16.1 Background

You can specify the survival distribution function either as a survival function of the form

\[S(t) = 1 - F(t) = pr(T > t), \hspace{3mm} 0 < t < \infty\]

or as a hazard function, the instantaneous failure rate given survival up to time \(t\)

\[h(t) = \lim_{\delta \rightarrow 0}{\frac{pr(t < T < t + \delta|T > 1)}{\delta}}.\]

The survival function is the compliment of the the cumulative distribution function. The Kaplan-Meier estimator for the survival function is

\[\hat{S} = \prod_{i: t_i < t}{\frac{n_i - d_i}{n_i}}\]

where \(n_i\) is the number of persons under observation at time \(i\) and \(d_i\) is the number of individuals dying at time \(i\). Calculate the Kaplan-Meier estimate with the survfit() function.

library(survival)



tm <- c(

    0, # birth

    1 / 365,  # first day of life

    7 / 365,  # seventh day of life

    28 / 365, # fourth week of life

    1:110     # subsequent years

)

hazMale = survexp.us[, "male", "2004"]

hazFemale = survexp.us[, "female", "2004"]

#plot(tm, hazMale)