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# survstan

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<!-- badges: end --> 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](https://CRAN.R-project.org) with:

``` r
install.packages("survstan")
```

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

``` r
# 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 $-\infty < \varphi < \infty$\$.

<!-- When $\varphi = 0$, $f(t | \mu, \sigma, \varphi)$ reduces to the f.d.p of the lognormal distribution. -->

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.
