[R] survConcordance with 'counting' type Surv()

[Terry Therneau]

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I was impressed to discover that the 'survConcordance' now handles
Surv() objects in counting format (example below to clarify what I
mean). This is not documented in the help page for the function. I am
very curious to see how a c-index is estimated in this case, using just
the linear predictors. It was my impression that with left truncation
the ordering of individuals might change over time as the baseline
hazard is no longer monotonic. So ordering based on just the linear
predictors would not be appropriate but risk to t should be used (after
choosing a t). Am I wrong?

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  This is a case where the software is ahead of the paper (still being
written).  Note also that summary(coxph) now prints the concordance
stististic by default.  The heart of the extension is: 
   a. consider a Cox model with the time dependent covariate rank(x),
where ranks are recomputed anew at each death time using only those
subjects still at risk at that death time
   b. the Cox score statistic under H0 for such a model is (C-D)/2,
where C and D are the number of concordant and discordant pairs, as
defined in the "concordance" papers. (Breslow approximation for ties).  
   c. this gives a natural extension of the concordance measure to
time-dependent covariates, subjects dropping into and out of the risk
sets, etc. etc.  
  d. It also revealed a path towards an O(n log n) computation for
concordance, as opposed to the usual O(n^2) one.  Compare the timing of
survConcordance to rcorr.cens on a really large data set....

  For those who want lots more details see the source code.  This part
was written using noweb to generate both a latex document and the R
source.

  Terry Therneau