Case-Cohort Sampling Likelihoods

How a Case-Cohort Sample Is Formed

A case-cohort design starts with a defined cohort. A subcohort is sampled at baseline, before the outcome period, and outcomes are then ascertained for the entire cohort. The analysis data consist of the sampled subcohort together with every case in the cohort. Thus, cases outside the subcohort are included, while noncases outside it are not. A case who belongs to the subcohort appears only once in the analysis data.

The design affects how much information each observed record represents. A sampled noncase stands in for noncases who were not sampled. By contrast, case ascertainment includes every case, so an observed case does not need to stand in for cases omitted by the design: none were omitted. This difference determines the inverse-probability weights for the full-cohort likelihood criterion.

The last statement assumes the subcohort is sampled before outcomes and that all cases in the defined cohort are ascertained. The subcohort sampling probability \(\pi_i\) may be common to everyone or may depend on a sampling stratum known under the design. If a case is also in the subcohort, its analysis-sample inclusion probability is still one: it is observed through the union, not counted twice.

From Inclusion Probabilities to Weighted Likelihood

Suppose the full-cohort model gives individual \(i\) a log-likelihood contribution \(\ell_i(\theta)\), based on the outcome and covariates that would be available for that individual in a complete-cohort analysis. The full-cohort log-likelihood is \(\ell(\theta)=\sum_{i=1}^N\ell_i(\theta)\). When only the case-cohort sample is observed, an inverse-probability weighted criterion replaces each missing contribution by an observed contribution multiplied by the reciprocal of its inclusion probability.

This is a Horvitz–Thompson weighted log-likelihood criterion. It is generally a weighted or pseudo-likelihood criterion, not the ordinary likelihood of the observed sample conditional on the sampling design. Its purpose here is to recover the full-cohort likelihood criterion in expectation over the sampling design.

Why the Weighted Criterion Recovers the Full-Cohort Criterion

The key calculation is design-based: condition on the fixed cohort outcomes, covariates, and their likelihood contributions, and take expectation over the subcohort sampling. For any individual, the expected weighted inclusion indicator is one.

Proof. Conditional on the cohort data, \(\ell_i(\theta)\) and \(\rho_i\) are fixed, while \(I_i\) is random under the sampling design. Since \(E_{\mathrm{design}}[I_i]=P_{\mathrm{design}}(I_i=1)=\rho_i\),

Summing this equality over the finite cohort and using linearity of expectation gives \(E_{\mathrm{design}}[\ell_{\mathrm{IPW}}(\theta)]=\sum_{i=1}^N\ell_i(\theta)=\ell(\theta)\). No independence among the inclusion indicators is needed for this expectation identity. In particular, fixed-size sampling can make subcohort indicators dependent without invalidating the result. \(\square\)

The result is about the sampling-design expectation of the criterion at a fixed \(\theta\). It does not say that a realized weighted criterion equals the full-cohort criterion, nor by itself guarantee that its maximizer is unbiased or has a particular variance.

Worked Example: Weights in a Simple Random Subcohort

Worked Example: Different Sampling Fractions by Stratum

Worked Example: Weighted Logistic Contributions and Score

Deriving the Logistic Score

For clarity, the score formula used above follows directly from the weighted criterion. Write \(x_i^{*}\) for an individual’s design vector, \(\eta_i=(x_i^{*})^\mathsf{T}\beta\), and \(p_i=\exp(\eta_i)/(1+\exp(\eta_i))\). The weighted Bernoulli log-likelihood is

Derivation. For one record, use \(\log(p_i)=\eta_i-\log(1+\exp(\eta_i))\) and \(\log(1-p_i)=-\log(1+\exp(\eta_i))\). Its unweighted contribution becomes \(y_i\eta_i-\log(1+\exp(\eta_i))\). Differentiating with respect to \(\beta\), and using \(\partial\eta_i/\partial\beta=x_i^{*}\), gives

Multiplication by the fixed design weight \(w_i\) multiplies this derivative by \(w_i\). Summing over observed records gives \(\partial\ell_{\mathrm{IPW}}(\beta)/\partial\beta=\sum_i w_i x_i^{*}(y_i-p_i)\). This derivation concerns the weighted criterion; uncertainty estimation must also account for the sampling design.

Interpretation and Common Pitfalls

Inverse-probability weighting restores the full-cohort contribution in expectation by expanding the contribution of sampled noncases. It does not mean that the observed sample has become a simple random sample of the cohort. Nor does the weighted criterion automatically provide ordinary likelihood-based standard errors: design dependence, including dependence induced by fixed-size subcohort sampling, matters for variance estimation.

• Giving every case weight \(1/\pi\). This incorrectly treats cases as observed only when sampled into the subcohort. Under complete case ascertainment their overall inclusion probability is one, so their weight is one.
• Counting subcohort cases twice. The observed data are a union. A case appearing in both the subcohort and the case list contributes once.
• Giving noncases weight one. This underrepresents cohort noncases when only a fraction \(\pi\) is sampled. Their inverse-probability weight is \(1/\pi\).
• Using the same weight across unequal sampling strata. When \(\pi_i\) differs by stratum, use the reciprocal of the individual’s applicable inclusion probability.
• Calling the weighted criterion the exact selected-sample likelihood. It is an inverse-probability weighted full-cohort criterion. A likelihood conditional on a sampling scheme is a different object.