Competing Risks Analysis

Competing risks analysis addresses situations where a subject is at risk of experiencing more than one type of event, but only one event can occur first and doing so removes the subject from further observation. A subject “competes” among several possible failure causes.

Classic examples:

A patient may die from cancer, heart disease, or another cause; the first to occur ends their observation period.
A mechanical component may fail by fatigue, corrosion, or overload; which failure mode occurs first determines both the failure time and its cause.
A customer may churn, upgrade, or downgrade; the event that happens first changes the analysis for the remaining outcomes.

Competing risks require special treatment because the causes are not independent; accounting for the wrong event type leads to biased estimates. This is sometimes called the “identifiability problem” in competing risks.

Relationship to Univariate Analysis

Standard survival methods (Kaplan-Meier, parametric MLE) applied to a single cause — ignoring others — estimate the cause-specific survival function. Naïvely applying KM while censoring the competing events yields a quantity that cannot be interpreted as the probability of experiencing the event in the real world because the competing events are not truly independent censoring mechanisms.

The correct marginal quantity of interest is the Cumulative Incidence Function (CIF), also called the sub-distribution survival function. For cause \(k\):

\[F_k(t) = P(T \leq t,\; K = k)\]

where \(T\) is the event time and \(K\) is the event type. The CIFs sum to the overall failure probability:

\[\sum_{k=1}^{K} F_k(t) = F(t) = 1 - S(t)\]

Non-Parametric CIF Estimation

The empirical CIF for cause \(k\) is estimated from the cause-specific hazard rates:

\[\hat{F}_k(t) = \sum_{x_i \leq t} \hat{h}_k(x_i)\, \hat{S}(x_i^-)\]

where \(\hat{h}_k(x_i) = d_{k,i} / r_i\) is the cause-specific hazard at event time \(x_i\), \(d_{k,i}\) is the number of cause-\(k\) events at \(x_i\), \(r_i\) is the total at-risk count, and \(\hat{S}(x_i^-)\) is the overall survival estimate just before \(x_i\).

Note

Non-parametric CIF estimation is not yet implemented in SurPyval. Use the parametric CompetingRisks model or an external library such as lifelines for this calculation currently.

Parametric Competing Risks

A parametric competing risks model specifies a separate parametric distribution for each cause. The overall survival function is the product of the cause-specific survival functions (assuming independent latent failure times):

\[S(t) = \prod_{k=1}^{K} S_k(t)\]

The overall density is:

\[f(t) = \sum_{k=1}^{K} f_k(t) \prod_{j \neq k} S_j(t)\]

SurPyval provides the CompetingRisks class for this model. Parameters for each cause distribution are estimated jointly by MLE.

Warning

The CompetingRisks model is implemented but has known issues. See DEVELOPMENT.md §1 for details before using in production analyses.

Regression: Fine-Gray and Cause-Specific PH

Two main regression approaches are used in competing risks:

Cause-specific proportional hazards — fits a separate Cox or parametric PH model for each cause, with all other cause events treated as censored:

\[h_k(t \mid Z) = h_{k,0}(t)\, e^{Z \beta_k}\]

This estimates the effect of covariates on the hazard of each cause independently.

Fine-Gray sub-distribution hazards — models the effect of covariates directly on the CIF via a proportional hazards model on the sub-distribution hazard:

\[h_k^*(t \mid Z) = h_{k,0}^*(t)\, e^{Z \gamma_k}\]

This is the natural choice when the scientific question is about the probability of a cause occurring in the presence of competing risks (e.g. clinical risk scores).

SurPyval provides the FineGray and CompetingRisksProportionalHazard classes.

Warning

Both FineGray.fit() and the competing risks regression model are currently broken (crash on first call). See DEVELOPMENT.md §2 for status. This functionality is under active development.