Utility functions

Since survival analysis regularly defines things in term of hazard of (positive) random variables, we defined the following functions here purely for convenience:

Hazard function

SurvivalDistributions.hazard — Function

hazard(X::UnivariateDistribution, t)

Provide the hazard fucntion of the random variable X (supposed to be a Distributions.ContinuousUnivariateDistributions) at point t. The default implementation is simply $vh(t) = \frac{f(t)}{S(t)}$ where $f$ and $S$ are the density and survival function of X.

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Log Hazard Function

SurvivalDistributions.loghazard — Function

loghazard(X, t)

Provide the log of the hazard fucntion of the random variable X. See hazard for the formal definition.

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Cumulative Hazard Function

SurvivalDistributions.cumhazard — Function

loghazard(X::UnivariateDistribution, t)

Provide the cumulative hazard function of the random variable X. It is defined as $H(t) = - \ln S(t)$ where $S$ is the survival fucntion of the random variable X.

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Censored Loglikelyhood.

SurvivalDistributions.censored_loglikelihood — Function

censored_loglikelihood(X::UnivariateDistribution, t, δ)

Provide the censored logliklyhood of the distribution X at point t, with status indicatrix δ. Is if defined as

\[δ * loghazard(X,t) - cumhazard(X,t)\]

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Examples of hazard and cumulative hazard functions

The hazard and the cumulative hazard functions play a crucial role in survival analysis. These functions define the likelihood function in the presence of censored observations. Thus, they are important in many context.

LogNormal

using SurvivalDistributions, Distributions, Plots, StatsBase
function mkplot(d)
    h = plot(t -> hazard(d, t), xlims = (0,10),  ylabel = "Hazard")
    H = plot(t -> cumhazard(d,t), xlims = (0,10), ylabel = "Cumulative Hazard",)
    return plot(h,H,plot_title = "$d")
end
mkplot(LogNormal(0.5, 1))

LogLogistic

mkplot(LogLogistic(1, 0.5))

Weibull

mkplot(Weibull(3, 0.5))

Gamma

mkplot(Gamma(3, 0.5))