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Mixture cure

model="mixture_cure" — a fraction of the population is cured and will never experience the event, no matter how long you watch. The rest fail normally.

Use it for long-term-survivor data: a Kaplan-Meier curve that flattens onto a plateau well above zero and stays there.

The model

Two components. First, whether a subject is cured, through a logistic model:

\[ \Pr(\text{cured} \mid X) = \frac{\exp(\alpha_0 + X^\top \beta_{\text{cure}})}{1 + \exp(\alpha_0 + X^\top \beta_{\text{cure}})}, \]

where \(\alpha_0\) is set from the cure_fraction you ask for. Then, for the uncured, an exponential failure time whose hazard is scaled by the covariates:

\[ T_i \mid \text{uncured} \sim \mathrm{Exponential}\big(\lambda \exp(X_i^\top\beta_{\text{surv}})\big). \]

Cured subjects are censored at max_time or at their random censoring time, whichever comes first — from the outside they are indistinguishable from someone who simply had not failed yet. That is exactly the inference problem a cure model has to solve, and the generated frame hands you the answer key in the cured column.

Parameters

gen_mixture_cure(n, cure_fraction, baseline_hazard=0.5, betas_survival=None,
                 betas_cure=None, n_covariates=2, covariate_dist="normal",
                 covariate_params=None, model_cens="uniform", cens_par=5.0,
                 max_time=10.0, seed=None)
Parameter Type Default Meaning
n int — number of subjects
cure_fraction float — target proportion cured, in (0, 1)
baseline_hazard float 0.5 exponential hazard for the uncured
betas_survival list[float] | None None covariate effects on the uncured failure hazard
betas_cure list[float] | None None covariate effects on the probability of being cured
n_covariates int 2 number of covariates when the betas_* are omitted
covariate_dist "normal" | "uniform" | "binary" "normal" see Covariates
covariate_params dict | None None distribution parameters
model_cens str "uniform" censoring mechanism
cens_par float 5.0 censoring parameter
max_time float | None 10.0 administrative censoring
seed int | Generator | None None reproducibility

With betas_cure=[0.0, 0.0] the cure probability is the same for everyone and equals cure_fraction exactly. With non-zero values it varies by subject, and cure_fraction becomes the population average rather than an individual probability.

Example

from gen_surv import generate

df = generate(model="mixture_cure", n=6, cure_fraction=0.3, seed=42)
print(df)

   id      time  status  cured        X0        X1
0   0  0.973194       0      0 -1.951035 -1.302180
1   1  2.041874       1      0  0.127840 -0.316243
2   2  0.219019       0      1 -0.016801 -0.853044
3   3  0.771447       0      1  0.879398  0.777792
4   4  3.415245       0      1  0.066031  1.127241
5   5  0.781287       1      0  0.467509 -0.859292

The cured column is ground truth you would never have in real data. Every cured subject has status == 0; the converse is not true, because an uncured subject can also be censored before failing.

import pandas as pd

pd.crosstab(df["cured"], df["status"])

Check: is the cure fraction the one you asked for?

from gen_surv import generate, cure_fraction_estimate

for target in (0.2, 0.4, 0.6):
    df = generate(model="mixture_cure", n=20000, cure_fraction=target,
                  baseline_hazard=1.0, betas_cure=[0.0, 0.0],
                  betas_survival=[0.0, 0.0], model_cens="uniform",
                  cens_par=20.0, max_time=20.0, seed=7)
    print(f"target={target}  actually cured={df['cured'].mean():.3f}  "
          f"estimated={cure_fraction_estimate(df):.3f}")

target=0.2  actually cured=0.197  estimated=0.197
target=0.4  actually cured=0.398  estimated=0.399
target=0.6  actually cured=0.602  estimated=0.602

cure_fraction_estimate reads the plateau off the tail of the Kaplan-Meier curve, so it needs follow-up long enough for the curve to flatten — here max_time=20 against a hazard of 1.0. Shorten the follow-up and the estimate drifts upward, because uncured survivors are still mixed in with the cured.

Seeing the plateau

from gen_surv import generate, plot_survival_curve

df = generate(model="mixture_cure", n=2000, cure_fraction=0.4,
              baseline_hazard=1.0, betas_cure=[0.0, 0.0],
              betas_survival=[0.0, 0.0], cens_par=20.0, max_time=20.0, seed=1)

fig, ax = plot_survival_curve(df, title="Cure fraction 0.4")

The curve drops steeply while the uncured fail, then flattens at roughly 0.4 and stays there. A survival model without a cure component will insist the curve must eventually reach zero and will fit the tail badly — which is the point of generating this data.