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Hazard functions

Posted by ubpdqn on February 14, 2011

Most of my posts are exercises  to increase my understanding of the topic. This post is no exception.

The hazard function h(t)  can be conceptualized as the conditional probability of failure (or other relevant event)  in an infinitesimal time \delta t,  given survival to time t.

h(t)= \lim_{\delta t\rightarrow 0} P(t<T<t+\delta t | T>t) =\frac{f(t)}{1-F(t)}=\frac{f(t)}{S(t)}

where f(t) denotes the probability density function of the random variable time to failure or event, F(t) represents the corresponding cumulative distribution function and S(t) denotes the survival function.

Noting, f(t)= - S'(t)

yields:

\ln(S(t))=-\int _0^t h(s) ds or equivalently,

S(t)=e^{-\int_0^t h(s) ds}

I find it instructional to explore the effects of different hazard functions:

Constant hazard function:

h(t)=\nu.

This yields S(t)= e^{-\nu t}\leftrightarrow F(t) = 1-e^{-\nu t}, i.e. exponential distribution of time to failure.

Weibull distribution of time to failure or event

The Weibull probability density function (f_W(t)) is frequently used to model hazard function.

f_W (t)= \frac{\alpha}{\beta}{(\frac{t}{\beta})}^{\alpha -1} e^{-{(\frac{t}{\beta})}^{\alpha}}

This yields:

h_W(t)= \frac{\alpha}{\beta}{(\frac{t}{\beta})}^{\alpha-1}

or equivalently:

\ln (h_W(t)) = \rho + \mu \ln (t)

where

\rho = \ln \alpha -\alpha\ln\beta

\mu = \alpha - 1

\alpha < 1 yields a decreasing hazard function

\alpha =1 reduces to constant hazard

\alpha > 1 is an increasing hazard function

Bathtub hazard function

The bathtub function is so named for the shape of the function. It is commonly considered as a hazard function, i.e. decreasing hazard->constant hazard->increasing hazard.

A piecewise continuous linear function can be used, e.g.

h(t)

= c_1 - c_0 t + \lambda , 0\leq t \leq c_0/c_1

= \lambda , c_1/c_0 < t \leq t_0

= c_2 (t-t_0) + \lambda , t > t_0

An alternative model is to mix Weibull hazard functions:

h(t) = h_W(t, \alpha_1,\beta_1)+h_W(t,\alpha_2,\beta_2)

where \alpha_1<1  and \alpha_2>1

Cox Proportional Hazard Model
Cox proportional hazard model assumes a hazard function of the form:
\ln(h_i(t)) = \alpha (t) + \sum_{j=1}^n \beta_j x_{ij}

The covariates are assumed to be independent of time. Baseline hazard function can be defined as:

\ln (h_0(t)) =\alpha (t)

Hazard ratio for  each unit covariate k, is

\frac{h_k(t)}{h_0(t)}=e^{\beta_k}

leading to the justification of proportional hazard.

If we let S_0(t) =e^{-\int_0^t h_0(s) ds}, then

S(t)= e^{-e^{x^T\beta}\int_0^t h_0(s) ds}=S_0(t)^{e^{x^T\beta}}

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