Concordance is defined as Pr(agreement) for any two randomly chosen
observations, where in this case agreement means that the observation
with the shorter survival time of the two
also has the larger risk score.
The predictor (or risk score) will often be the result of
a Cox model or other regression.For continuous covariates concordance is equivalent to Kendall's tau,
and for logistic regression is is equivalent to the area under the ROC
curve. A value of 1 signifies perfect agreement, .6-.7
a common result for survival data, .5 an agreement that is no
better than chance, and .3-.4 the performace of some stock market analysts.
The computation involves all n(n-1)/2 pairs of data points in the sample.
For survival data, however, some of the pairs are incomparable.
For instance a pair of times (5+, 8), the first being a censored value.
We do not know whether the first survival time is greater than or less than
the second.
Among observations that are comparable, pairs may also be tied on
survival time (but only if both are uncensored) or on the predictor.
The final concondance is (agree + tied/2)/(agree + disagree + tied).
There is, unfortunately, one aspect of the formula above that is unclear.
Should the count of ties include observations that are tied on survival time y,
tied on the predictor x, or both?
In 2 by 2 tables, Kendall's tau-b is equal to (agree - disagree)/ n,
where n is the geometric mean of (agree + disagree + tied.x) and
(agree + disagree + tied.y). The Goodman-Kruskal Gamma statistic is
(agree-disagree)/(agree + disagree), ignoring ties. It ranges from -1 to
+1 similar to a correlation coefficient.
In this program we count those observations which were tied on x, but not
tied on y (survival time), as `ties'. All of the components are returned
in the result, however, so people can compute other combinations if interested.
(If two observations have the same survival and the same x, they are counted
in the tied.time category).