Computes quantile treatment effects comparable to those of crq model from a coxph object.
QTECox(x, smooth = TRUE)
points of evaluation of the QTE.
matrix of QTEs, the ith column contains the QTE for the
ith covariate effect. Note that there is no intercept effect.
see plot.summary.crqs for usage.
An object of class coxph produced by coxph.
Logical indicator if TRUE (default) then Cox survival function is smoothed.
Roger Koenker Stephen Portnoy & Tereza Neocleous
Estimates of the Cox QTE, \(\frac{dQ(t|x)}{dx_{j}}\) at \(x=\bar{x}\), can be expressed as a function of t as follows:
$$\frac{dQ(t|x)}{dx_{j}}=\frac{dt}{dx_{j}}\frac{dQ(t|x)}{dt}$$
The Cox survival function, \(S(y|x)=\exp \{-H_{0}(y)\exp (b^{\prime }x)\}\)
$$\frac{dS(y|x)}{dx_{j}}=S(y|x)log \{S(y|x)\}b_{j}$$
where \(\frac{dQ(t|x)}{dx_{j}}\)
can be estimated by \(\frac{\Delta (t)}{\Delta (S)}
(1-t)\)
where $S$ and $t$ denote the surv and time components
of the survfit object.
Note that since \(t=1-S(y|x)\), the above is the
value corresponding to the argument $(1-t)$; and furthermore
$$\frac{dt}{dx_{j}}=-\frac{dS(y|x)}{dx_{j}}=-(1-t) log (1-t)b_{j}$$
Thus the QTE at the mean of x's is:
$$(1-S)= \frac{\Delta (t)}{\Delta (S)}S ~log (S)b_{j}$$
Since \(\Delta S\) is negative and $log (S)$ is also negative this has the same sign as \(b_{j}\) The crq model fits the usual AFT form Surv(log(Time),Status), then
$$\frac{d log (Q(t|x))}{dx_{j}}=\frac{dQ(t|x)}{dx_{j}}/ Q(t|x)$$
This is the matrix form returned.
Koenker, R. and Geling, O. (2001). Reappraising Medfly longevity: a quantile regression survival analysis, J. Amer. Statist. Assoc., 96, 458-468
crq