QTECox: Function to obtain QTE from a Cox model

Description

Computes quantile treatment effects comparable to those of crq model from a coxph object.

Usage

QTECox(x, smooth = TRUE)

Arguments

An object of class coxph produced by coxph.

smooth

Logical indicator if TRUE (default) then Cox survival function is smoothed.

Value

tauspoints of evaluation of the QTE.
QTEmatrix 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.

Details

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.

References