# survstan

The aim of the R package survstan is to provide a toolkit for fitting survival models using Stan.

The R package survstan can be used to fit right-censored survival data under independent censoring. The implemented models allow the fitting of survival data in the presence/absence of covariates. All inferential procedures are currently based on the maximum likelihood (ML) approach.

## Installation

You can install the released version of survstan from CRAN with:

install.packages("survstan")

You can install the development version of survstan from GitHub with:

# install.packages("devtools")
devtools::install_github("fndemarqui/survstan")

## Inference procedures

Let $$(t_{i}, \delta_{i})$$ be the observed survival time and its corresponding failure indicator, $$i=1, \cdots, n$$, and $$\boldsymbol{\theta}$$ be a $$k \times 1$$ vector of parameters. Then, the likelihood function for right-censored survival data under independent censoring can be expressed as:

$L(\boldsymbol{\theta}) = \prod_{i=1}^{n}f(t_{i}|\boldsymbol{\theta})^{\delta_{i}}S(t_{i}|\boldsymbol{\theta})^{1-\delta_{i}}.$

The maximum likelihood estimate (MLE) of $$\boldsymbol{\theta}$$ is obtained by directly maximization of $$\log(L(\boldsymbol{\theta}))$$ using the rstan::optimizing() function. The function rstan::optimizing() further provides the hessian matrix of $$\log(L(\boldsymbol{\theta}))$$, needed to obtain the observed Fisher information matrix, which is given by:

$\mathscr{I}(\hat{\boldsymbol{\theta}}) = -\frac{\partial^2}{\partial \boldsymbol{\theta}\boldsymbol{\theta}'} \log L(\boldsymbol{\theta})\mid_{\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}},$

Inferences on $$\boldsymbol{\theta}$$ are then based on the asymptotic properties of the MLE, $$\hat{\boldsymbol{\theta}}$$, that state that:

$\hat{\boldsymbol{\theta}} \asymp N_{k}(\boldsymbol{\theta}, \mathscr{I}^{-1}(\hat{\boldsymbol{\theta}})).$

## Baseline Distributions

Some of the most popular baseline survival distributions are implemented in the R package survstan. Such distributions include:

• Exponential
• Weibull
• Lognormal
• Loglogistic
• Gamma,
• Generalized Gamma (original Stacy’s parametrization)
• Generalized Gamma (alternative Prentice’s parametrization)
• Gompertz
• Rayleigh
• Birnbaum-Saunders (fatigue)

The parametrizations adopted in the package survstan are presented next.

### Exponential Distribution

If $$T \sim \mbox{Exp}(\lambda)$$, then

$f(t|\lambda) = \lambda\exp\left\{-\lambda t\right\}I_{[0, \infty)}(t),$ where $$\lambda>0$$ is the rate parameter.

The survival and hazard functions in this case are given by:

$S(t|\lambda) = \exp\left\{-\lambda t\right\}$ and $h(t|\lambda) = \lambda.$

### Weibull Distribution

If $$T \sim \mbox{Weibull}(\alpha, \gamma)$$, then

$f(t|\alpha, \gamma) = \frac{\alpha}{\gamma^{\alpha}}t^{\alpha-1}\exp\left\{-\left(\frac{t}{\gamma}\right)^{\alpha}\right\}I_{[0, \infty)}(t),$ where $$\alpha>0$$ and $$\gamma>0$$ are the shape and scale parameters, respectively.

The survival and hazard functions in this case are given by:

$S(t|\alpha, \gamma) = \exp\left\{-\left(\frac{t}{\gamma}\right)^{\alpha}\right\}$ and $h(t|\alpha, \gamma) = \frac{\alpha}{\gamma^{\alpha}}t^{\alpha-1}.$

### Lognormal Distribution

If $$T \sim \mbox{LN}(\mu, \sigma)$$, then

$f(t|\mu, \sigma) = \frac{1}{\sqrt{2\pi}t\sigma}\exp\left\{-\frac{1}{2}\left(\frac{log(t)-\mu}{\sigma}\right)^2\right\}I_{[0, \infty)}(t),$ where $$-\infty < \mu < \infty$$ and $$\sigma>0$$ are the mean and standard deviation in the log scale of $$T$$.

The survival and hazard functions in this case are given by:

$S(t|\mu, \sigma) = \Phi\left(\frac{-log(t)+\mu}{\sigma}\right)$ and $h(t|\mu, \sigma) = \frac{f(t|\mu, \sigma)}{S(t|\mu, \sigma)},$ where $$\Phi(\cdot)$$ is the cumulative distribution function of the standard normal distribution.

### Loglogistic Distribution

If $$T \sim \mbox{LL}(\alpha, \gamma)$$, then

$f(t|\alpha, \gamma) = \frac{\frac{\alpha}{\gamma}\left(\frac{t}{\gamma}\right)^{\alpha-1}}{\left[1 + \left(\frac{t}{\gamma}\right)^{\alpha}\right]^2}I_{[0, \infty)}(t), ~ \alpha>0, \gamma>0,$

where $$\alpha>0$$ and $$\gamma>0$$ are the shape and scale parameters, respectively.

The survival and hazard functions in this case are given by:

$S(t|\alpha, \gamma) = \frac{1}{1+ \left(\frac{t}{\gamma}\right)^{\alpha}}$ and $h(t|\alpha, \gamma) = \frac{\frac{\alpha}{\gamma}\left(\frac{t}{\gamma}\right)^{\alpha-1}}{1 + \left(\frac{t}{\gamma}\right)^{\alpha}}.$

### Gamma Distribution

If $$T \sim \mbox{Gamma}(\alpha, \lambda)$$, then

$f(t|\alpha, \lambda) = \frac{\lambda^{\alpha}}{\Gamma(\alpha)}t^{\alpha-1}\exp\left\{-\lambda t\right\}I_{[0, \infty)}(t),$

where $$\Gamma(\alpha) = \int_{0}^{\infty}u^{\alpha-1}\exp\{-u\}du$$ is the gamma function.

The survival function is given by

$S(t|\alpha, \lambda) = 1 - \frac{\gamma^{*}(\alpha, \lambda t)}{\Gamma(\alpha)},$ where $$\gamma^{*}(\alpha, \lambda t)$$ is the lower incomplete gamma function, which is available only numerically. Finally, the hazard function is expressed as:

$h(t|\alpha, \lambda) = \frac{f(t|\alpha, \lambda)}{S(t|\alpha, \lambda)}.$

### Generalized Gamma Distribution (original Stacy’s parametrization)

If $$T \sim \mbox{ggstacy}(\alpha, \gamma, \kappa)$$, then

$f(t|\alpha, \gamma, \kappa) = \frac{\kappa}{\gamma^{\alpha}\Gamma(\alpha/\kappa)}t^{\alpha-1}\exp\left\{-\left(\frac{t}{\gamma}\right)^{\kappa}\right\}I_{[0, \infty)}(t),$ for $$\alpha>0$$, $$\gamma>0$$ and $$\kappa>0$$.

It can be show that the survival function can be expressed as:

$S(t|\alpha, \gamma, \kappa) = S_{G}(x|\nu, 1),$ where $$x = \displaystyle\left(\frac{t}{\gamma}\right)^\kappa$$, and $$F_{G}(\cdot|\nu, 1)$$ corresponds to the distribution function of a gamma distribution with shape parameter $$\nu = \alpha/\gamma$$ and scale parameter equals to 1.

Finally, the hazard function is expressed as:

$h(t|\alpha, \gamma, \kappa) = \frac{f(t|\alpha, \gamma, \kappa)}{S(t|\alpha, \gamma, \kappa)}.$

### Generalized Gamma Distribution (alternative Prentice’s parametrization)

If $$T \sim \mbox{ggprentice}(\mu, \sigma, \varphi)$$, then

$f(t | \mu, \sigma, \varphi) = \begin{cases} \frac{|\varphi|(\varphi^{-2})^{\varphi^{-2}}}{\sigma t\Gamma(\varphi^{-2})}\exp\{\varphi^{-2}[\varphi w - \exp(\varphi w)]\}I_{[0, \infty)}(t), & \varphi \neq 0 \\ \frac{1}{\sqrt{2\pi}t\sigma}\exp\left\{-\frac{1}{2}\left(\frac{log(t)-\mu}{\sigma}\right)^2\right\}I_{[0, \infty)}(t), & \varphi = 0 \end{cases}$ where $$w = \frac{\log(t) - \mu}{\sigma}$$, for $$-\infty < \mu < \infty$$, $$\sigma>0$$ and $$-\infty < \varphi < \infty$$\$.

It can be show that the survival function can be expressed as:

$S(t|\mu, \sigma, \varphi) = \begin{cases} S_{G}(x|1/\varphi^2, 1), & \varphi > 0 \\ 1-S_{G}(x|1/\varphi^2, 1), & \varphi < 0 \\ S_{LN}(x|\mu, \sigma), & \varphi = 0 \end{cases}$ where $$x = \frac{1}{\varphi^2}\exp\{\varphi w\}$$, $$S_{G}(\cdot|1/\varphi^2, 1)$$ is the distribution function of a gamma distribution with shape parameter $$1/\varphi^2$$ and scale parameter equals to 1, and $$S_{LN}(x|\mu, \sigma)$$ corresponds to the survival function of a lognormal distribution with location parameter $$\mu$$ and scale parameter $$\sigma$$.

Finally, the hazard function is expressed as:

$h(t|\alpha, \gamma, \kappa) = \frac{f(t|\alpha, \gamma, \kappa)}{S(t|\alpha, \gamma, \kappa)}.$

### Gompertz Distribution

If $$T \sim \mbox{Gamma}(\alpha, \gamma)$$, then

$f(t|\alpha, \lambda) = \alpha\exp\left\{\gamma x-\frac{\alpha}{\gamma}\left(e^{\gamma x} - 1\right)\right\}I_{[0, \infty)}(t).$

The survival and hazard functions are given, respectively, by

$S(t|\alpha, \lambda) = \exp\left\{-\frac{\alpha}{\gamma}\left(e^{\gamma x} - 1\right)\right\}.$ and

$h(t|\alpha, \lambda) = \alpha\exp\{\gamma x}.$

### Rayleigh Distribution

Let $$T \sim \mbox{rayleigh}(\sigma)$$, where $$\sigma>0$$ is a scale parameter. Then, the density, survival and hazard functions are respectively given by:

$f(t|\sigma) = \frac{x}{\sigma^2}\exp\left\{-\frac{x^2}{2\sigma^2}\right\},$ $S(t|\sigma) = \exp\left\{-\frac{x^2}{2\sigma^2}\right\}$ and

$h(t|\sigma) = \frac{x}{\sigma^2}.$

### Birnbaum-Saunders (fatigue) Distribution

If $$T \sim \mbox{fatigue}(\alpha, \gamma)$$, then

$f(t|\alpha, \gamma) = \frac{\sqrt{\frac{t}{\gamma}}+\sqrt{\frac{\gamma}{t}}}{2 \alpha t}\phi\left(\sqrt{\frac{t}{\gamma}}+\sqrt{\frac{\gamma}{t}}\right)(t), ~ \alpha>0, \gamma>0,$

where $$\phi(\cdot)$$ is the probability density function of a standard normal distribution, $$\alpha>0$$ and $$\gamma>0$$ are the shape and scale parameters, respectively.

The survival function in this case is given by:

$S(t|\alpha, \gamma) =\Phi\left(\sqrt{\frac{t}{\gamma}}-\sqrt{\frac{\gamma}{t}}\right)(t)$,

where $$\Phi(\cdot)$$ is the cumulative distribution function of a standard normal distribution. The hazard function is given by $h(t|\mu, \sigma) = \frac{f(t|\alpha, \gamma)}{S(t|\alpha, \gamma)}.$

## Regression models

When covariates are available, it is possible to fit six different regression models with the R package survstan:

• accelerated failure time (AFT) models;
• proportional hazards (PH) models;
• proportional odds (PO) models;
• accelerated hazard (AH) models.
• Yang and Prentice (YP) models.
• extended hazard (EH) models.

The regression survival models implemented in the R package survstan are briefly described in the sequel. Denote by $$\mathbf{x}$$ a $$1\times p$$ vector of covariates, and let $$\boldsymbol{\beta}$$ and $$\boldsymbol{\phi}$$ be $$p \times 1$$ vectors of regression coefficients, and $$\boldsymbol{\theta}$$ a vector of parameters associated with some baseline survival distribution. To ensure identifiability of the implemented regression models, we shall assume that the linear predictors $$\mathbf{x} \boldsymbol{\beta}$$ and $$\mathbf{x} \boldsymbol{\phi}$$ do not include an intercept term.

### Accelerate Failure Time Models

Accelerated failure time (AFT) models are defined as

$T = \exp\{\mathbf{x} \boldsymbol{\beta}\}\nu,$ where $$\nu$$ follows a baseline distribution with survival function $$S_{0}(\cdot|\boldsymbol{\theta})$$ so that

$f(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = e^{-\mathbf{x} \boldsymbol{\beta}}f_{0}(te^{-\mathbf{x} \boldsymbol{\beta}}|\boldsymbol{\theta})$ and

$S(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = S_{0}(t e^{-\mathbf{x} \boldsymbol{\beta}}|\boldsymbol{\theta}).$

### Proportional Hazards Models

Proportional hazards (PH) models are defined as

$h(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = h_{0}(t|\boldsymbol{\theta})\exp\{\mathbf{x} \boldsymbol{\beta}\},$ where $$h_{0}(t|\boldsymbol{\theta})$$ is a baseline hazard function so that

$f(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = h_{0}(t|\boldsymbol{\theta})\exp\left\{\mathbf{x} \boldsymbol{\beta} - H_{0}(t|\boldsymbol{\theta})e^{\mathbf{x} \boldsymbol{\beta}}\right\},$ and

$S(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = \exp\left\{ - H_{0}(t|\boldsymbol{\theta})e^{\mathbf{x} \boldsymbol{\beta}}\right\}.$

### Proportional Odds Models

Proportional Odds (PO) models are defined as

$R(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = R_{0}(t|\boldsymbol{\theta})\exp\{\mathbf{x} \boldsymbol{\beta}\},$ where $$\displaystyle R_{0}(t|\boldsymbol{\theta}) = \frac{1-S_{0}(t|\boldsymbol{\theta})}{S_{0}(t|\boldsymbol{\theta})} = \exp\{H_{0}(t|\boldsymbol{\theta})\}-1$$ is a baseline odds function so that

$f(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = \frac{h_{0}(t|\boldsymbol{\theta})\exp\{\mathbf{x} \boldsymbol{\beta} + H_{0}(t|\boldsymbol{\theta})\}}{[1 + R_{0}(t|\boldsymbol{\theta})e^{\mathbf{x} \boldsymbol{\beta}}]^2}.$

and

$S(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = \frac{1}{1 + R_{0}(t|\boldsymbol{\theta})e^{\mathbf{x} \boldsymbol{\beta}}}.$

### Accelerated Hazard Models

Accelerated hazard (AH) models can be defined as

$h(t|\boldsymbol{\theta}, \boldsymbol{\beta},\mathbf{x}) = h_{0}\left(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta}\right)$

so that

$S(t|\boldsymbol{\theta}, \boldsymbol{\beta},\mathbf{x}) = \exp\left\{- H_{0}\left(t/ e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta}\right)e^{\mathbf{x}\boldsymbol{\beta}} \right\}$ and $f(t|\boldsymbol{\theta}, \boldsymbol{\beta}, \mathbf{x}) = h_{0}\left(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta}\right)\exp\left\{- H_{0}\left(t/ e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta}\right)e^{\mathbf{x}\boldsymbol{\beta}} \right\}.$

### Extended hazard Models

The survival function of the extended hazard (EH) model is given by:

$S(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = \exp\left\{-H_{0}(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta})\exp(\mathbf{x}(\boldsymbol{\beta} + \boldsymbol{\phi}))\right\}.$

The hazard and the probability density functions are then expressed as:

$h(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = h_{0}(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta})\exp\{\mathbf{x}\boldsymbol{\phi}\}$ and

$f(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = h_{0}(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta})\exp\{\mathbf{x}\boldsymbol{\beta}\}\exp\left\{-H_{0}(t/e^{\mathbf{x}\boldsymbol{\beta}}|\boldsymbol{\theta})\exp(\mathbf{x}(\boldsymbol{\beta}+ \boldsymbol{\phi}))\right\},$

respectively.

The EH model includes the AH, AFT and PH models as particular cases when $$\boldsymbol{\phi} = \mathbf{0}$$, $$\boldsymbol{\phi} = -\boldsymbol{\beta}$$, and $$\boldsymbol{\beta} = \mathbf{0}$$, respectively.

### Yang and Prentice Models

The survival function of the Yang and Prentice (YP) model is given by:

$S(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = \left[1+\frac{\kappa_{S}}{\kappa_{L}}R_{0}(t|\boldsymbol{\theta})\right]^{-\kappa_{L}}.$

The hazard and the probability density functions are then expressed as:

$h(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = \frac{\kappa_{S}h_{0}(t|\boldsymbol{\theta})\exp\{H_{0}(t|\boldsymbol{\theta})\}}{\left[1+\frac{\kappa_{S}}{\kappa_{L}}R_{0}(t|\boldsymbol{\theta})\right]}$ and

$f(t|\boldsymbol{\theta},\boldsymbol{\beta}, \boldsymbol{\phi}) = \kappa_{S}h_{0}(t|\boldsymbol{\theta})\exp\{H_{0}(t|\boldsymbol{\theta})\}\left[1+\frac{\kappa_{S}}{\kappa_{L}}R_{0}(t|\boldsymbol{\theta})\right]^{-(1+\kappa_{L})},$

respectively, where $$\kappa_{S} = \exp\{\mathbf{x}\boldsymbol{\beta}\}$$ and $$\kappa_{L} = \exp\{\mathbf{x}\boldsymbol{\phi}\}$$.

The YO model includes the PH and PO models as particular cases when $$\boldsymbol{\phi} = \boldsymbol{\beta}$$ and $$\boldsymbol{\phi} = \mathbf{0}$$, respectively.