Case–Cohort Study Design

Case–cohort study design (also known as the Prentice case–cohort design) is a two-phase epidemiological sampling architecture and survival-analysis methodology in public health and global health introduced by Ross L. Prentice in 1986 (Biometrika) and extended by Steven G. Self and Ross L. Prentice (1988, Annals of Statistics), Sholom Wacholder et al. (1989), and William E. Barlow (1994; Barlow, Ichikawa, Rosner, and Black, 1999). Developed for large prospective cohort studies of N = 10,000 to 500,000 participants in which baseline biological specimens (frozen serum, plasma, DNA, or tissue biopsies) or high-dimensional records are archived for all participants, the case–cohort design avoids the prohibitive cost of assaying expensive exposures (such as whole-genome sequencing, proteomics, metabolomics, or environmental toxin panels) on the full cohort when the cumulative disease incidence over follow-up is small (typically 1% to 5%). Instead of matching controls to individual cases at each failure time as in a nested case–control study, the investigator selects a single random (or stratified random) subcohort 𝒮̃ of size ñ = αN at baseline—independently of future disease status—and assays the expensive exposure covariates Z_i(t) only on members of the subcohort 𝒮̃ plus all incident cases 𝒟 arising anywhere in the full cohort N during follow-up.

The Rare-Disease Resource Bottleneck in Full-Cohort Cox Regression

In a classical prospective cohort study of N disease-free individuals followed over time t ∈ [0, τ], let T_i⁰ denote the true time to disease onset and C_i denote the right-censoring time for individual i ∈ {1, …, N}, yielding the observed follow-up time T_i = min(T_i⁰, C_i), event indicator Δ_i = I(T_i⁰ ≤ C_i), and counting-process functions N_i(t) = I(T_i ≤ t, Δ_i = 1) and at-risk indicator Y_i(t) = I(T_i ≥ t). Under David R. Cox's (1972) proportional hazards model, the conditional hazard rate λ_i(t | Z_i(t)) for individual i with p-dimensional covariate vector Z_i(t) is:

Maximizing Cox's full-cohort partial likelihood:

Notice that in the numerator of L_Cox(β), only the D = ∑_i=1^N Δ_i incident cases contribute directly at their failure times T_i, whereas the N − D non-cases enter solely through the risk-set denominators ∑_j=1^N Y_j(T_i) exp(β^⊤ Z_j(T_i)). In a cohort of N = 100,000 individuals where D = 500 develop a specific cancer over 10 years, the precision (Fisher information) of β̂ is limited by the number of events D = 500 rather than the number of non-cases (N − D = 99,500). Assaying a 500 USD genomic or proteomic panel on all 100,000 archived baseline blood samples would cost 50,000,000 USD, whereas assaying all D = 500 cases plus a random baseline subcohort of ñ = 1,500 subjects (totaling fewer than 2,000 assays) costs under 1,000,000 USD while retaining over 75% of the asymptotic statistical efficiency of the full 100,000-person cohort!

Formal Construction of the Case–Cohort Sample

The case–cohort sample 𝒞 = 𝒮̃ ∪ 𝒟 is formed in two stages:

1. Phase I (Full Cohort Assembly and Baseline Archiving): Enrol the full cohort ℱ = {1, …, N} at baseline t = 0, record inexpensive Phase I covariates W_i (age, sex, study center, smoking status), collect and cryopreserve baseline biological specimens, and follow all N subjects prospectively to ascertain event times T_i and failure indicators Δ_i.
2. Phase II (Subcohort Plus All Incident Cases):
   • The Subcohort (𝒮̃): Draw a random sample 𝒮̃ ⊂ ℱ of size ñ (with sampling fraction α = ñ / N ∈ (0, 1)) at baseline t = 0 without regard to follow-up time T_i or event status Δ_i. Because 𝒮̃ is a representative random cross-section of the baseline cohort ℱ, approximately αD of the incident cases will naturally fall inside the subcohort (𝒮̃ ∩ 𝒟 ≠ ∅).
   • Non-Subcohort Cases (𝒟 \ 𝒮̃): Identify all remaining (1 − α)D incident cases who developed the outcome during follow-up outside the subcohort.
   • Biospecimen Assay Set (𝒞 = 𝒮̃ ∪ 𝒟): Measure the expensive Phase II exposure vector Z_i on all individuals in 𝒞 = 𝒮̃ ∪ 𝒟. The total number of Phase II assays required is:

Before Prentice introduced the case–cohort design in 1986, the standard two-phase survival design was the nested case–control design (Thomas, 1977; Langholz and Thomas, 1990), which performs incidence density (risk-set) sampling: whenever a case i fails at time T_i, m controls are randomly drawn from the risk set R(T_i) = {j : T_j ≥ T_i} of individuals still disease-free at exact time T_i.

The case–cohort design provides five major epidemiological and logistical advantages over the nested case–control design:

1. Simultaneous Reuse of One Subcohort Across Multiple Disease Endpoints: Because the subcohort 𝒮̃ is selected at baseline t = 0 independently of any failure time or disease definition, the same assayed subcohort 𝒮̃ serves as the comparison group for multiple distinct endpoints (such as coronary heart disease, stroke, type 2 diabetes, and cancers in EPIC-InterAct, ARIC, and the Women's Health Initiative). In a nested case–control study, controls are matched to the failure time T_i of a specific disease's cases and cannot be reused for a second disease without reweighting.
2. Baseline Assay Execution: Because 𝒮̃ is chosen at baseline, laboratory assays and genomic sequencing on 𝒮̃ can be performed at the start of the study for quality control and cross-sectional biomarker studies without waiting years for cases to accrue.
3. Unbiased Estimation of Population Exposure Prevalence: Because 𝒮̃ is a random sample of the baseline cohort ℱ, it provides design-unbiased estimates of baseline exposure prevalence, biomarker quantiles, and covariate correlations in the source population.
4. Direct Estimation of Baseline Hazard and Absolute Risk: Because the subcohort sampling fraction α = ñ / N is known, weighting each subcohort member by 1/α reconstructs the full-cohort risk-set size at every time t, enabling direct Breslow estimation of the cumulative baseline hazard Λ₀(t) = ∫₀^t λ₀(u) du and absolute t-year disease risk P(T⁰ ≤ t | Z).
5. Time-Scale Flexibility: Unlike nested case–control studies—which commit to a primary time scale (time-on-study vs. attained age) when controls are risk-set matched—a case–cohort dataset can be analyzed on any time scale during statistical modeling.

Because the subcohort 𝒮̃ is a subset of the full cohort ℱ rather than a series of independent risk-set draws conditioned on failure order, replacing the full-cohort risk set R(T_i) in Cox's partial likelihood yields a pseudolikelihood. Three canonical weightings of the risk set R̃(t) at failure time t = T_i have been formulated in the biostatistical literature:

Differentiating log L̃(β) with respect to β yields the pseudolikelihood score vector U(β) = ∑_i=1^N ∫₀^τ [ Z_i(t) − Ē(β, t) ] dN_i(t) = 0, where the weighted risk-set covariate average Ē(β, t) is:

The three classical estimators differ in how they define the time-dependent risk-set weight w_j(t) for subcohort non-cases (j ∈ 𝒮̃ \ 𝒟), subcohort cases (j ∈ 𝒮̃ ∩ 𝒟), and non-subcohort cases (j ∈ 𝒟 \ 𝒮̃):

1. Self–Prentice Estimator (1988)

Steven G. Self and Ross L. Prentice (1988) defined the risk-set denominator at every failure time t = T_i over members of the subcohort 𝒮̃:

• w_j^SP(t) = 1 if j ∈ 𝒮̃ (whether j is a case or non-case);
• w_j^SP(t) = 0 if j ∈ 𝒟 \ 𝒮̃ (non-subcohort cases never enter the denominator, even at their own failure time t = T_i).

Because the denominator S̃_SP⁽⁰⁾(β, t) = ∑_j∈𝒮̃ Y_j(t) exp(β^⊤ Z_j(t)) depends only on the random subcohort 𝒮̃, (ñ)⁻¹ S̃_SP^(r)(β, t) is an unbiased empirical average of the full-cohort quantity N⁻¹ S^(r)(β, t) at every time t.

2. Prentice Original Estimator (1986)

Ross L. Prentice (1986) noted that when a non-subcohort case i ∈ 𝒟 \ 𝒮̃ fails at t = T_i, omitting i from its own risk-set denominator allows the individual ratio to exceed 1 in small samples. Prentice therefore included each non-subcohort case i ∈ 𝒟 \ 𝒮̃ in the risk set only at its failure instant t = T_i (and with weight 0 at all prior times t < T_i):

• w_j^P(t) = 1 if j ∈ 𝒮̃;
• w_j^P(t) = 1 if j ∈ 𝒟 \ 𝒮̃ and t = T_j;
• w_j^P(t) = 0 if j ∈ 𝒟 \ 𝒮̃ and t < T_j.

Self and Prentice (1988) proved that the Prentice (1986) and Self–Prentice (1988) estimators are asymptotically first-order equivalent: √N(β̂_P − β̂_SP) →_P 0 as N → ∞, with Prentice (1986) exhibiting slightly smaller finite-sample bias when the subcohort size ñ is modest.

3. Barlow Inverse-Probability-Weighted (Horvitz–Thompson) Estimator (1994, 1999)

William E. Barlow (1994) and Kalbfleisch and Lawless (1988) reformulated the case–cohort risk set from a survey-sampling Horvitz–Thompson inverse-probability-of-sampling perspective. At any time t < T_i prior to failure, all incident cases are sampled with probability 1 (or can be weighted dynamically), while non-cases in the subcohort represent 1/α non-cases in the full cohort. In Barlow's time-dependent weighting:

• w_j^B(t) = 1 / α̂(t) = N₀(t) / ñ₀(t) for all subcohort members j ∈ 𝒮̃ at risk prior to failure (where α = ñ/N is the baseline sampling fraction, or dynamically the ratio of at-risk non-cases in the full cohort to at-risk non-cases in the subcohort);
• w_j^B(t) = 1 at the instant of failure t = T_j for every case j ∈ 𝒟 (and w_j^B(t) = 0 for t < T_j if j ∈ 𝒟 \ 𝒮̃, whereas in the Kalbfleisch–Lawless time-invariant variant all cases j ∈ 𝒟 receive weight w_j^KL(t) = 1 at all times t ≤ T_j and subcohort non-cases j ∈ 𝒮̃ \ 𝒟 receive weight 1/α₀).

Why Standard Cox Partial-Likelihood Standard Errors Are Invalid

In a full-cohort Cox regression, the score process U(β, t) is a zero-mean martingale with respect to the natural filtration ℱ_t generated by the counting processes N_i(s) (s ≤ t), so the score increments across distinct failure times are uncorrelated and the asymptotic variance of β̂_Cox is simply the inverse of the observed Fisher information I_Cox(β̂)⁻¹.

In a case–cohort study, the pseudolikelihood score process U(β, t) is NOT a martingale! Why?

1. Whether an individual j belongs to the subcohort 𝒮̃ depends on the random baseline sampling indicator ξ_j ∈ {0, 1}, while inclusion of non-subcohort cases i ∈ 𝒟 \ 𝒮̃ depends on their future failure indicator Δ_i = I(T_i⁰ ≤ C_i) over the entire follow-up window [0, τ], violating predictability with respect to ℱ_t.
2. Because the same subcohort 𝒮̃ of size ñ = αN is used repeatedly in the denominator at every failure time T_i, random sampling fluctuations of Z_j within the subcohort 𝒮̃ induce positive covariance Cov(U_i(β), U_k(β)) > 0 between score contributions at different failure times T_i and T_k.

Consequently, naive Cox model standard errors Ĩ(β̂)⁻¹ severely underestimate the true variance of β̂ in a case–cohort study.

Self–Prentice Asymptotic Variance Decomposition and Robust Sandwich Variance

Self and Prentice (1988) proved that as N → ∞ with ñ / N → α ∈ (0, 1), the scaled score N^−1/2 U(β₀) converges in distribution to a normal vector N(0, Σ₀ + ((1 − α) / α) Δ_sub), yielding the Self–Prentice asymptotic variance matrix:

Notice the clean structural separation:

• The first term, Σ₀⁻¹, is the full-cohort Cox asymptotic variance that would have been obtained had all N subjects been assayed.
• The second term, ((1 − α) / α) Σ₀⁻¹ Δ_sub Σ₀⁻¹, is the extra variance due to Phase II subcohort sampling at fraction α = ñ/N. Notice that as α → 1 (the subcohort approaches the full cohort), (1 − α)/α → 0 and the case–cohort variance reduces smoothly to Σ₀⁻¹!

In applied biostatistical software (such as the survival cch routine in R and SAS PHREG), the asymptotic variance is computed via the Lin–Wei / Barlow robust jackknife (influence function) sandwich estimator:

where Ĩ(β̂) = −∂U(β)/∂β^⊤ |_β=β̂ is the pseudolikelihood information matrix and D̂_j(β̂) is the empirical dfbeta score residual of individual j ∈ 𝒞:

Stratified Case–Cohort Sampling and Phase I Calibration (Borgan et al., 2000)

When informative Phase I covariates W_i (such as age strata, study center, or an inexpensive surrogate biomarker correlated with Z_i) are recorded on all N cohort members, Ornulf Borgan et al. (2000) and Norman E. Breslow et al. (2009) showed that efficiency increases via two extensions:

1. Exposure-Stratified Subcohort Sampling (Borgan I, II, and III Estimators): Partition the baseline cohort ℱ into L strata ℱ₁, …, ℱ_L based on W_i and draw a stratified random subcohort 𝒮̃_l of size ñ_l = α_l N_l from each stratum l = 1, …, L. Weighting each subcohort member in stratum l by w_j = 1/α_l yields Borgan Estimator II (time-fixed weights) or Estimator III (score-unbiased weights).
2. Two-Phase Survey Calibration (Breslow et al., 2009): Using full-cohort Phase I data W_i (i = 1, …, N) to predict unmeasured Phase II covariates Z_i via an auxiliary imputation model and calibrating the inverse-probability weights w_j against full-cohort auxiliary totals achieves near-full-cohort efficiency on the main exposure effect and full-cohort precision on Phase I confounders.

Why are some incident cases already inside the random subcohort?

Because the subcohort 𝒮̃ of size ñ = αN is drawn as a random sample of the baseline cohort N at t = 0 without regard to future disease status, members of 𝒮̃ experience the same disease incidence rate as the rest of the cohort. Thus, if the cohort experiences D total incident cases over follow-up, an expected αD cases already belong to the subcohort (𝒮̃ ∩ 𝒟). Excluding future cases when drawing the baseline subcohort would turn the comparison group into a cumulative-incidence survivor group rather than a true baseline subcohort, biasing hazard and baseline-risk estimates.

How large should the subcohort be relative to the number of cases?

For a single primary endpoint with D incident cases, the asymptotic relative efficiency of a case–cohort study with subcohort size ñ = m · D compared to assaying the entire cohort of N individuals under β = 0 is m / (m + 1). A subcohort equal in size to the case series (m = 1) achieves ~50% relative efficiency; m = 2 achieves ~67% efficiency; and m = 3 to 4 achieves 75% to 80% of full-cohort efficiency.

Which weighting scheme (Prentice, Self–Prentice, or Barlow) should be used in practice?

Simulation studies (Onland-Moret et al., 2007; Cologne et al., 2012) show that Prentice's original weighting (1986) coupled with the Self–Prentice / Lin–Wei robust variance exhibits the smallest finite-sample bias when the subcohort sampling fraction α is small (< 5%). When estimating absolute t-year disease risks, cumulative baseline hazards Λ₀(t), or analyzing stratified case–cohort designs with stratum-specific sampling fractions α_l, Barlow's / Borgan's inverse-probability weighting (1/α_l) with survey-calibrated robust standard errors is preferred.

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2. Self, S. G., & Prentice, R. L. (1988). Asymptotic distribution theory and efficiency results for case-cohort studies. The Annals of Statistics, 16(1), 64–81. https://doi.org/10.1214/aos/1176350691
3. Barlow, W. E., Ichikawa, L., Rosner, D., & Black, S. (1999). Analysis of case-cohort designs. Journal of Clinical Epidemiology, 52(12), 1165–1172. https://doi.org/10.1016/S0895-4356(99)00102-X00102-X)
4. Borgan, Ø., Langholz, B., Samuelsen, S. O., Goldstein, L., & Breslow, N. (2000). Exposure stratified case-cohort designs. Lifetime Data Analysis, 6(1), 39–58. https://doi.org/10.1023/A:1009661900674
5. Breslow, N. E., Lumley, T., Ballantyne, C. M., Chambless, L. E., & Kulich, M. (2009). Using the whole cohort in the analysis of case-cohort data. American Journal of Epidemiology, 169(11), 1398–1405. https://doi.org/10.1093/aje/kwp055