Chapter 17: Causal Survival Analysis

This chapter extends causal inference methods to survival analysis and time-to-event outcomes. We define causal effects for survival data, address challenges posed by censoring and competing events, and show how to estimate causal survival curves using IP weighting and the parametric g-formula.

This chapter is based on Hernán and Robins (2020, chap. 17, pp. 247-264).

Key challenge: Survival outcomes involve time, and individuals may be censored before experiencing the event. This requires careful definition of potential outcomes and attention to competing risks.

1 17.1 Hazards and Risks (pp. 247-250)

We begin by reviewing key concepts from survival analysis.

Basic Definitions

Definition 1 (Survival Analysis Terms) Event time \(T\): Time from baseline until an event occurs (e.g., death, disease onset)

Censoring time \(C\): Time from baseline until censoring (e.g., loss to follow-up, study end)

Observed time \(Y = \min(T, C)\): The time we actually observe

Event indicator \(D = I(T \leq C)\): 1 if event occurred, 0 if censored

Risk at time \(t\): \(\Pr[T \leq t]\) (cumulative incidence)

Survival at time \(t\): \(S(t) = \Pr[T > t] = 1 - \Pr[T \leq t]\)

Hazard at time \(t\): \(\lambda(t) = \lim_{dt \to 0} \frac{\Pr[t \leq T < t + dt \mid T \geq t]}{dt}\)

Hazard vs Risk

Hazard: Instantaneous rate of event occurrence among those at risk

• Ranges from 0 to \(\infty\)
• Can increase, decrease, or remain constant over time
• Related to survival: \(S(t) = \exp\left(-\int_0^t \lambda(s) ds\right)\)

Risk (cumulative incidence): Probability of event by time \(t\)

• Ranges from 0 to 1
• Always non-decreasing
• More interpretable for most purposes

Relationship:

\[\text{Risk}(t) = 1 - \exp\left(-\int_0^t \lambda(s) ds\right)\]

Intuition: The hazard is like a velocity (rate of change), while risk is like a distance traveled (cumulative probability).

For causal inference, we typically focus on causal risks rather than causal hazards, though both can be defined.

2 17.2 From Hazards to Risks (pp. 250-252)

The survival function \(S(t)\) and cumulative hazard \(\Lambda(t)\) are key quantities.

Cumulative Hazard

\[\Lambda(t) = \int_0^t \lambda(s) ds\]

Interpretation: Total accumulated hazard up to time \(t\).

Relationships

Survival from cumulative hazard:

\[S(t) = \exp(-\Lambda(t))\]

Risk from survival:

\[\Pr[T \leq t] = 1 - S(t) = 1 - \exp(-\Lambda(t))\]

Kaplan-Meier Estimator

Nonparametric estimator of survival function:

\[\hat{S}(t) = \prod_{t_i \leq t} \left(1 - \frac{d_i}{n_i}\right)\]

where: - \(t_i\) are the ordered event times - \(d_i\) is the number of events at \(t_i\) - \(n_i\) is the number at risk just before \(t_i\)

Assumption: Censoring is independent of event time (given covariates).

Kaplan-Meier provides a nonparametric estimate of the survival curve without assuming a parametric model for the hazard.

Limitation: Doesn’t adjust for confounders. For causal inference, we need IP weighting or parametric g-formula.

3 17.3 Why Censoring Matters (pp. 252-255)

Censoring creates missing data problems in survival analysis.

Types of Censoring

Administrative censoring: Study ends before all events occur

• Predictable, often at a fixed time
• Relatively benign if independent of event risk

Loss to follow-up: Participants drop out before event or study end

• Potentially problematic
• May be related to prognosis (informative censoring)

Competing risks: Individuals experience a different event that precludes the event of interest

• e.g., death from other causes when studying heart attack
• Requires special treatment

Independent Censoring Assumption

Definition 2 (Independent Censoring) Censoring is independent if:

\[T \perp\!\!\!\perp C \mid L\]

where \(T\) is event time, \(C\) is censoring time, \(L\) are measured covariates.

Interpretation: Among individuals with the same covariate values, censoring is unrelated to their (unobserved) event time.

When violated: Standard survival methods (Kaplan-Meier, Cox regression) are biased.

Solution: Use IP weighting for censoring (Section 17.5).

Informative censoring: When censoring is related to prognosis even after adjusting for covariates.

Example: Patients who feel sicker may be more likely to drop out AND more likely to have events. If we can measure “feeling sick”, we can adjust. If not, we have a problem.

Practical approach: Measure as many predictors of censoring as possible, then assume independence given those predictors.

4 17.4 The Hazard Ratio (pp. 255-257)

The hazard ratio is commonly used to quantify treatment effects in survival analysis.

Definition

Definition 3 (Hazard Ratio) The hazard ratio comparing treatment \(a\) to \(a'\) is:

\[HR^{a,a'}(t) = \frac{\lambda^a(t)}{\lambda^{a'}(t)}\]

where \(\lambda^a(t)\) is the hazard function under treatment \(a\).

Cox proportional hazards model: Assumes \(HR^{a,a'}(t) = HR^{a,a'}\) (constant over time).

Interpretation Challenges

Collapsibility: Unlike risk differences and risk ratios, hazard ratios are non-collapsible.

• Marginal HR \(\neq\) average of conditional HRs
• Adjusted HR from Cox model is not a causal marginal effect
• Conditional on covariates, HR has complex interpretation

Built-in selection bias: Hazards condition on survival to time \(t\), creating selection bias when effects are heterogeneous.

Why hazard ratios are tricky:

1. Non-collapsibility: Even under no confounding, adjusted HR \(\neq\) unadjusted HR
2. Selection bias: By conditioning on \(T \geq t\), we select a subset of the population that changes over time
3. No causal interpretation: The HR from a Cox model is not generally a causal parameter

Better approach for causal inference: Focus on causal risk differences or causal risk ratios, not hazard ratios.

5 17.5 IP Weighting of Survival Curves (pp. 257-260)

We can use IP weighting to estimate causal survival curves.

Causal Survival Curve

Definition 4 (Causal Survival Function) The causal survival function under treatment \(a\) is:

\[S^a(t) = \Pr[T^a > t]\]

where \(T^a\) is the potential event time under treatment \(a\).

Causal risk at time \(t\): \(\Pr[T^a \leq t] = 1 - S^a(t)\)

IP Weighted Kaplan-Meier

Goal: Estimate \(S^a(t)\) adjusting for confounding.

Method: IP weighted Kaplan-Meier estimator

1. Compute IP weights for treatment: \(W^A = \frac{1}{\Pr[A \mid L]}\)
2. Apply weighted Kaplan-Meier:

\[\hat{S}^a(t) = \prod_{t_i \leq t} \left(1 - \frac{\sum_{j: A_j = a} W_j^A I(Y_j = t_i, D_j = 1)}{\sum_{j: A_j = a} W_j^A I(Y_j \geq t_i)}\right)\]

Result: Estimate of causal survival curve under treatment \(a\).

Handling Censoring

Joint weights for treatment and censoring:

\[W^{A,C}_i = \frac{1}{\Pr[A_i \mid L_i]} \times \frac{1}{\Pr[C_i > t \mid A_i, L_i, \bar{Y}_i(t)]}\]

where \(\bar{Y}_i(t)\) represents the history of being at risk up to time \(t\).

Stabilized weights can improve stability:

\[SW^{A,C} = \frac{\Pr[A]}{\Pr[A \mid L]} \times \frac{\Pr[C > t \mid A]}{\Pr[C > t \mid A, L, \bar{Y}(t)]}\]

Time-dependent weights: Censoring weights can change over time if censoring depends on time-varying covariates or outcome history.

Estimation:

1. Model \(\Pr[C > t \mid A, L, \bar{Y}(t)]\) using pooled logistic regression
2. For each individual at each time, compute censoring probability
3. Create weights as product of treatment weights and censoring weights
4. Apply weighted Kaplan-Meier

This is implemented in various R packages (e.g., ipw, WeightIt).

6 17.6 The Parametric G-Formula for Survival Data (pp. 260-262)

The parametric g-formula can also be used for survival outcomes.

Discrete-Time Approach

Model: Hazard at each time point \(t\)

\[\Pr[T = t \mid T \geq t, A, L] = \text{expit}(\alpha_0(t) + \alpha_1 A + \alpha_2^{\top} L)\]

This can be fit using pooled logistic regression:

• Create one record per person per time period
• Include time as a predictor (e.g., dummy variables for each \(t\))
• Fit logistic regression

G-formula algorithm:

1. Fit pooled logistic model for \(\Pr[T = t \mid T \geq t, A, L]\)
2. For each individual \(i\) and each time \(t\):
   • Predict \(\hat{p}_{it}^a = \Pr[T = t \mid T \geq t, A = a, L_i]\)
3. Compute survival probabilities:
   • \(\hat{S}_i^a(t) = \prod_{s=1}^t (1 - \hat{p}_{is}^a)\)
4. Average over individuals:
   • \(\hat{S}^a(t) = \frac{1}{n} \sum_{i=1}^n \hat{S}_i^a(t)\)

Continuous-Time Approach

For continuous time, use parametric survival models:

• Weibull, exponential, log-normal, etc.
• Specify hazard function with parameters depending on \(A, L\)
• Predict survival curves for each individual under each treatment
• Average to get causal survival curves

Pooled logistic regression:

Approximates the continuous-time hazard when time intervals are small. As intervals \(\to 0\), converges to Cox model.

Advantages over IP weighting:

• May be more efficient when outcome model is correct
• Easier to incorporate time-varying covariates
• Natural for discrete-time data

Disadvantages:

• Requires correct specification of outcome model
• More complex with time-varying treatments

7 17.7 Competing Risks (pp. 262-264)

Competing risks occur when multiple types of events can prevent observation of the event of interest.

Definition

Definition 5 (Competing Risks) A competing risk is an event that precludes the occurrence of the event of interest.

Example: When studying heart attack incidence, death from cancer is a competing risk.

If someone dies from cancer, they can never have a heart attack (or at least we can’t observe it).

Challenges

Censoring is not independent: Individuals who experience competing events may have different risk of the event of interest.

Standard survival methods fail: Treating competing events as censoring leads to biased estimates.

Approaches

1. Cause-specific hazards:

• Model hazard for each type of event separately
• Use cause-specific Cox models
• Provides hazard ratios, not easily interpretable as causal effects

2. Subdistribution hazards (Fine-Gray model):

• Models cumulative incidence function directly
• Individuals with competing events remain in risk set
• More interpretable but still has issues

3. Parametric g-formula:

• Model all event types jointly
• Simulate outcomes under each treatment
• Compute cumulative incidence of each event type
• Recommended for causal inference

Why g-formula for competing risks:

The g-formula naturally handles competing risks by modeling the joint distribution of all outcomes. Under an intervention to set \(A = a\):

1. Predict which event type each person would experience first
2. Compute the proportion experiencing each event type by time \(t\)
3. These are the causal cumulative incidence functions

Practical implementation:

• Use multinomial logistic regression for multiple event types
• Or separate models for each event type with appropriate risk sets
• Simulate individual trajectories under each treatment
• Summarize to get causal curves

8 Summary

Key concepts:

1. Survival analysis: Study of time-to-event outcomes with censoring
2. Hazard vs risk: Instantaneous rate vs cumulative probability
3. Censoring: Missing data problem requiring careful assumptions
4. Causal survival curves: \(S^a(t) = \Pr[T^a > t]\) under treatment \(a\)
5. IP weighted Kaplan-Meier: Adjusts for confounding via weighting
6. Parametric g-formula: Models hazards, predicts survival curves, averages
7. Competing risks: Multiple event types that preclude each other

Methods for causal survival analysis:

Method	Approach	Advantages	Disadvantages
IP weighted KM	Weight observations	Nonparametric, robust	Needs correct treatment/censoring models
Parametric g-formula	Model hazards	Efficient, handles competing risks	Needs correct outcome model
Cox model	Model conditional hazard	Familiar, flexible	HR not causally interpretable

Causal estimands:

• Causal risk difference: \(\Pr[T^{a=1} \leq t] - \Pr[T^{a=0} \leq t]\)
• Causal risk ratio: \(\frac{\Pr[T^{a=1} \leq t]}{\Pr[T^{a=0} \leq t]}\)
• Causal survival ratio: \(\frac{S^{a=1}(t)}{S^{a=0}(t)}\)

NOT generally causal:

• Hazard ratio from Cox model with covariates (non-collapsible, built-in selection)

Assumptions:

1. Conditional exchangeability: \(T^a \perp\!\!\!\perp A \mid L\)
2. Independent censoring: \(T \perp\!\!\!\perp C \mid A, L\)
3. Positivity: \(\Pr[A = a \mid L] > 0\) and \(\Pr[C > t \mid A, L] > 0\)
4. Correct models: Treatment, censoring, and/or outcome models

Practical recommendations:

1. Focus on risks, not hazards, for causal inference
2. Adjust for confounding using IP weighting or g-formula
3. Handle censoring via IP weighting or modeling
4. Use g-formula for competing risks
5. Check assumptions, especially independent censoring
6. Report survival curves, not just summary measures

Common mistakes:

1. Interpreting adjusted hazard ratios as causal effects
2. Treating competing events as independent censoring
3. Ignoring time-varying confounding (see Part III)
4. Using methods that assume independent censoring when it’s violated

Looking ahead: Part III extends these ideas to time-varying treatments, where survival analysis becomes even more complex but the g-formula and IP weighting remain powerful tools.

Hernán, Miguel A, and James M Robins. 2020. Causal Inference: What If. Boca Raton: Chapman & Hall/CRC. https://miguelhernan.org/whatifbook.