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Accelerated failure time

Three generators — aft_ln, aft_weibull and aft_log_logistic — where covariates act on the event time rather than on the hazard.

Use them when the proportional-hazards assumption is what you want to violate, or when a time-scale interpretation ("this treatment doubles survival") is more natural than a hazard ratio.

beta does not mean the same thing in all three

Only aft_ln puts beta directly on log time. In aft_weibull and aft_log_logistic, beta enters through exp(-eta / shape), so its effect on log time is \(-\beta/\texttt{shape}\) — different sign, different magnitude. The section below shows the measurements. Getting this wrong will make a correct estimator look broken.

The three models

Log-normal — aft_ln

\[ \log T_i = X_i^\top \beta + \varepsilon_i, \qquad \varepsilon_i \sim \mathcal{N}(0, \sigma^2) \]

so \(T_i\) is log-normal with median \(\exp(X_i^\top\beta)\). The survival function is

\[ S(t \mid X) = 1 - \Phi\!\left(\frac{\log t - X^\top \beta}{\sigma}\right). \]

The hazard rises then falls — useful precisely because it is not proportional.

Weibull — aft_weibull

\[ T_i = \texttt{scale} \cdot \big(-\log U_i \cdot e^{-\eta_i}\big)^{1/\texttt{shape}}, \qquad U_i \sim \mathrm{Uniform}(0,1), \quad \eta_i = X_i^\top\beta \]

The Weibull is the one distribution that is both proportional-hazards and accelerated-failure-time, and this parameterisation is the PH one: \(\beta\) is a log hazard ratio. The hazard is monotone — falling for shape < 1, constant at shape = 1, rising for shape > 1.

Log-logistic — aft_log_logistic

\[ T_i = \texttt{scale} \cdot \left(\frac{U_i}{1 - U_i}\right)^{1/\texttt{shape}} e^{-\eta_i/\texttt{shape}} \]

A unimodal hazard: it rises to a peak, then decays. The standard choice when risk is highest some way into follow-up.

Parameters

gen_aft_log_normal(n, beta, sigma, model_cens, cens_par, seed=None)
gen_aft_weibull(n, beta, shape, scale, model_cens, cens_par, seed=None)
gen_aft_log_logistic(n, beta, shape, scale, model_cens, cens_par, seed=None)
Parameter Type Applies to Meaning
n int all number of subjects
beta list[float] all one coefficient per covariate; its length sets the number of covariates
sigma float aft_ln standard deviation of the log-time error
shape float aft_weibull, aft_log_logistic shape parameter — controls the hazard's direction
scale float aft_weibull, aft_log_logistic time scale
model_cens str all "uniform" or "exponential"
cens_par float all censoring parameter
seed int | Generator | None all reproducibility

Covariates are drawn as independent standard normals, X0 … X{p-1} where p = len(beta). Pass beta=[0.5, -0.3] and you get two covariates; pass five coefficients and you get five.

Example

from gen_surv import generate

df = generate(model="aft_ln", n=6, beta=[0.5, -0.3], sigma=1.0,
              model_cens="exponential", cens_par=3.0, seed=42)
print(df)

   id      time  status        X0        X1
0   0  1.699586       1  0.304717 -1.039984
1   1  1.238956       0  0.750451  0.940565
2   2  0.889270       1 -1.951035 -1.302180
3   3  0.496337       1  0.127840 -0.316243
4   4  1.851995       1 -0.016801 -0.853044
5   5  0.471177       1  0.879398  0.777792

What beta actually does

Generate with almost no censoring and regress log(time) on the covariates. For beta = [0.5, -0.3], shape = 1.5, n = 40000:

Model Fitted effect on log(time) Matches
aft_ln X0 0.500, X1 −0.298 \(\beta\) itself — [0.5, -0.3]
aft_weibull X0 −0.332, X1 0.209 \(-\beta/\texttt{shape}\) — [-0.333, 0.200]
aft_log_logistic X0 −0.336, X1 0.187 \(-\beta/\texttt{shape}\) — [-0.333, 0.200]

For aft_weibull the natural estimator is therefore a Cox model, which reads beta off directly:

from gen_surv import generate
from lifelines import CoxPHFitter

df = generate(model="aft_weibull", n=40000, beta=[0.5, -0.3], shape=1.5,
              scale=2.0, model_cens="uniform", cens_par=1e6, seed=7)

cph = CoxPHFitter().fit(df[["time", "status", "X0", "X1"]],
                        duration_col="time", event_col="status")
print(cph.params_.round(3).to_dict())

{'X0': 0.499, 'X1': -0.311}

And for aft_ln, ordinary least squares on log time recovers beta and sigma:

import numpy as np
from gen_surv import generate

df = generate(model="aft_ln", n=20000, beta=[0.5, -0.3], sigma=1.0,
              model_cens="exponential", cens_par=100.0, seed=7)

X = np.column_stack([df["X0"], df["X1"]])
coef, *_ = np.linalg.lstsq(X, np.log(df["time"]), rcond=None)
resid_sd = np.std(np.log(df["time"]) - X @ coef)
print(coef.round(3), round(float(resid_sd), 3))

[ 0.491 -0.299] 1.002

(That shortcut ignores the 1.8% of rows that are censored. With heavier censoring, fit a proper AFT model — lifelines.LogNormalAFTFitter — instead of regressing on observed times.)

Choosing shape

from gen_surv import generate, plot_hazard_comparison

models = {
    f"shape={s}": generate(model="aft_weibull", n=2000, beta=[0.5, -0.3],
                           shape=s, scale=2.0, model_cens="uniform",
                           cens_par=5.0, seed=42)
    for s in (0.5, 1.0, 2.0)
}
fig, ax = plot_hazard_comparison(models, title="Weibull hazard by shape")
shape Hazard over time Typical use
< 1 falling early failures, infant mortality
= 1 constant equivalent to an exponential model
> 1 rising wear-out, ageing