---
title: "Probability distribution functions"
output: rmarkdown::html_vignette
bibliography: references.bib
csl: proc_b.csl
vignette: >
  %\VignetteIndexEntry{Probability distribution functions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
header-includes: 
  \usepackage{amsmath}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(anovir)
```

### Parameters {#top}

The expressions for the probability distributions used in this package are given below. 
In each case, at time _t_,

* _S_(_t_) is the cumulative survival function,

* _f_(_t_) is the probability distribution function,

* _h_(_t_) is the hazard function, where

* _a_ is the location parameter, and 

* _b_ the scale parameter.

### Weibull distribution

\(\quad S(t) = \exp \left[ - \exp (z) \right]\)

\(\quad f(t) = \frac{1}{bt} \exp \left[ z - \exp(z)  \right]\)

\(\quad h(t) = \frac{1}{bt} \exp \left[ z  \right]\)

where

\(\quad z = \frac{\log t - a}{b} \\ \)


### Gumbel distribution

\(\quad S(t) = \exp \left[ - \exp (z) \right]\)

\(\quad f(t) = \frac{1}{b} \exp \left[ z - \exp(z)  \right]\)

\(\quad h(t) = \frac{1}{b} \exp \left[ z  \right]\)

where

\(\quad z = \frac{t - a}{b} \\\)

### Fréchet distribution

\(\quad S(t) = 1 - \exp \left[ - \exp (-z) \right]\)

\(\quad f(t) = \frac{1}{bt} \exp \left[ -z - \exp(-z)  \right]\)

\(\quad h(t) = \frac{f(t)}{S(t)}\)

where

\(\quad z = \frac{\log t - a}{b}\)

### Exponential distribution

This is a special case of the Weibull distribution when _b = 1_

\(\quad S(t) = \exp \left[ - \exp (z) \right]\)

\(\quad f(t) = \frac{1}{t} \exp \left[ z - \exp(z)  \right]\)

\(\quad h(t) = \frac{1}{t} \exp \left[ z  \right]\)

where

\(\quad z = \log t - a\)

The hazard function is constant over time at, _h_(_t_) = _exp_(_-a_)


#### [back to top](#top)

### Notes

The Weibull distribution is frequently expressed in terms of _scale_ and _shape_ parameters.


\(\quad S(t) = \exp \left[ - \left( \frac{t}{\alpha} \right) ^{\beta} \right]\)

\(\quad f(t) = \frac{\beta}{\alpha} \left( \frac{t}{\alpha} \right)^{\beta-1}  \exp \left[ - \left( \frac{t}{\alpha} \right) ^{\beta} \right]\)

\(\quad h(t) = \frac{\beta}{\alpha} \left( \frac{t}{\alpha} \right)^{\beta-1} \\\)

where \(\alpha\) is the scale parameter and \(\beta\) the shape parameter. 

This can be converted to the location and scale form with the substitutions, \(\alpha\) = _exp_(_a_) and \(\beta\) = 1/_b_

\(S(t) = \exp \left( - \left[ \frac{t}{\alpha} \right]^\beta \right)\)

\(S(t) = \exp \left( - \left[ \frac{\exp\left(\log t\right)}{\exp(a)} \right]^\beta \right)\)

\(S(t) = \exp \left( - \left[ \exp \left( \log t - a \right) \right]^\beta \right)\)

\(S(t) = \exp \left( - \left[ \exp \left( \log t - a \right) \right]^{1/b} \right)\)

\(S(t) = \exp \left( - \left[ \exp \left( \frac{\log t - a}{b} \right) \right] \right) \\ \)


### Linear transformations of cumulative survival 

The cumulative survival functions of the Weibull, Gumbel, Fréchet and exponential distributions can be tranformed into linear functions of time;

\begin{align}
\log\left(-\log \left[S \left(t \right) \right] \right) &= \frac{1}{b} \log t - \frac{a}{b} \\ \\
\log\left[-\log S \left(t \right) \right] &= \frac{1}{b} t - \frac{a}{b} \\  \\
-\log\left(-\log \left[1 - S \left(t \right) \right] \right) &= \frac{1}{b} \log t - \frac{a}{b} \\ \\
\log\left(-\log \left[S \left(t \right) \right] \right) &= \log t - a \\ \\
\end{align}

where _a_ and _b_ are the location and scale parameters, respectively. 

If plotting transformed cumulative survival data against the appropriate measure of time yields an approximately linear relationship, it indicates the data follow the particular probability distribution in question. 
When this is the case, the coefficients for the intercept and slope of a linear regression can be used to get approximate estimates of the distributions' location and scale parameters.




#### [back to top](#top)