Chapter_17

### Chapter 17: Causal Survival Analysis

Introduction

• Context: Extends causal inference to time-to-event outcomes (e.g., time until death) where events may occur after study end, necessitating specialized methods for administrative censoring.
• Key shift: Previous chapters focused on outcomes at fixed time points; survival analysis handles events occurring at any time during follow-up.
• Administrative censoring: Intrinsic to survival studies (e.g., study end precludes observing all events). Other censoring types (e.g., loss to follow-up, competing events) require adjustment but are deferred to Part III.

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17.1 Hazards and Risks

Technical Points:

• Survival probability: $Pr[T > k]$ = Proportion surviving beyond time $k$.
• Risk (cumulative incidence): $Pr[T \leq k] = 1 - Pr[T > k]$.
• Hazard: Discrete-time hazard defined as $Pr[T = k \mid T > k-1]$ (probability of event in interval $k$ given survival until $k-1$).
• Key difference:
  • Risk uses fixed baseline denominator (cumulative events).
  • Hazard uses time-varying denominator (at-risk individuals at $k$).

Fine Points:

• Hazard ratio limitations:
  1. Varies over time; single summary (e.g., Cox model) is a weighted average, obscuring interpretation.
  2. Subject to selection bias (conditioning on survival introduces collider bias).
• Preferred contrasts: Survival/risk differences/ratios at specific times (e.g., 5-year survival difference).

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17.2 From Hazards to Risks

Technical Points:

• Data structures:
  • Wide format: One row per individual.
  • Person-time format: One row per person-time interval (enables hazard estimation).
• Time-varying indicator: $D_k = 1$ if $T \leq k$ (event by $k$), else $0$.
• Survival from hazards:
  $Pr[D_k = 0] = \prod_{m=1}^k Pr[D_m = 0 \mid D_{m-1} = 0] = \prod_{m=1}^k (1 - \text{hazard at } m)$.
• Estimation:
  • Nonparametric: Kaplan-Meier estimator (product-limit).
  • Parametric: Logistic model for hazards (e.g., $\text{logit } Pr[D_{k+1}=1 \mid D_k=0, A] = \theta_{0,k} + \theta_1 A + \theta_2 A \times k$).

Fine Points:

• Model choices:
  • Parametric models (e.g., logistic) smooth unstable nonparametric hazards.
  • Semiparametric models (e.g., Cox, AFT) avoid distributional assumptions but impose proportional hazards or other constraints.

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17.3 Why Censoring Matters

Technical Points:

• Goal: Estimate $Pr[D_k = 0 \mid A=a]$ (survival if no censoring).
• Challenge: Naive estimation using uncensored individuals introduces selection bias.
• Solution under random censoring:
  $Pr[D_k = 0 \mid A=a] = \prod_{m=1}^k Pr[D_m = 0 \mid D_{m-1}=0, C_m=0, A=a]$.
• Staggered entry: Adjust for baseline calendar time and confounders.

Fine Points:

• Artificial censoring: Exclude individuals whose event time would be unobserved under alternative treatments to preserve exchangeability.

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17.4 IP Weighting of Marginal Structural Models

Technical Points:

• Assumptions: Exchangeability, positivity, consistency.
• Weights: Stabilized IP weights $SW^A = \frac{Pr[A=1]}{Pr[A=1 \mid L]}$ for treated, $\frac{1 - Pr[A=1]}{1 - Pr[A=1 \mid L]}$ for untreated.
• MSM for hazards:
  $\text{logit } Pr[D^{a}_{k+1} = 0 \mid D^{a}_k = 0] = \beta_{0,k} + \beta_1 a + \beta_2 a \times k$.
• Survival estimation: Multiply estimated conditional survivals over time.

Results (example):

• 120 months: 80.7% survival under smoking cessation vs. 80.5% under no cessation (difference 0.2%; 95% CI: -4.1%, 3.7%).

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17.5 The Parametric g-Formula

Technical Points:

• Standardized survival:
  $Pr[D^{a}_{k+1} = 0] = \sum_l Pr[D_{k+1} = 0 \mid L=l, A=a] Pr[L=l]$.
• Steps:
  1. Fit parametric hazards model conditional on $L$ and $A$.
  2. Standardize by averaging predictions over covariate distribution $L$.
• G-computation: Predict survival under treatment scenarios using outcome model.

Results (example):

• 120-month survival: 80.4% (cessation) vs. 80.6% (no cessation); difference -0.2% (95% CI: -4.6%, 4.1%).

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17.6 G-Estimation of Structural Nested Models

Technical Points:

• Structural AFT model:
  $T^{a=0}_i = T_i \exp(\psi A_i)$ (rank-preserving).
• G-estimation with censoring:
  • Define artificial censoring indicator $\Delta(\psi)$ to handle administrative censoring.
  • Find $\psi$ such that $\Delta(\psi) \perp\!\!\!\perp A \mid L$ (using logistic model).
• Interpretation: $\exp(-\psi)$ = Ratio of median survival times under treatment vs. control.

Results (example):

• $\hat{\psi} = -0.047$ (95% CI: -0.223, 0.333); survival time ratio = 1.05.

Fine Points:

• Limitations:
  • Rank preservation unrealistic.
  • Computationally intensive; few software implementations.
• Alternatives: Structural cumulative failure/time models for rare events.

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Key Themes

• Censoring: Central challenge; addressed via hazard/product-limit estimators, IP weighting, g-methods.
• Contrasts: Prefer survival/risk differences over hazard ratios due to interpretability and bias risks.
• Model trade-offs:
  • IP weighting: Robust to outcome model misspecification if treatment model correct.
  • g-formula: Robust to treatment model misspecification if outcome model correct.
  • G-estimation: Theoretically appealing but rarely used due to complexity.

Note: Figures/tables referenced (e.g., 17.1–17.7) illustrate survival curves and causal diagrams but could not be summarized due to lack of visual data.