📘 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)

This semi-parametric approach models the hazard as a baseline component multiplied by an exponential term involving the covariates. 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)

This extension of the Cox model allows covariate values to vary during follow-up, accommodating exposures or treatments that 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)

This framework captures transitions between a finite set of states where waiting times are exponentially distributed. 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 handles situations where the process evolves through unobserved states that generate the observed responses. It simulates observed states with latent 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

These fully parametric models relate covariates to the logarithm of the survival time. They assume the effect of a covariate speeds up or slows down the event time directly, rather than acting on 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}) )

6. Competing Risks Models

These models handle scenarios where several distinct failure types can occur. Each cause has its own hazard function, and the observed status indicates which event occurred (1, 2, …). The package includes constant-hazard and Weibull-hazard versions.

7. Mixture Cure Models

These models posit that a subset of the population is cured and will never experience the event of interest. The generator mixes a logistic cure component with an exponential hazard for the uncured, returning a cured indicator column alongside the usual time and status.

8. Piecewise Exponential Model

Here the baseline hazard is assumed constant within each of several user-specified intervals. This allows flexible hazard shapes over time while remaining easy to simulate.