laplace-survival

A Laplace library of parametric survival models for Stan — Gompertz, log-logistic, Gompertz–Makeham, and piecewise-exponential — with right-censored likelihoods, pointwise log-likelihoods for LOO, survival curves, and event-time simulators. Import it into any .laplace model and call it with namespaced calls (survival::function_name(...)).

Like all Laplace libraries, survival compiles down to plain, readable Stan functions. Nothing about how you use it hides what actually ends up in your .stan file.

Models

Every model is described by two functions: the log hazard $\log h(t)$, the instantaneous risk at time $t$, and the cumulative hazard $H(t)$, the total risk accumulated up to $t$. Everything else in the library — likelihoods, survival curves, simulation — is built from those two, using $S(t) = e^{-H(t)}$ and the right-censored log likelihood

$$ \log L = \sum_{i=1}^{N} \Big[ d_i \log h(t_i) - H(t_i) \Big], $$

where $d_i = 1$ if the event was observed and $d_i = 0$ if the observation was right-censored.

Model	Hazard $h(t)$	Cumulative hazard $H(t)$	Parameters
Gompertz	$e^{\eta + \gamma t}$	$e^{\eta}\frac{e^{\gamma t}-1}{\gamma}$	gamma log-hazard slope, eta log hazard at $t=0$
Log-logistic	$\frac{(\beta/\alpha)(t/\alpha)^{\beta-1}}{1+(t/\alpha)^{\beta}}$	$\log\left(1+(t/\alpha)^{\beta}\right)$	alpha scale (median), beta shape
Gompertz–Makeham	$a + e^{\eta + \gamma t}$	$at + e^{\eta}\frac{e^{\gamma t}-1}{\gamma}$	a background hazard, gamma, eta as Gompertz
Piecewise exponential	$\lambda_{j} e^{\eta}$ for $t \in (s_j, s_{j+1}]$	$e^{\eta}\sum_j \lambda_j E_j$	log_lambda baseline log hazard per interval, eta linear predictor

In the Gompertz, Makeham, and PWE models, eta is where your covariates go (e.g. eta0 + X * beta), giving a proportional-hazards model. The Gompertz and Makeham cumulative hazards are computed in a form that stays numerically stable as $\gamma \to 0$, where both reduce to an exponential model.

What's included

All functions live in survival_hazard.laplacelib and follow the naming pattern srv_<model>_<quantity>, with <model> one of gompertz, log_logistics, makeham, pwe.

Function	Returns	Gompertz	Log-logistic	Makeham	PWE
srv_<model>_log_hazard	$\log h(t)$	✓	✓	✓	✓
srv_<model>_cumulative_hazard	$H(t)$	✓	✓	✓	✓
srv_<model>_lpdf	Summed right-censored log likelihood	✓	✓	✓	srv_pwe_loglik_sum
srv_<model>_lccdf	Summed log survival, $\sum \log S(t_i)$	✓	✓	✓	—
srv_<model>_loglik	Per-observation log likelihood (for LOO / WAIC)	✓	✓	✓	✓
srv_<model>_survival_curve	$S(t)$	✓	✓	✓	✓
srv_<model>_rng	One simulated event time	✓	✓	✓	—

The piecewise-exponential model adds two data-preparation helpers, meant to be called once in transformed data:

Function	Returns
srv_pwe_interval(t, starts)	Interval index of each observation: the largest $j$ with $s_j < t$
srv_pwe_exposure(t, starts)	$N \times J$ matrix of time spent at risk in each interval

Most functions are overloaded so shared parameters can be passed as real and per-observation parameters as vector. Every function carries @brief, @param, @return, @example, and @math documentation, so you can read it from the terminal without leaving your model:

laplace doc survival::srv_gompertz_lpdf

Argument order

Argument order is not identical across models — in particular, where the event indicator d goes — so here it is in one place:

Model	Hazards / survival curve	lpdf	lccdf	loglik	rng
Gompertz	(t, gamma, eta)	(t | gamma, eta, d)	(t | gamma, eta)	(t, d, gamma, eta)	(gamma, eta)
Log-logistic	(t, alpha, beta)	(t | alpha, beta, d)	(t | alpha, beta)	(t, d, alpha, beta)	(alpha, beta)
Makeham	(t, a, gamma, eta)	(t | a, gamma, eta, d)	(t | a, gamma, eta)	(t, a, gamma, eta, d)	(log_c, gamma, eta)
PWE	(interval, log_lambda, eta) / (E, log_lambda, eta)	loglik_sum(interval, E, d, log_lambda, eta)	—	(interval, E, d, log_lambda, eta)	—

Installation

survival is distributed as a git-hosted Laplace library — there's no published registry entry yet, so it's added by pointing laplace (or cmdlaplacer, if you're working from R) directly at the repository. The package lives in the repository's laplace/ subdirectory, so pass it as the subdir.

Via the laplace CLI

From inside a Laplace project (a directory with its own laplace.toml):

laplace add survival --git https://github.com/mlatinov/laplace-survival --tag 0.1.0 --subdir laplace

Via R (cmdlaplacer)

library(cmdlaplacer)

laplace_install_git(
  "survival",
  "https://github.com/mlatinov/laplace-survival",
  tag = "0.1.0",
  subdir = "laplace"
)

Either way, this pins the dependency in your project's laplace.toml/laplace.lock at tag 0.1.0. Check the releases for newer tags as they become available.

Usage

Import the library in a library { } block and call its functions with the survival:: namespace prefix.

Gompertz proportional-hazards model

A Gompertz model with covariates, right censoring, pointwise log likelihoods for loo, a baseline survival curve, and a simulated event time:

library {
    import survival
}

data {
  int<lower=1> N;                      // subjects
  int<lower=1> K;                      // covariates
  vector<lower=0>[N] t;                // follow-up time
  vector<lower=0, upper=1>[N] d;       // 1 = event, 0 = censored
  matrix[N, K] X;
  int<lower=1> G;
  vector<lower=0>[G] t_grid;           // times for the survival curve
}

parameters {
  real eta0;                           // baseline log hazard at t = 0
  vector[K] beta;                      // log hazard ratios
  real gamma;                          // change in log hazard per unit time
}

model {
  vector[N] eta = eta0 + X * beta;

  eta0 ~ normal(-3, 2);
  beta ~ normal(0, 1);
  gamma ~ normal(0, 0.5);

  target += survival::srv_gompertz_lpdf(t | gamma, eta, d);
}

generated quantities {
  vector[N] log_lik = survival::srv_gompertz_loglik(t, d, gamma, eta0 + X * beta);
  vector[G] S_baseline = survival::srv_gompertz_survival_curve(t_grid, gamma, rep_vector(eta0, G));
  real t_new = survival::srv_gompertz_rng(gamma, eta0);
}

Swapping in another parametric model is a matter of changing the function family and its parameters, e.g. survival::srv_log_logistics_lpdf(t | alpha, beta, d) or survival::srv_makeham_lpdf(t | a, gamma, eta, d).

Piecewise-exponential model

The interval lookup and exposure matrix depend only on data, so they are computed once in transformed data:

library {
    import survival
}

data {
  int<lower=1> N;
  int<lower=1> K;
  int<lower=1> J;                      // number of intervals
  vector<lower=0>[N] t;
  vector<lower=0, upper=1>[N] d;
  matrix[N, K] X;
  vector<lower=0>[J] starts;           // interval start points, starts[1] = 0
}

transformed data {
  array[N] int interval = survival::srv_pwe_interval(t, starts);
  matrix[N, J] E = survival::srv_pwe_exposure(t, starts);
}

parameters {
  vector[J] log_lambda;                // baseline log hazard per interval
  vector[K] beta;
}

model {
  log_lambda ~ normal(-3, 1);
  beta ~ normal(0, 1);

  target += survival::srv_pwe_loglik_sum(interval, E, d, log_lambda, X * beta);
}

generated quantities {
  vector[N] log_lik = survival::srv_pwe_loglik(interval, E, d, log_lambda, X * beta);
}

From R

With cmdlaplacer, the .laplace file compiles straight to a cmdstanr model, and the generated .stan file stays on disk next to it:

library(cmdlaplacer)

mod <- laplace_model("gompertz.laplace")
fit <- mod$sample(data = stan_data)

fit$loo()   # uses the log_lik vector from generated quantities

Things to know

• Use target +=, not ~. Laplace rewrites survival::func( calls, so write target += survival::srv_gompertz_lpdf(t | ...). The t ~ survival::srv_gompertz(...) form won't resolve.
• The lpdf functions already handle censoring through d, so pass every observation — events and censored — in one call. The lccdf functions are for the alternative pattern where censored rows are passed separately; never use both for the same rows, or censored observations are counted twice.
• d is a vector, not an integer array. If your indicator is array[N] int, convert it once in transformed data with to_vector(...).
• Right censoring only. Left censoring, interval censoring, and delayed entry (left truncation) are not supported yet.
• Log-logistic times must be strictly positive, since the hazard involves $\log t$.
• Gompertz with $\gamma < 0$ implies a cure fraction: some subjects never experience the event, and srv_gompertz_rng returns positive_infinity() for them. Account for this when summarising simulated times.
• srv_makeham_rng takes the background hazard on the log scale (log_c), unlike the other Makeham functions, which take a on the natural scale. Pass log(a).
• PWE intervals should start at 0 (starts[1] = 0). A time exactly on a boundary belongs to the earlier interval, and the last interval is open-ended.
• Survival curves take one eta per time point. For a single covariate profile over a time grid, use rep_vector(eta0, G).

License

See LICENSE.