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 $\hat Q_{T} (\tau |x)$ .

The QUANTLIFE Procedure

Relationship of Quantile Function and Survival Function