The QUANTLIFE Procedure

Relationship of Quantile Function and Survival Function

Both quantile function and survival function are useful in characterizing a lifetime distribution.

By the definition of the quantile function $Q_{T} (\tau |x)$,

\begin{equation*}  F(Q_{T} (\tau |x))=P(T \le Q_{T} (\tau |x) )= \tau \end{equation*}

In other words, the cumulative distribution function $F_ T(t|x)$ maps $Q_{T} (\tau |x)$ to $\tau $, and thus the corresponding survival function $S_ T(t|x)$ maps $Q_{T} (\tau |x)$ to $1-\tau $.

When you specify the LOG option, the QUANTLIFE procedure fits a linear quantile regression model for a log transformation of the lifetime as

\[ Q_{\mr{log}(T)} (\tau |x)=x’\bbeta (\tau ) \]

where $Q_{\mr{log}(T)} (\tau |x)$ is the $\tau $th quantile of $\mbox{log}(T)$ at x. The estimated quantile function for $T$ given x is $\hat Q_{T} (\tau |x)=e^{x'\hat\bbeta (\tau )}$, because the quantile function is invariant under a monotone transformation.

You can specify the covariates x in the COVARIATES= data set of the BASELINE statement and the PLOTS=(QUANTILE SURVIVAL) option in the PROC statement. Then the conditional quantile function at x is plotted as $\hat Q_{T} (\tau |x)$ against $\tau $, and the conditional survival function at x is plotted as $1-\tau $ against $\hat Q_{T} (\tau |x)$.