📘 Mathematical Foundations of gen_surv¶

This page presents the mathematical formulation behind the survival models implemented in the gen_surv package.

1. Cox Proportional Hazards Model (CPHM)¶

The hazard function conditioned on covariates is:

$$ h(t \mid X) = h_0(t) \exp(X \beta) $$

Where:

  • ( h_0(t) ) is the baseline hazard

  • ( X \beta ) is the linear predictor

Weibull baseline hazard:¶

$$ h_0(t) = \lambda \rho t^{\rho - 1} $$

The cumulative hazard is:

$$ \Lambda_0(t) = \lambda t^{\rho} $$

And the survival function becomes:

$$ S(t \mid X) = \exp\left(-\Lambda_0(t) \exp(X \beta)\right) $$

2. Time-Dependent Covariate Model (TDCM)¶

A generalization of CPHM where covariates change over time:

$$ h(t \mid Z(t)) = h_0(t) \exp(Z(t) \beta) $$

In this package, piecewise covariate values are simulated with dependence across segments using correlated normal draws.

3. Continuous-Time Multi-State Markov Model (CMM)¶

Markov model with generator matrix ( Q ). The transition probability matrix is given by:

$$ P(t) = \exp(Qt) $$

Where:

  • ( Q ) is the rate matrix

  • ( P(t)_{ij} ) gives the probability of being in state j at time t given starting in state i

4. Time-Homogeneous Hidden Markov Model (THMM)¶

This model simulates observed states with unobserved latent state transitions.

Let:

  • ( S_t ) be the latent state at time t

  • ( Y_t ) be the observed variable conditional on ( S_t )

The transition structure is governed by a homogeneous Markov chain with transition matrix ( P ), and emissions are Gaussian:

$$ Y_t \mid S_t = k \sim \mathcal{N}(\mu_k, \sigma_k^2) $$

5. Accelerated Failure Time (AFT) Models¶

AFT models assume that the effect of covariates accelerates or decelerates time to event directly, rather than the hazard.

Log-Normal AFT¶

The model assumes:

$$ \log(T_i) = X_i \beta + \varepsilon_i, \quad \varepsilon_i \sim \mathcal{N}(0, \sigma^2) $$

Thus:

$$ T_i \sim \text{Log-Normal}(X_i \beta, \sigma^2) $$

The survival function is:

$$ S(t \mid X) = 1 - \Phi\left(\frac{\log(t) - X \beta}{\sigma}\right) $$

Where:

  • ( \Phi ) is the standard normal cumulative distribution function (CDF)

This model is especially useful when the proportional hazards assumption is not valid and provides interpretable effects in the time domain.

Notes¶

All models support censoring:

  • Uniform: ( C_i \sim U(0, \text{cens\_par}) )

  • Exponential: ( C_i \sim \text{Exp}(\text{cens\_par}) )