). Its reciprocal, \(8/5\), is not the correct weight for either event-specific total: the two totals concern different risk sets and have inclusion probabilities \(1/4\) and \(1/2\), respectively. A person selected as a control earlier and later observed as a case likewise contributes as a sampled control to the earlier risk set and as the case to the later event.

Using Weights Without Overclaiming

Inverse-probability weights provide a principled way to estimate finite-cohort quantities that are sums over risk-set members. They can also be used in estimating procedures designed for the sampling structure, provided the relevant inclusion probabilities are known and the estimating procedure accounts for how the cases and controls were selected. When sampling probabilities depend on covariates, the weights must reflect that dependence.

A weighted total is design-unbiased under the conditions stated above. A ratio of two estimated totals, a nonlinear function of estimated risk-set totals, or a regression coefficient does not automatically inherit that exact unbiasedness. In particular, plugging estimated weighted totals into a nonlinear likelihood does not by itself prove that the resulting estimator is unbiased or consistent. The model, sampling design, and estimating procedure must be considered together.

Nested case-control analyses often use the matched risk sets directly to estimate relative effects. Under uniform risk-set sampling, the conditional comparison of each case with its sampled controls can preserve information about relative rates. That does not mean that a conditional comparison estimates every cohort quantity, such as an absolute risk or a full-population covariate total. Design weights are useful when the target involves the underlying cohort, but they do not turn every weighted regression into a valid design-based analysis automatically.

Common Mistake

A frequent error is to assign a control a single weight equal to the inverse probability that the person was ever selected in the study. This confuses a subject’s inclusion in the union of all sampled sets with inclusion in a particular event-specific risk set. The correction is to define the target quantity first, identify the risk set to which it refers, and use the inclusion probability for that subject in that risk set. A second error is to use \(m_j\) divided by the risk-set size when the case is excluded from control sampling; the denominator must be the eligible control-pool size \(R_j\), not a count that includes the case.