Algorithm Overview

Cox Proportional Hazards Model (CPHM)

The hazard at time $t$ is proportional to a baseline hazard multiplied by the exponential of covariate effects. It is widely used for modelling relative risks under the proportional hazards assumption. See Cox (1972) in the References for the seminal paper.

Accelerated Failure Time Models (AFT)

These parametric models directly relate covariates to survival time. gen_surv includes log-normal, log-logistic and Weibull variants allowing different baseline distributions. They are convenient when the effect of covariates accelerates or decelerates event times.

Continuous-Time Multi-State Markov Model (CMM)

Simulates the illness-death process over states 1 (healthy), 2 (illness) and 3 (death), with Weibull transition intensities scaled by a covariate. The three rate pairs and three coefficients map one-to-one onto the 1 -> 2, 1 -> 3 and 2 -> 3 transitions. Output is in counting-process form: while a subject occupies state 1 it is at risk of both 1 -> 2 and 1 -> 3, so it contributes a row for each over the same interval, and a subject that reaches state 2 contributes a further 2 -> 3 row. Sojourn times are drawn on a reset clock, making the model semi-Markov. The mathematical formulation follows the counting-process approach of Andersen et al. Andersen et al. (1993).

Time-Dependent Covariate Model (TDCM)

Extends the Cox model to covariates that vary during follow-up. Covariates are simulated in a piecewise fashion with optional correlation across segments.

Time-Homogeneous Markov Model (THMM)

Simulates a three-state model (1 healthy, 2 illness, 3 death) whose transition intensities are constant in time, which is what makes it time-homogeneous. Each intensity is scaled by a covariate through rate * exp(beta * X0), so the three rates and three coefficients are matched one-to-one with the 1 -> 2, 1 -> 3 and 2 -> 3 transitions. Output is a panel of state observations: each subject starts in state 1 at time 0 and contributes a further observation at each transition, or at censoring in whichever state it then occupies. This layout differs from the counting-process form used by CMM, matching the distinction drawn by the R package between genTHMM and genCMM. For background on multistate survival models see Andersen et al. Andersen et al. (1993).

Competing Risks

Allows multiple failure types with cause-specific hazards. gen_surv supports constant and Weibull hazards for each cause. The subdistribution approach of Fine and Gray Fine and Gray (1999) is commonly used for analysis.

Mixture Cure Model

Assumes a proportion of individuals will never experience the event. A logistic component determines who is cured, while uncured subjects follow an exponential failure distribution. Mixture cure models were introduced by Farewell Farewell (1982).

Piecewise Exponential Model

Approximates complex hazard shapes by dividing follow-up time into intervals with constant hazard within each interval. This yields a flexible baseline hazard while remaining computationally simple.

For additional reading on these methods please see the References.