The QUANTLIFE Procedure (Experimental)

Quantile Regression

Suppose a data set contains observations $(Y_ i, x_ i)$, where $Y_ i$ is a dependent variable of interest (such as the survival time or some monotone transformation of the survival time) and $x_ i$ is a $p\times 1$ vector of covariates.

You can use regression analysis to explore the relationship between a response $Y_ i$ and its predictor $x_ i$. Classical linear regression estimates the conditional mean function $E(Y_ i|x_ i)$ with a linear predictor $x_ i^{\prime } \beta $; a linear quantile regression estimates the $\tau $th conditional quantile function $Q_{\tau }(Y_ i|x_ i)$ with a different linear predictor $x_ i^{\prime } \beta (\tau )$, where the quantile level $\tau $ ranges between 0 and 1. For example, $x_ i^{\prime } \beta (0.95)$ is the linear predictor for the 0.95th quantile (commonly referred to as the 95th percentile).

The quantile regression coefficient ${\beta }(\tau )$ can be estimated by minimizing the following objective function over b:

\[  r(b) = \sum _{i=1}^ n \rho _\tau ( Y_ i - x_ i^{\prime } \,  b ) \, ,  \]

The loss function $\rho _\tau ( u )$ is defined as $u ( \tau - I(u<0))$, in contrast to the square loss function for classical linear regression.

When $\tau =0.5$, the coefficient $\bbeta (0.5)$ minimizes the sum of absolute residuals, which corresponds to median regression (or $L_1$ regression).

The following set of regression quantiles is referred to as the quantile process, and it completely describes the conditional distribution of $Y_ i$ given the predictor $x_ i$:

\[  \{ \beta (\tau ): \tau \in (0, 1) \}   \]

When all the observations are observed, you can use the QUANTREG procedure to estimate the quantile function $Q(\tau |X=x)$ and draw statistical inference about the regression parameters $ \beta (\tau )$. See Chapter 77: The QUANTREG Procedure, for details.

However, when the observations are incomplete, as is the case with censored data in survival analysis, the classical quantile regression method is not appropriate. The QUANTLIFE procedure implements appropriate quantile regression methods to model the relationship between the response $Y_ i$ and the predictor $x_ i$.