IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Estimation procedures for lifetime distributions: formula sheet

Full chapter guide

Key formulas

Right-censored observation: Contribution to likelihood = S(c) = P(T > c); Use when you only know the life survived beyond c. In terms of the hazard: exp(−∫₀^c μ(t) dt).
Uncensored (death) observation: Contribution to likelihood = f(t) = S(t) × μ(t); Use when the exact time of death t is observed.
Left-censored observation: Contribution to likelihood = F(c) = P(T < c) = 1 − S(c); Use when you know the event occurred before c but not when.
Interval-censored observation: Contribution to likelihood = S(a) − S(b) = P(a < T ≤ b); Use when death is known to fall between a and b.
Left-truncated observation (death at t, entry at x): Contribution to likelihood = f(t) ÷ S(x); Condition on survival to entry time x. If instead the life is right-censored at c, the contribution is S(c) ÷ S(x).
Right-truncated observation (included only if T ≤ r): Contribution to likelihood = f(t) ÷ F(r); Condition on the event having happened by r.
Constant hazard with right censoring: μ̂ = d ÷ v, where d = number of deaths and v = total time observed; Total time includes time lived by censored lives. This is the MLE when the hazard is constant.
Exact death contribution: f(t) = S(t) × μ(t); Use when the death time is observed exactly.
Right-censored contribution: P(T > c) = S(c); Use when the life is known only to survive beyond c.
Left-truncated contribution: f(t) ÷ S(x₀) for a death; S(c) ÷ S(x₀) for a censored life; x₀ is the time the life enters observation. Condition on survival to x₀.
General likelihood: L = Π [μ(tᵢ)]^δᵢ × exp(−∫ μ(s) ds from entry to tᵢ); δᵢ = 1 for death, 0 for censored. Entry is 0 if there is no truncation.
Constant hazard likelihood: L(μ) = μ^d × e^(−μv); d = number of deaths; v = total observed time for all lives.
Constant hazard MLE: μ̂ = d ÷ v; Check it is a maximum: d²ln L/dμ² = −d ÷ μ² < 0.
Variance of MLE: Var(μ̂) ≈ μ² ÷ d, estimated by μ̂² ÷ d = d ÷ v²; Asymptotic result from the Cramér-Rao lower bound, using the information −E[d²ln L/dμ²].
Kaplan-Meier estimator: Ŝ(t) = Π over tⱼ ≤ t of (1 − dⱼ/nⱼ); nⱼ = number at risk just before tⱼ. dⱼ = deaths at tⱼ. Ŝ(t) = 1 before the first death time.
Risk set update: nⱼ₊₁ = nⱼ − dⱼ − cⱼ; cⱼ = number censored between tⱼ and tⱼ₊₁ (if censored at exactly a death time, the standard convention is that they are still in the risk set at that time).
Greenwood's formula: Var(Ŝ(t)) ≈ Ŝ(t)² × Σ over tⱼ ≤ t of dⱼ ÷ [nⱼ(nⱼ − dⱼ)]; An estimate of the variance. The standard error is its square root.
Approximate confidence interval: Ŝ(t) ± z × √Var(Ŝ(t)); z = 1.96 for 95%. This is a normal approximation and can go outside [0, 1] near the ends.
Conditional survival from KM: Ŝ(t₂) ÷ Ŝ(t₁) = Π over t₁ < tⱼ ≤ t₂ of (1 − dⱼ/nⱼ); Use this for estimates of survival from t₁ to t₂.
Nelson-Aalen cumulative hazard: Λ̂(t) = Σ over tⱼ ≤ t of dⱼ ÷ nⱼ; dⱼ = deaths at time tⱼ. nⱼ = number at risk just before tⱼ. Λ̂(t) = 0 before the first death.
Survival from Nelson-Aalen: Ŝ(t) = exp(−Λ̂(t)); Uses the relation S(t) = exp(−Λ(t)). Gives a survival estimate that differs slightly from Kaplan-Meier.
Kaplan-Meier estimator: Ŝ(t) = Π over tⱼ ≤ t of (1 − dⱼ ÷ nⱼ); Use it to compare. It is a product, whereas Nelson-Aalen is a sum.
Cumulative hazard from Kaplan-Meier: Λ̂(t) = −ln Ŝ(t); Gives an alternative cumulative hazard estimate from Kaplan-Meier. It differs from the Nelson-Aalen estimate.
Relationship between survival and cumulative hazard: S(t) = exp(−Λ(t)), where Λ(t) = ∫₀ᵗ μₛ ds; μₛ is the force of mortality (hazard rate).
Variance of Nelson-Aalen estimate: Var[Λ̂(t)] ≈ Σ over tⱼ ≤ t of dⱼ ÷ nⱼ²; Use only if the question asks for it or gives a result in the core reading style. State it is an approximation.
Cox model: h(t; z) = h₀(t) × exp(β₁z₁ + β₂z₂ + ... + βₚzₚ) = h₀(t) exp(β'z); h₀(t) is the baseline hazard for z = 0. There is no separate intercept term, as it would be absorbed into h₀(t).
Proportional hazards: h(t; z₁) ÷ h(t; z₂) = exp(β'(z₁ − z₂)); Independent of t. This is the assumption you must be able to state and test.
Partial likelihood (no tied deaths): L(β) = Π over death times tᵢ of [ exp(β'zᵢ) ÷ Σ over j in R(tᵢ) of exp(β'zⱼ) ]; zᵢ is the covariate vector of the life that dies at tᵢ. R(tᵢ) is the risk set: all lives still under observation just before tᵢ, including the one who dies. Censored lives contribute only through the risk sets.
Log partial likelihood: l(β) = Σ over death times of [ β'zᵢ − ln( Σ over j in R(tᵢ) of exp(β'zⱼ) ) ]; Differentiate this and set equal to zero to find β̂. Usually solved numerically.
Hazard ratio: Hazard ratio for a one-unit rise in covariate i = exp(βᵢ); Other covariates fixed. For a rise of k units, use exp(kβᵢ).
Survival function under the model: S(t; z) = [S₀(t)]^exp(β'z); S₀(t) is the baseline survival function. This follows because the cumulative hazard is multiplied by exp(β'z).
Central exposed to risk estimate of force of mortality: μ̂_x = d_x ÷ E^c_x; Valid as the MLE when μ is constant over the age and period. E^c_x is the total time under observation.
Initial exposed to risk estimate of q_x: q̂_x = d_x ÷ E_x; E_x counts each life from the start of the year, including those who die or leave part-way if the deaths are treated as full-year lives.
Link between exposures: E_x ≈ E^c_x + ½ d_x; Assumes deaths occur on average half-way through the year. For a death at fraction t of the year, the exact addition is (1 − t).
Poisson approximation for deaths: D_x ~ Poisson(E^c_x μ_x); Var(μ̂_x) ≈ μ_x ÷ E^c_x; Mean and variance of D_x are both E^c_x μ_x. Good when μ is constant.
Binomial model for deaths: D_x ~ Binomial(E_x, q_x); Var(q̂_x) = q_x(1 − q_x) ÷ E_x; Needs initial exposure and independent lives with the same q_x.
Constant force relationship: q_x = 1 − e^(−μ); Holds for one year when μ is constant over the year. q ≈ μ if μ is small.
Census approximation of exposure: E^c_x ≈ ∫ P_x(t) dt ≈ average of census counts × period length; P_x(t) is the number of lives aged x last birthday at time t. The trapezium rule is the usual approximation.

Quick revision

Right censoring: the life is known to survive past a time, but the exact lifetime is unknown. Left truncation: lives enter observation only after surviving to some age.
Likelihood: deaths contribute f(t), right-censored lives contribute S(t), and left-truncated lives are conditioned on survival to entry.
Kaplan-Meier: Ŝ(t) = Π over death times tj ≤ t of (1 − dj ÷ nj).
nj is the number at risk just before tj, so it includes lives censored at tj but excludes those who left earlier.
Nelson-Aalen: Ĥ(t) = Σ over tj ≤ t of dj ÷ nj, and Ŝ(t) = exp(−Ĥ(t)) is the matching survival estimate.
Kaplan-Meier changes only at death times and is a step function that is constant between them.
Cox model: h(t; z) = h0(t) exp(β·z), where the baseline hazard h0(t) is left unspecified.
The hazard ratio for two lives is exp(β·(z1 − z2)) and does not depend on time under proportional hazards.
The Cox partial likelihood at each death compares the failing life's hazard with the total over the risk set.
Central exposed to risk Ec is the total time lived in the age group, and the estimate of μ is deaths ÷ Ec.
Initial exposed to risk E is used for qx, and the estimate of q is deaths ÷ E.
Always state the assumption you use, such as constant force of mortality over the age interval.

Common mistakes

Using the words censoring and truncation as if they mean the same thing. Fix: Ask whether the life is in the data. Censored lives are observed with partial information. Truncated lives are included only if they meet a condition, and others are never seen.
Ignoring left truncation and counting exposure from birth. Fix: Count exposure only from the age at which the life entered observation. Condition the likelihood on survival to that age by dividing by S(entry age).
Leaving out censored lives' time from v. Fix: Add every life's observed time. Censored lives add to v but not to d.
Using f(c) for a censored life instead of S(c). Fix: Ask: did I see the death? If not, use S.
Dropping the curve at censoring times Fix: Only deaths cause a step. A censored life just leaves the risk set after its censoring time.
Using the wrong risk set Fix: nⱼ is the number with observed time ≥ tⱼ. Lives censored before tⱼ are out. Lives censored at or after tⱼ are in.
Using the wrong risk set, for example forgetting to remove censored lives. Fix: After every event, subtract both deaths and censorings. Check that n at the next row equals the previous n minus all exits.
Multiplying (1 − d ÷ n) terms when asked for Nelson-Aalen. Fix: Remember: Nelson-Aalen adds hazard increments. Kaplan-Meier multiplies survival factors.
Leaving the dying life out of its own risk set, or leaving in lives who were censored earlier. Fix: Define R(tᵢ) as all lives with observed time ≥ tᵢ. The life who dies at tᵢ is always in it. A life censored before tᵢ is never in it.
Including a factor in the partial likelihood for each censored life. Fix: Only death times give factors. Censored lives appear only in the denominators of the earlier death times where they were still at risk.

Exam tips

Begin every answer by stating whether each life is censored or truncated, with a one-line reason. This earns marks even if the later calculation slips.
Write the likelihood contribution of each type of life in a short list before multiplying. Examiners award method marks for each correct contribution.
When the question mentions that lives enter at a certain age, think left truncation and condition on survival to that age.
For discussion questions, always state the assumption that censoring is non-informative, and say what goes wrong if it is not.
In MCQs, the wrong options are often the swapped terms. Re-read the definition: partial information means censoring, missing lives means truncation.
Write the contribution of each life in a short table (entry, exit, death or censored) before writing L. It earns method marks.
State assumptions clearly: independent lives, non-informative censoring, and constant hazard if used.
In MCQs, compute d and v first. The answer is usually d ÷ v and the trap is leaving out censored time.