Three Populations, Three Different Questions
A case-control data set is not usually a miniature version of the population from which the cases and controls came. Cases and controls may have been sampled at different rates, and eligibility rules may already have restricted the population under study. Interpretation therefore begins by naming the population a quantity describes.
The source population is the population to which the study's scientific question refers. The eligible population consists of source-population units that meet the study's inclusion criteria. The sampled population is the distribution of the eligible units that were actually selected. These populations can differ in both their composition and their outcome prevalence.
This definition allows selection to depend on outcome, covariates, or both. In a standard case-control design, \(Y\) is binary and cases and controls are sampled at different rates. If selection is random within outcome groups, then \(\pi(y,x)=\pi_y\): the inclusion probability depends on case-control status but not on \(X\) within that status. More complicated designs can have covariate-dependent inclusion probabilities.
How Inclusion Changes the Distribution
Let \(f_E(y,x)\) denote the joint probability mass function or density of \((Y,X)\) in the eligible population. Suppose the sampling mechanism is described by \(\pi(y,x)\), and define the overall eligible-population inclusion probability \(\alpha=P_E(R=1)\). Then the distribution among sampled units is proportional to the eligible distribution multiplied by the inclusion probability.
The formula is a statement about what distribution the sample represents, not a method for fitting a model. It explains why the observed case fraction generally is not the eligible-population case fraction: cases and controls may have different inclusion probabilities. It also shows why the covariate distribution can change when inclusion depends on \(X\).
\(f_E(y,x)\) describes the outcomes and covariates among all eligible units, whether sampled or not.
The joint probability of being eligible with values \((y,x)\) and being sampled is proportional to \(\pi(y,x)f_E(y,x)\).
Divide by the overall inclusion probability \(\alpha\) so that the sampled distribution has total probability one.
Worked Example: Eligibility Changes the Target Population
A source population contains 10,000 people: 1,000 have the outcome and 9,000 do not. A study requires participants to have at least one year of follow-up. Of the 1,000 people with the outcome, 800 meet this criterion; of the 9,000 without the outcome, 7,200 do.
The source-population outcome prevalence is \(1{,}000/10{,}000=0.10\), or 10%.
The eligible population contains \(800+7{,}200=8{,}000\) people. Its outcome prevalence is \(800/8{,}000=0.10\), also 10%. In this example eligibility leaves the prevalence unchanged, because the same proportion of each outcome group is eligible.
If instead only 4,500 of the 9,000 people without the outcome met the follow-up criterion, the eligible population would contain \(800+4{,}500=5{,}300\) people, and its outcome prevalence would be \(800/5{,}300\approx0.151\). The source prevalence would still be 10%. Thus the eligible population is not automatically representative of the source population; that depends on how eligibility relates to outcome and covariates.
What Outcome-Only Sampling Preserves
A useful special case occurs when inclusion depends on \(Y\) but not on \(X\) after conditioning on \(Y\) and eligibility. For each outcome value with positive inclusion probability, the distribution of covariates within that outcome group is then unchanged by sampling. This fact concerns the eligible population; it does not by itself say that the eligible population represents the broader source population.
Proof. By the definition of conditional probability, the left-hand side is the probability of \(X\in B\) among eligible units with \(Y=y\) that are sampled. The joint probability of those events is \(\pi_yP_E(X\in B,Y=y)\). The probability of sampling an eligible unit with \(Y=y\) is \(\pi_yP_E(Y=y)\). Therefore,
The cancellation requires positive inclusion probability for that outcome group and a sampling probability that does not vary with \(X\) within the group. The result allows sampled cases and sampled controls to describe the eligible covariate distributions among cases and controls, respectively. It does not make the sample case fraction a valid estimate of eligible-population prevalence.
Worked Example: Different Case and Control Sampling Rates
Among 1,000 eligible people, 200 are cases and 800 are controls. There are 120 exposed cases and 160 exposed controls. Cases are sampled with probability \(0.5\), while controls are sampled with probability \(0.1\), independently of exposure within outcome groups.
The eligible exposure proportions are \(120/200=0.60\) among cases and \(160/800=0.20\) among controls. The expected sampled counts are 100 cases and 80 controls. Among sampled cases, the expected exposed and unexposed counts are \(120(0.5)=60\) and \(80(0.5)=40\), so the exposure proportion is \(60/100=0.60\). Among sampled controls, these counts are \(160(0.1)=16\) and \(640(0.1)=64\), giving \(16/80=0.20\).
The sample case fraction is \(100/(100+80)=5/9\approx0.556\), whereas the eligible case fraction is \(200/1{,}000=0.20\). The covariate distributions within each outcome group agree, as the theorem predicts; the overall outcome fraction does not.
Recovery, Positivity, and Nonidentification
The distribution formula also gives a precise condition for recovering the eligible distribution from a sampled distribution. If inclusion probabilities are known and positive throughout the eligible support, then the sampling distortion can be undone at the level of distributions. If probabilities are unknown, the sampled distribution alone generally cannot determine the eligible outcome prevalence.
Derivation. From the sampling identity, \(f_R(y,x)/\pi(y,x)=f_E(y,x)/\alpha\). Integrating both sides over the full support gives \(\int f_R(u,v)/\pi(u,v)\,d(u,v)=1/\alpha\), because \(f_E\) integrates to one. Dividing \(f_R(y,x)/\pi(y,x)\) by this integral yields \(f_E(y,x)\). Positivity matters: if some eligible units have zero chance of inclusion, their distribution cannot be learned from the sample by this inversion. \(\square\)
Worked Example: Correcting a Covariate-Dependent Sample
Suppose an eligible population has four outcome-covariate cells. The first two rows below are cases; the last two are controls. Inclusion probabilities vary by both outcome and exposure.
| Outcome \(Y\) | Exposure \(X\) | Eligible count | Inclusion probability | Sampled count |
|---|---|---|---|---|
| 1 | 1 | 40 | 0.50 | 20 |
| 1 | 0 | 60 | 0.25 | 15 |
| 0 | 1 | 100 | 0.10 | 10 |
| 0 | 0 | 300 | 0.20 | 60 |
Among eligible cases, the exposed proportion is \(40/(40+60)=0.40\). Among sampled cases, it is \(20/(20+15)=4/7\approx0.571\). The difference occurs because exposed cases have inclusion probability 0.50, compared with 0.25 for unexposed cases.
When the probabilities are known, divide each sampled cell count by its inclusion probability: \(20/0.50=40\), \(15/0.25=60\), \(10/0.10=100\), and \(60/0.20=300\). These recover the eligible cell counts in this illustration, totaling 500. In general, the inversion identifies the eligible distribution, not necessarily the source-population distribution. To reach the source population, the relationship between eligibility and source membership must also be known or otherwise justified.
Unknown inclusion probabilities create a genuine identification limit, not merely a computational inconvenience. For example, suppose the sampled case fraction is \(q=0.5\). If the eligible prevalence is \(p=0.2\), inclusion probabilities \(\pi_1=0.8\) for cases and \(\pi_0=0.2\) for controls produce \(\pi_1p/\{\pi_1p+\pi_0(1-p)\}=0.5\). If instead \(p=0.8\), probabilities \(\pi_1=0.2\) and \(\pi_0=0.8\) produce the same sampled case fraction. Thus a sampled fraction of one-half is compatible with distinct eligible prevalences. Without information about the sampling mechanism, it cannot identify which prevalence is correct.
Interpreting What the Design Can Reveal
The population parameter a design can reveal depends on the target and on what is known about both selection stages. The following distinctions are central:
| Quantity of interest | What the sampled data reveal | What additional information may be needed |
|---|---|---|
| Covariate distribution within outcome groups in the eligible population | Directly represented under outcome-only sampling | Selection must not vary by covariates within outcome groups |
| Eligible outcome prevalence | Not generally represented by the sampled case fraction | Known inclusion probabilities or known sampling fractions |
| Eligible joint distribution of outcome and covariates | Distorted according to the inclusion probabilities | Known positive inclusion probabilities across the eligible support |
| Source-population distribution | Not automatically identified by an eligible sample | Information about how eligibility relates to source-population membership |
A common mistake is to read the proportion of cases in a case-control sample as the disease prevalence in the population. That reasoning silently treats sampled cases and controls as though they entered at the same rate. The correction is to state the target population first, then check whether the inclusion probabilities justify using the sample proportion for that target. Another mistake is to assume that outcome-only sampling leaves every feature of the data unchanged. It preserves covariate distributions within outcome groups under the stated condition, but changes the relative frequency of those groups.
Check Your Understanding
For each question, specify the population being discussed and the assumption, if any, that supports the conclusion.
- In a case-control study, cases are sampled at twice the rate of controls. Can the sampled case fraction estimate eligible-population prevalence? Explain.
- State the condition on \(\pi(y,x)\) under which the covariate distribution among sampled cases matches that among eligible cases.
- Why does knowing positive inclusion probabilities allow recovery of the eligible joint distribution, while a zero inclusion probability prevents that recovery?
- A study excludes people without a year of follow-up. What additional question must be addressed before its results are interpreted as describing the source population?
- Can two different eligible prevalences produce the same sampled case fraction when the case and control inclusion probabilities are unknown? Explain briefly.