[R] Concordance Index - interpretation

K F Pearce k.f.pearce at newcastle.ac.uk
Fri Dec 12 11:06:59 CET 2008

Hello everyone.
This is a question regarding generation of the concordance index (c
index) in R using the function rcorr.cens.  In particular about
interpretation of its direction and form of the 'predictor'.
One of the arguments is a "numeric predictor variable" ( presumably
this is just a *single* predictor variable).  Say this variable takes
numeric values....  Am I correct in thinking that if the c index is >
0.5 (with Somers D positive) then  this tells us that the higher the
numeric values of the 'predictor', the  greater the survival probability
and similarly if the c index is <0.5 (with Somers D negative) then  this
tells us that the higher the numeric values of the 'predictor' the
lower  the survival probability ?
The c index  estimates the "probability of concordance between predicted
and observed responses"....Harrel et al (1996) says "in predicting time
until death, concordance is calculated by considering all possible pairs
of patients, at least one of whom has died.  If the *predicted* survival
time (probability) is larger for the patient who (actually) lived
longer, the predictions for that pair are said to be concordant with the
(actual) outcomes.  ".  I have read that "the c index is defined by the
proportion of all usable patients in which the predictions and outcomes
are concordant".
Now, secondly, I'd like to ask what form the predictor can take.
Presumably if the predictor was a continuous-type variable e.g. 'age'
then predicted survival probability (calculated internally via Cox
regression?) would be compared with actual survival time for each
specific age to get the c index?  Now, if the predictor was an *ordinal
categorical variable* where 1=worst group and 5=best group - I presume
that the c index would be calculated similarly but this time there would
be many ties in the predictor (as regards predicted survival
probability) - hence  if I wanted to count all ties in such a case I
would keep the default argument outx=FALSE? 

Does anyone have a clear reference which gives the formula used to
generate the concordance index (with worked examples)? 
Many thanks for your help on these interpretations
Kind Regards,

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