Competing risks and cumulative incidence

Verified for scikit-survival 0.28.0 on 2026-10-01.

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 is documented as conditional survival. In released 0.28.0 it fails with a broadcasting error when it removes any unique times; the bundled helper rejects that setting. Do not claim a conditional/landmark analysis from this helper. A different verified method and explicit target population are needed.

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). Native 0.28.0 returns cause-specific rows in lower/upper order, but **total-risk row 0 is reversed and ignores conf_level**. Repair that row with the public all-event KM estimator:

from sksurv.nonparametric import kaplan_meier_estimator

km_time, _, km_ci = kaplan_meier_estimator(
    event > 0, time, conf_type="log-log", conf_level=0.95,
)
assert np.array_equal(km_time, time_points)
confidence_interval[0] = 1.0 - km_ci[::-1]
assert np.all(confidence_interval[:, 0] <= confidence_interval[:, 1])

The bundled helper applies this repair using the requested level and rejects non-finite or reversed intervals. This repairs the total-risk row, not general small-sample coverage. 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

For prespecified groups where every original cause occurs (the native function requires contiguous observed causes), estimate point curves:

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

If a subgroup has no event for a cause, do not silently relabel its rows or invent an event to satisfy the API. Preserve a global cause mapping and use a method that supports absent causes; an all-censored group needs separate handling.

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 within observed follow-up;
• repairs total-risk intervals and guards the upstream conditional-time defect;
• 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 2026-10-01:

• Released CIF implementation — native defects above reproduced on a six-row synthetic fixture.

• 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