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Competing risks

model="competing_risks" and model="competing_risks_weibull" — several distinct failure types, where the first one to occur is the one you observe and the others are never seen for that subject.

Use them when "death" is not one thing: relapse versus death in remission, device failure by mode, discharge versus in-hospital death.

The model

Each cause \(k\) has its own cause-specific hazard

\[ h_k(t \mid X) = h_{0k}(t) \exp(X^\top \beta_k), \]

with an independent set of coefficients per cause. A latent time is drawn for each cause and the smallest wins:

\[ T = \min_k T_k, \qquad \delta = \arg\min_k T_k . \]

The observed status is the winning cause — 1, 2, … — or 0 if censoring came first, or if nothing happened before max_time.

The two variants differ only in the baseline:

Model Baseline hazard Parameters
competing_risks constant, \(h_{0k}(t) = \lambda_k\) baseline_hazards
competing_risks_weibull Weibull, monotone per cause shape_params, scale_params

Parameters

gen_competing_risks(n, n_risks=2, baseline_hazards=None, betas=None,
                    covariate_dist="normal", covariate_params=None,
                    max_time=10.0, model_cens="uniform", cens_par=5.0, seed=None)

gen_competing_risks_weibull(n, n_risks=2, shape_params=None, scale_params=None,
                            betas=None, covariate_dist="normal",
                            covariate_params=None, max_time=10.0,
                            model_cens="uniform", cens_par=5.0, seed=None)
Parameter Type Default Meaning
n int — number of subjects
n_risks int 2 number of competing causes
baseline_hazards list[float] | None None one constant hazard per cause (competing_risks)
shape_params, scale_params list[float] | None None one per cause (competing_risks_weibull)
betas list[list[float]] | None None one row per cause, one column per covariate
covariate_dist "normal" | "uniform" | "binary" "normal" see Covariates
covariate_params dict | None None distribution parameters
max_time float | None 10.0 administrative censoring: nothing is observed beyond this
model_cens str "uniform" random censoring mechanism
cens_par float 5.0 censoring parameter
seed int | Generator | None None reproducibility

betas is a matrix: betas[k][j] is the effect of covariate j on cause k + 1. Leave it out and coefficients are drawn at random, which is fine for a smoke test and useless for validation.

Example

Cause 1 driven by X0, cause 2 by X1, with cause 1 twice as fast:

from gen_surv import generate

df = generate(model="competing_risks", n=20000, n_risks=2,
              baseline_hazards=[0.4, 0.2],
              betas=[[0.8, 0.0],      # cause 1: X0 matters
                     [0.0, -0.5]],    # cause 2: X1 matters
              model_cens="uniform", cens_par=50.0, max_time=50.0, seed=7)

df["status"].value_counts().sort_index()

status is the cause, so count it rather than summing it:

0      716     # censored
1    12408     # cause 1
2     6876     # cause 2

Check: does a cause-specific Cox model recover each row of betas?

Analyse one cause at a time, treating the other causes as censored — that is what "cause-specific" means:

from lifelines import CoxPHFitter

for cause in (1, 2):
    d = df.assign(event=(df["status"] == cause).astype(int))
    fit = CoxPHFitter().fit(d[["time", "event", "X0", "X1"]],
                            duration_col="time", event_col="event")
    print(f"cause {cause}: {fit.params_.round(3).to_dict()}")

cause 1: {'X0': 0.8, 'X1': -0.002}
cause 2: {'X0': 0.014, 'X1': -0.513}

Both rows come back where they were set, and neither cause picks up the other's covariate.

Cause-specific hazards are not subdistribution hazards

A Fine-Gray model estimates something different — the effect on the cumulative incidence, keeping subjects who failed from other causes in the risk set. Its coefficients will not match the betas you passed here, and that is correct rather than a bug. betas are cause-specific by construction.

Administrative censoring with max_time

max_time truncates follow-up for everybody, on top of the random censoring from model_cens. Set it to None for no administrative cut-off, or to the length of your hypothetical study to mimic one:

df = generate(model="competing_risks", n=1000, n_risks=3,
              baseline_hazards=[0.3, 0.2, 0.1],
              max_time=2.0, cens_par=100.0, seed=1)
# nothing is observed after t = 2