Mathematical foundations¶

The formulation behind each generator. Every model page repeats the part relevant to it; this page puts them side by side and fixes notation.

Throughout, \(T\) is the event time, \(C\) the censoring time, \(X\) a covariate vector, and the observed pair is \((\min(T, C), \mathbb{1}\{T \le C\})\).

Three equivalent ways to describe a survival distribution:

\[ S(t) = \Pr(T > t), \qquad h(t) = \lim_{\Delta \to 0}\frac{\Pr(t \le T < t + \Delta \mid T \ge t)}{\Delta}, \qquad S(t) = \exp\!\left(-\int_0^t h(u)\,du\right). \]

Sampling works through the last of these: draw \(U \sim \mathrm{Uniform}(0,1)\) and solve \(S(T) = U\). Every generator here is some version of that inversion.

1. Cox proportional hazards¶

The hazard splits into a baseline and a covariate multiplier:

\[ h(t \mid X) = h_0(t)\exp(X^\top\beta). \]

The ratio of hazards for two subjects is \(\exp((X_1 - X_2)^\top\beta)\) — constant in \(t\), which is what "proportional" means. cphm uses a constant baseline \(h_0(t) = 1\), so

\[ T \mid X \sim \mathrm{Exponential}\big(\exp(X^\top\beta)\big), \]

drawn by inversion as \(T = -\log(U)/\exp(X^\top\beta)\).

With a Weibull baseline \(h_0(t) = \lambda\rho t^{\rho-1}\) the cumulative hazard is \(\Lambda_0(t) = \lambda t^{\rho}\) and

\[ S(t \mid X) = \exp\!\big(-\lambda t^{\rho}\exp(X^\top\beta)\big), \]

which is what the Weibull AFT generator produces in its PH parameterisation.

→ cphm

2. Accelerated failure time¶

Instead of scaling the hazard, scale time:

\[ \log T = X^\top\beta + \sigma\varepsilon, \]

so \(S(t \mid X) = S_0\!\left(t e^{-X^\top\beta}\right)\): a covariate stretches or compresses the whole survival curve. The distribution of \(\varepsilon\) picks the family.

Log-normal. \(\varepsilon \sim \mathcal{N}(0,1)\), giving

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

with \(\Phi\) the standard normal CDF. The hazard rises then falls.

Weibull. The only family that is both AFT and PH. gen_surv draws it as

\[ T = \texttt{scale}\cdot\big(-\log U \cdot e^{-X^\top\beta}\big)^{1/\texttt{shape}}, \]

which is the PH parameterisation — \(\beta\) is a log hazard ratio, and its effect on \(\log T\) is \(-\beta/\texttt{shape}\).

Log-logistic. \(S(t) = \big(1 + (t/\texttt{scale})^{\texttt{shape}}\big)^{-1}\), giving a unimodal hazard: rising to a peak, then decaying. Not a PH family.

→ AFT models

3. Piecewise exponential¶

Partition follow-up at \(0 < \tau_1 < \dots < \tau_k\) and hold the hazard constant on each piece:

\[ h_0(t) = \lambda_j, \quad t \in [\tau_j, \tau_{j+1}). \]

The cumulative hazard is piecewise linear,

\[ \Lambda_0(t) = \sum_{j} \lambda_j \big(\min(t, \tau_{j+1}) - \tau_j\big)_+, \]

and inversion walks the intervals, consuming exponential "budget" until it runs out inside one of them. With enough pieces this approximates any hazard shape.

→ piecewise_exponential

4. Competing risks¶

With \(K\) causes, each has a cause-specific hazard

\[ h_k(t \mid X) = \lim_{\Delta \to 0} \frac{\Pr(t \le T < t+\Delta,\; \delta = k \mid T \ge t)}{\Delta} = h_{0k}(t)\exp(X^\top\beta_k). \]

The all-cause hazard is \(\sum_k h_k\), and the observed pair is \((T, \delta) = (\min_k T_k, \arg\min_k T_k)\).

The quantity usually reported is the cumulative incidence

\[ F_k(t) = \int_0^t h_k(u) S(u)\, du, \]

which depends on all the hazards through \(S\) — this is why a Fine-Gray subdistribution coefficient does not equal the cause-specific \(\beta_k\) that generated the data.

→ Competing risks

5. Mixture cure¶

A latent indicator \(Y\) marks the uncured, with \(\Pr(Y = 1 \mid X)\) logistic in \(X\). The population survival function mixes a point mass at infinity with a proper distribution:

\[ S_{\text{pop}}(t \mid X) = \pi(X) + \big(1 - \pi(X)\big)S_u(t \mid X), \]

where \(\pi(X)\) is the cure probability and \(S_u\) the survival function of the uncured. As \(t \to \infty\), \(S_{\text{pop}} \to \pi(X)\) — the plateau. gen_surv takes \(S_u\) exponential with hazard \(\lambda\exp(X^\top\beta_{\text{surv}})\).

→ Mixture cure

6. Multi-state models¶

States \(\{1, 2, 3\}\) = healthy, ill, dead, with transition intensities

\[ \alpha_{ij}(t \mid X) = \lim_{\Delta \to 0} \frac{\Pr\big(Z(t + \Delta) = j \mid Z(t) = i, X\big)}{\Delta}. \]

For a time-homogeneous chain the intensities are constant, the generator matrix \(Q\) has \(Q_{ij} = \alpha_{ij}\) and \(Q_{ii} = -\sum_{j \ne i}\alpha_{ij}\), and transition probabilities follow from the matrix exponential:

\[ P(t) = \exp(Qt). \]

Sojourns are exponential and memoryless. That is thmm, with \(\alpha_{ij} = \lambda_{ij}\exp(\beta_{ij}X)\).

cmm instead draws Weibull sojourns,

\[ T_{ij} = \left(\frac{-\log(1-U)}{\lambda_{ij}\exp(\beta_{ij}X)}\right)^{1/\rho_{ij}}, \]

on a clock that resets when a subject enters state 2. The intensity of dying then depends on time since illness rather than time since entry, making the process semi-Markov rather than Markov.

7. Time-dependent covariates¶

When a covariate changes during follow-up the hazard is

\[ h(t \mid Z(t)) = h_0(t)\exp(Z(t)^\top\beta), \]

and the partial likelihood needs \(Z(t)\) evaluated at each event time — hence the (start, stop] layout that survival software expects. gen_surv simulates a single switch from 0 to 1 at a crossover time correlated with the baseline covariate.

→ tdcm

Censoring¶

All generators apply independent right-censoring: \(C \perp T \mid X\), with

\[ C \sim \mathrm{Uniform}(0, \texttt{cens\_par}) \quad\text{or}\quad C \sim \mathrm{Exponential}(\text{mean} = \texttt{cens\_par}). \]

Independence is what makes the standard estimators unbiased here. To break it deliberately, see Censoring.

Some models add administrative censoring at max_time, which is deterministic and also independent of \(T\).