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Cox proportional hazards

model="cphm" — the textbook proportional-hazards setup: one covariate, one coefficient, an exponential baseline hazard.

Use it when you want a dataset whose log hazard ratio you know, to check that a Cox implementation recovers it.

The model

The hazard for subject \(i\) with covariate \(X_i\) is

\[ h(t \mid X_i) = h_0(t) \exp(\beta X_i), \]

with a constant baseline \(h_0(t) = 1\). Event times therefore come from an exponential distribution whose rate is scaled by the covariate:

\[ T_i \mid X_i \sim \mathrm{Exponential}\big(\exp(\beta X_i)\big), \qquad X_i \sim \mathrm{Uniform}(0, \texttt{covariate\_range}). \]

A censoring time \(C_i\) is drawn independently, and the row records \(\min(T_i, C_i)\) together with which came first.

Because the baseline is constant, the hazard does not change over follow-up. For a hazard that rises or falls, use Weibull AFT; for an arbitrary shape, piecewise exponential.

Parameters

gen_cphm(n, model_cens, cens_par, beta, covariate_range, seed=None)
Parameter Type Constraint Meaning
n int > 0 number of subjects, one row each
model_cens str "uniform" or "exponential" censoring mechanism
cens_par float > 0 censoring parameter — see Censoring
beta float — log hazard ratio per unit of X0. A scalar, not a list
covariate_range float > 0 X0 is drawn from Uniform(0, covariate_range)
seed int | Generator | None — reproducibility

beta is scalar here

cphm is the only model whose beta is a single number. AFT models take a list of coefficients, and the multi-state models take exactly three.

Example

from gen_surv import generate

df = generate(model="cphm", n=6, beta=0.5, covariate_range=2.0,
              model_cens="uniform", cens_par=1.0, seed=42)
print(df)

       time  status        X0
0  0.438878     0.0  1.547912
1  0.094177     0.0  1.394736
2  0.037041     1.0  1.522279
3  0.370798     0.0  0.900772
4  0.646901     1.0  1.287730
5  0.251113     1.0  0.454477
Column dtype Meaning
time float64 min(event, censoring)
status float64 1.0 event, 0.0 censored
X0 float64 the covariate

There is no id column — this is the only generator without one. See Output schemas.

Controlling the censoring rate

With model_cens="uniform", censoring times are Uniform(0, cens_par), so a larger cens_par censors less:

for cens_par in (0.5, 1.0, 5.0, 100.0):
    df = generate(model="cphm", n=20000, beta=0.5, covariate_range=2.0,
                  model_cens="uniform", cens_par=cens_par, seed=1)
    print(f"cens_par={cens_par:>6}  event rate = {df['status'].mean():.3f}")

Check: does a Cox model recover beta?

from gen_surv import generate
from lifelines import CoxPHFitter

df = generate(model="cphm", n=5000, beta=0.5, covariate_range=2.0,
              model_cens="uniform", cens_par=1.0, seed=7)

cph = CoxPHFitter().fit(df, duration_col="time", event_col="status")
print(cph.summary[["coef", "se(coef)", "coef lower 95%", "coef upper 95%"]].round(3))

            coef  se(coef)  coef lower 95%  coef upper 95%
covariate
X0         0.501     0.035           0.433           0.569

At n=20000 and five seeds the estimates are 0.502, 0.489, 0.500, 0.489, 0.517 — centred on the truth, and unaffected by how heavily the data is censored.