πŸ“˜ 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 Markov Model (THMM)

This model describes a subject moving through three observed states, 1 (healthy), 2 (illness) and 3 (death), where every transition intensity is constant in time. That constancy is what β€œtime-homogeneous” refers to; the states themselves are observed, so this is not a hidden Markov model.

For a covariate ( X ), the intensity of the transition from state ( i ) to state ( j ) is

$$ \alpha_{ij}(t \mid X) = \lambda_{ij} \exp(\beta_{ij} X) $$

with ( \lambda_{ij} > 0 ) constant in ( t ). The three rates and three coefficients correspond to the ( 1 \to 2 ), ( 1 \to 3 ) and ( 2 \to 3 ) transitions.

Because the intensities do not depend on ( t ), each candidate sojourn time is exponential:

$$ T_{ij} \mid X \sim \mathrm{Exponential}!\left(\lambda_{ij} \exp(\beta_{ij} X)\right) $$

A subject leaves state 1 at ( \min(T_{12}, T_{13}) ), and the destination is whichever of the two came first. If censoring occurs before that, the subject is recorded in state 1.

The generator returns the full trajectory as a panel of state observations: an entry observation in state 1 at time 0, then one observation per transition, or an observation in the currently occupied state at the censoring time.

Note

This panel layout differs from the counting-process intervals returned by the CMM generator. The distinction is deliberate and follows the R package, where genTHMM returns states observed at times and genCMM returns transition intervals.

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.