Competing risks and cumulative incidence

Verified for scikit-survival 0.28.0 on 2026-07-23.

Estimand

Competing risks are mutually exclusive causes (J \in {1,\ldots,K}), where the first observed cause prevents observing the others as first events.

The cause-(k) cumulative incidence function (CIF) is:

[ F_k(t) = P(T \le t, J=k). ]

It is an absolute cause-specific event probability accounting for all competing causes. It is not:

• a cause-specific hazard;
• 1 - Kaplan-Meier after censoring other causes;
• a conditional probability among only those still event-free;
• a causal effect or clinical-utility measure.

The total risk is (\sum_k F_k(t)). Its complement is estimated all-cause event-free survival. Censoring is an observation mechanism, not an additional event-free state.

Event coding

The nonparametric CIF API takes two separate arrays:

# event: 0=censored; 1..K=mutually exclusive causes
event = frame["status"].to_numpy(dtype=int)
time = frame["time"].to_numpy(dtype=float)

Requirements:

• event is integer and non-negative;
• 0 always denotes right-censoring;
• positive codes 1..K are contiguous;
• the data contains observations for every code 1..K;
• time is finite and positive;
• event/time lengths match.

Do not pass a boolean Surv outcome to cumulative_incidence_competing_risks(). Surv intentionally collapses event status to event versus censoring and loses cause identity.

Nonparametric CIF API

from sksurv.nonparametric import cumulative_incidence_competing_risks

time_points, cumulative_incidence = (
    cumulative_incidence_competing_risks(event, time)
)

Current signature:

cumulative_incidence_competing_risks(
    event,
    time_exit,
    time_min=None,
    conf_level=0.95,
    conf_type=None,
    var_type="Aalen",
)

Returns:

• time_points: shape (n_times,);
• cumulative_incidence: shape (K + 1, n_times);
• row 0: total risk of any cause;
• row k: CIF for cause k.

total_risk = cumulative_incidence[0]
cause_1 = cumulative_incidence[1]
cause_2 = cumulative_incidence[2]

assert np.allclose(
    total_risk,
    cumulative_incidence[1:].sum(axis=0),
)

time_min estimates conditionally on surviving at least to that time. This changes the target population and must not be selected after viewing outcomes.

Confidence intervals

time_points, cumulative_incidence, confidence_interval = (
    cumulative_incidence_competing_risks(
        event,
        time,
        conf_type="log-log",
        conf_level=0.95,
        var_type="Aalen",
    )
)

confidence_interval has shape (K + 1, 2, n_times), where axis 1 is lower/upper. Current variance choices are:

• "Aalen"
• "Dinse"
• "Dinse_Approx"

Pointwise confidence intervals are not simultaneous confidence bands. Sparse causes and late follow-up can make estimates unstable even when the function returns a result.

Built-in competing-risk datasets

from sksurv.datasets import load_bmt, load_cgvhd

X_bmt, y_bmt = load_bmt()       # status codes 0, 1, 2
X_cgvhd, y_cgvhd = load_cgvhd() # status codes 0, 1, 2, 3

The first structured field is integer cause status, not boolean. These are real study datasets distributed for examples. The bundled tests do not use them; they use synthetic non-clinical outcomes only.

Why 1 - Kaplan-Meier is wrong for one cause

If cause 2 prevents cause 1, censoring cause 2 in a Kaplan-Meier curve treats those subjects as if they could still experience cause 1 later under non-informative censoring. That counterfactual risk set does not estimate the observed-world probability (F_1(t)) and typically overstates cause-1 probability.

Use CIF for cause-specific absolute probability:

time_points, cif = cumulative_incidence_competing_risks(event, time)
probability_cause_1_by_t = cif[1]

Kaplan-Meier remains appropriate for all-cause event-free survival after collapsing all causes to event, if that is the estimand and censoring assumptions hold.

Comparing groups

Estimate group-specific curves without fitting preprocessing on the full dataset:

curves = {}
for label in prespecified_groups:
    mask = group == label
    curves[label] = cumulative_incidence_competing_risks(
        event[mask],
        time[mask],
        conf_type="log-log",
    )

Plotting pointwise intervals does not test equality. scikit-survival 0.28 does not provide Gray's test in this API. Do not substitute an ordinary log-rank test: survival and CIF group hypotheses differ.

Group labels and comparison times should be prespecified. Report at-risk/event support; late visual separation with few rows can be misleading.

Cause-specific Cox hazards

For cause (k), a cause-specific hazard model encodes that cause as an event and other causes as censored at their occurrence time:

from sksurv.linear_model import CoxPHSurvivalAnalysis
from sksurv.util import Surv

y_cause_1 = Surv.from_arrays(
    event=(event == 1),
    time=time,
)
cause_1_hazard_model = CoxPHSurvivalAnalysis(alpha=0.1)
cause_1_hazard_model.fit(X_train, y_cause_1_train)

This estimates association with the instantaneous cause-specific hazard under a PH model. Other causes are censored for this hazard likelihood, which is different from pretending they are independent censoring when estimating absolute CIF.

To derive cause-specific CIF predictions from cause-specific hazards, all modeled causes must be combined:

[ F_k(t \mid x) = \int_0^t S(u^- \mid x),dH_k(u \mid x), \quad S(t \mid x)=\exp\left[-\sum_j H_j(t \mid x)\right]. ]

Therefore, 1 - cause_1_model.predict_survival_function(...) is not the cause-1 CIF. A set of separately fitted cause-specific models requires careful joint integration, common time grids, and external validation.

Fine-Gray regression

scikit-survival 0.28 does not implement Fine-Gray subdistribution-hazard regression. Do not invent an import or describe cumulative_incidence_competing_risks as Fine-Gray; it is a nonparametric CIF estimator.

If using another implementation:

• verify it is actively maintained and supports the required censoring/truncation;
• use its official API documentation;
• distinguish subdistribution from cause-specific hazard coefficients;
• keep preprocessing and tuning leakage-safe;
• validate cause-specific absolute probabilities, not only coefficients.

Neither hazard parameterization is universally "better." The estimand determines the method.

Prediction evaluation

Standard concordance_index_ipcw, cumulative_dynamic_auc, and Brier APIs in scikit-survival are documented for right-censored single-event outcomes. A competing-risk prediction question needs:

• a named cause;
• a case/control definition at each horizon;
• handling of other causes consistent with that definition;
• cause-specific probability predictions for calibration/Brier evaluation;
• censoring weights fitted on training data;
• evaluation times supported by training follow-up;
• nested tuning or an untouched holdout.

Do not label an all-event C-index as cause-specific discrimination, and do not use an all-event survival probability as a cause-specific CIF.

Bundled helper

The helper defaults to deterministic synthetic data:

python skills/scikit-survival/scripts/competing_risk_cif.py

For local CSV:

python skills/scikit-survival/scripts/competing_risk_cif.py \
  --input competing.csv \
  --event-column status \
  --time-column time \
  --horizons 2,5,10 \
  --confidence \
  --curve-output cif-curves.npz \
  --output cif-summary.json

It:

• rejects URLs, symlinks, missing/non-contiguous causes, and invalid times;
• bounds file size and row count;
• verifies that cause-specific rows sum to total CIF;
• writes numeric arrays without pickle;
• reports point estimates at requested horizons;
• makes no network calls.

Use only authorized, de-identified local data. Do not include row-level data or PHI in reports.

Reporting checklist

• cause definitions and code mapping;
• censoring definition and follow-up window;
• CIF versus cause-specific or subdistribution hazard estimand;
• number of rows/events for every cause;
• horizon-specific CIF with uncertainty and support;
• whether intervals are pointwise;
• handling of time_min, if any;
• competing-risk-specific prediction evaluation;
• no causal or clinical-utility claim from association/probability alone.

Sources

Official scikit-survival sources checked 2026-07-23:

• Competing-risks user guide
• CIF API
• Dataset API
• 0.24 release notes introducing CIF

Primary methods:

• Aalen O. "Nonparametric estimation of partial transition probabilities in multiple decrement models." Annals of Statistics 6 (1978), 534-545. Project Euclid record
• Gray RJ. "A class of K-sample tests for comparing the cumulative incidence of a competing risk." Annals of Statistics 16 (1988), 1141-1154. doi:10.1214/aos/1176350951