Probability distribution functions

library(anovir)

Parameters

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)

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.