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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

Three lives with a single binary covariate z are observed. Life A (z=1) dies at time 1, life B (z=0) dies at time 2, and life C (z=0) is censored at time 3. All were at risk from time 0. What is the Cox partial likelihood, with θ = exp(β)?

The partial likelihood is θ/(θ+2) multiplied by 1/2. At time 1 the risk set has all three lives, giving θ/(θ+2). At time 2 only B and C remain, both with z=0, giving 1/2. Censoring adds no factor.

  1. Aθ/(θ+2) × 1/2Correct
  2. Bθ/(θ+2) × 1/(1+θ)
  3. Cθ/(θ+2) × 1
  4. Dθ/(2θ+1) × 1/2
  5. θ/3 × 1/2

Explanation

At time 1 all three are at risk: A has relative risk θ, B and C have 1 each, so the factor is θ/(θ+2). After A dies, B and C remain at time 2, both with z=0, so B's factor is 1/2. C's censoring contributes no factor. The product is θ/(θ+2) × 1/2. Including θ in the second denominator wrongly keeps A in the risk set.

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