Incidence-Density Control Sampling

Why the Control’s Time of Selection Matters

A control selected from the population at the end of follow-up represents a person who did not become a case during that period. A control selected by incidence-density sampling, in contrast, represents a person who was at risk when a particular case occurred. That person may later become a case, and may therefore be selected as a control earlier in follow-up. This alignment of case and control time is the key to the estimand: the exposure odds ratio from properly analyzed risk sets targets an incidence rate ratio, not a ratio of cumulative disease odds.

The result does not depend on disease being rare. It follows from comparing the exposure of an event subject with exposure among people eligible to become that subject’s control at the same event time. The comparison must preserve the risk-set matching; simply pooling all cases and all sampled controls can discard that structure.

For the basic derivation, suppose exposure is binary, with values 0 and 1. In a particular risk set, let \(Y_0\) and \(Y_1\) denote the numbers of unexposed and exposed people. Suppose each person in group \(x\) has the same instantaneous event rate \(\lambda_x\) within this risk set. The group incidence rate is events per unit person-time; in a continuous-time formulation, \(\lambda_x\) is the event intensity at that time. The exposure incidence rate ratio is \(\lambda_1/\lambda_0\).

The Event Odds Within a Risk Set

Consider a risk set at a time when exactly one event occurs. For a small time interval of length \(dt\), the probability that an individual in exposure group \(x\) experiences the event is approximately \(\lambda_x dt\). With \(Y_x\) such individuals, the event contribution from that group is proportional to \(Y_x\lambda_x\). Equivalently, under the usual continuous-time competing-risks description, the probability that the next event comes from group \(x\) is proportional to the group’s total event intensity \(Y_x\lambda_x\).

Therefore, conditional on an event occurring in this risk set, the odds that the event subject is exposed rather than unexposed are

If a control were sampled uniformly from the risk set before the case were removed, the exposure odds for that draw would be \(Y_1/Y_0\). Dividing the event-subject odds by these risk-set exposure odds gives \(\lambda_1/\lambda_0\). In actual matched sampling the case is typically excluded from its own controls. The exact argument below accounts for that detail and shows why the matched comparison still identifies the rate ratio.

For completeness, write \(N=Y_0+Y_1\). The probabilities of the two discordant orderings are

The first factor in each expression is the probability of the case’s exposure group. The second is the probability of selecting a control from the other exposure group after removing the case. Their ratio is

This proves the theorem. The result is an odds ratio for the orientation of discordant matched pairs: it compares pairs in which the exposed person became the case with pairs in which the unexposed person became the case. It is not, in general, the odds ratio obtained by treating all sampled cases and controls as two unrelated groups. \(\square\)

Multiple Controls and Conditional Logistic Likelihood

The same logic works when a case has more than one control. Suppose a risk set \(R\) contains \(N\) people, and \(m\) controls are sampled uniformly without replacement from the \(N-1\) people other than the case. Let \(S\) be the observed set consisting of the case and those \(m\) controls. If person \(i\) is the case, the probability of the observed control set, conditional on that case, includes the factor \(1/\binom{N-1}{m}\). For each possible choice of case \(j\) within the same observed set \(S\), this factor is identical: the controls would be the other \(m\) members of \(S\).

Conditioning on the event having occurred in \(S\) and on the sampled set \(S\), the case probability is therefore proportional to that person’s rate. If person \(i\) has covariates \(x_i\) and rate \(\lambda_i=\lambda_0(t)\exp(\beta x_i)\) within the risk set, then

The common baseline rate \(\lambda_0(t)\) cancels. This is the conditional logistic form used for a matched risk set. With binary exposure \(x_i\in\{0,1\}\), the rate ratio is \(\exp(\beta)\). The likelihood conditions on the sampled set, so it does not require estimating the baseline rate from the case-control sample.

The assumptions matter. The risk set must correctly represent people eligible to experience the event at the case’s event time. Control selection must be uniform within that set, or its selection probabilities must be handled in the analysis. Exposure and covariates must be measured at the relevant time. For a single common rate ratio, the model must specify a common exposure effect across the risk sets being combined; if effects differ, a single coefficient need not represent every stratum-specific rate ratio.

A Sampling Detail That Can Mislead

When controls are sampled after the case is removed, their marginal exposure odds across the sampled pairs are not simply the exposure odds in the original risk set. Nor can the case exposure odds divided by those marginal control odds generally be used as the rate-ratio estimator. The dependence created by selecting a control from the same risk set as its case is part of the matched design. The exact rate-ratio result comes from the two discordant pair probabilities, or from the conditional likelihood, not from pretending that cases and controls were independently sampled.

For example, suppose there are \(N_1\) exposed and \(N_0\) unexposed people in one risk set, and let \(q\) be the probability that the case is exposed. The expected numbers left in the eligible pool after removing the case are \(N_1-q\) exposed and \(N_0-1+q\) unexposed. Consequently, for one control drawn uniformly from those \(N-1\) people, the probabilities of exposure and non-exposure are

These are probabilities for one control draw. The numerators are expected numbers remaining in the eligible pool, not expected counts among one control. This distinction is important: an expected number of exposed people remaining can exceed one, whereas the expected exposed count for a single draw is a probability and cannot.

Why the Result Matters

Incidence-density sampling is useful when a full cohort follow-up is expensive or when detailed exposure information is available only for a sample. Its central advantage is a clear estimand: under the stated conditions, the matched exposure odds ratio identifies the incidence rate ratio even when events are not rare over the entire follow-up period. A control can later become a case because control status means only “at risk at this event time,” not “never a case.”

The common mistake is to treat the resulting sample as an ordinary unmatched case-control table and interpret its pooled exposure odds ratio automatically as the target. The correction is to retain the risk-set matching in analysis and use the conditional comparison that corresponds to the sampling mechanism. Also keep the outcome measure straight: the result concerns an incidence rate ratio, not a cumulative risk ratio or a cumulative disease odds ratio.