The concordance statistic compute the agreement between an observed response and a predictor. It is closely related to Kendall's tau-a and tau-b, Goodman's gamma, and Somers' d, all of which can also be calculated from the results of this function.
concordance(object, ...)
# S3 method for formula
concordance(object, data, weights, subset, na.action,
cluster, ymin, ymax, timewt= c("n", "S", "S/G", "n/G2", "I"),
influence=0, ranks = FALSE, reverse=FALSE, timefix=TRUE, keepstrata=10, ...)
# S3 method for lm
concordance(object, ..., newdata, cluster, ymin, ymax,
influence=0, ranks=FALSE, timefix=TRUE, keepstrata=10)
# S3 method for coxph
concordance(object, ..., newdata, cluster, ymin, ymax,
timewt= c("n", "S", "S/G", "n/G2", "I"), influence=0,
ranks=FALSE, timefix=TRUE, keepstrata=10)
# S3 method for survreg
concordance(object, ..., newdata, cluster, ymin, ymax,
timewt= c("n", "S", "S/G", "n/G2", "I"), influence=0,
ranks=FALSE, timefix=TRUE, keepstrata=10)An object of class concordance containing the following
components:
the estimated concordance value or values
a vector containing the number of concordant pairs, discordant, tied on x but not y, tied on y but not x, and tied on both x and y
the number of observations
a vector containing the estimated variance of the concordance based on the infinitesimal jackknife (IJ) method. If there are multiple models it contains the estimtated variance/covariance matrix.
a vector containing the estimated variance(s) of the
concordance values, based on the variance formula for the associated
score test from a proportional hazards model. (This was the primary
variance used in the survConcordance function.)
optional, the vector of leverage estimates for the concordance
optional, the matrix of leverage values for each of the counts, one row per observation
optional, a data frame containing the Somers' d rank at each event time, along with the time weight, and the case weight of the observation. The time weighted sum of the ranks will equal concordant pairs - discordant pairs.
a fitted model or a formula. The formula should be of
the form y ~x or y ~ x + strata(z) with a single
numeric or survival response and a single predictor.
Counts of concordant, discordant and tied pairs
are computed separately per stratum, and then added.
a data.frame in which to interpret the variables named in
the formula, or in the subset and the weights
argument. Only applicable if object is a formula.
optional vector of case weights.
Only applicable if object is a formula.
expression indicating which subset of the rows of data should be used in
the fit. Only applicable if object is a formula.
a missing-data filter function. This is applied to the model.frame
after any subset argument has been used. Default is
options()\$na.action. Only applicable if object is a formula.
multiple fitted models are allowed. Only applicable if
object is a model object.
optional, a new data frame in which to evaluate (but not refit) the models
optional grouping vector for calculating the robust variance
compute the concordance over the restricted range ymin <= y <= ymax. (For survival data this is a time range.)
the weighting to be applied. The overall statistic is a weighted mean over event times.
1= return the dfbeta vector, 2= return the full influence matrix, 3 = return both
if TRUE, return a data frame containing the scaled ranks that make up the overall score.
if TRUE then assume that larger x values predict
smaller response values y; a proportional hazards model is
the common example of this, larger hazard = shorter survival.
correct for possible rounding error. See the vignette on tied times for more explanation. Essentially, exact ties are an important part of the concordance computatation, but "exact" can be a subtle issue with floating point numbers.
either TRUE, FALSE, or an integer value.
Computations are always done within stratum, then added. If the
total number of strata greater than keepstrata, or
keepstrata=FALSE, those subtotals are not kept in the output.
Terry Therneau
The concordance is an estimate of \(Pr(x_i < x_j | y_i < y_j)\), for a model fit replace \(x\) with \(\hat y\), the predicted response from the model. For a survival outcome some pairs of values are not comparable, e.g., censored at time 5 and a death at time 6, as we do not know if the first observation will or will not outlive the second. In this case the total number of evaluable pairs is smaller.
Relatations to other statistics: For continuous x and y, 2C- 1 is equal to Somers' d. If the response is binary, C is equal to the area under the receiver operating curve or AUC. For a survival response and binary predictor C is the numerator of the Gehan-Wilcoxon test.
A naive compuation requires adding up over all n(n-1)/2 comparisons,
which can be quite slow for large data sets.
This routine uses an O(n log(n)) algorithm.
At each uncensored event time y, compute the rank of x for the subject
who had the event as compared to the x values for all others with a longer
survival, where the rank has value between 0 and 1.
The concordance is a weighted mean of these ranks,
determined by the timewt option. The rank vector can be
efficiently updated as subjects are added to the risk set.
For further details see the vignette.
The variance is based on an infinetesimal jackknife. One advantage of this approach is that it also gives a valid covariance for the covariance based on multiple different predicted values, even if those predictions come from quite different models. See for instance the example below which has a poisson and two non-nested Cox models. This has been useful to compare a machine learning model to a Cox model fit, say. It is absolutely critical, however, that the predicted values line up exactly, with the same observation in each row; otherwise the result will be nonsense. (Be alert to the impact of missing values.)
The timewt option is only applicable to censored data. In this
case the default corresponds to Harrell's C statistic, which is
closely related to the Gehan-Wilcoxon test;
timewt="S" corrsponds to the Peto-Wilcoxon,
timewt="S/G" is suggested by Schemper, and
timewt="n/G2" corresponds to Uno's C.
It turns out that the Schemper and Uno weights are computationally
identical, we have retained both option labels as a user convenience.
The timewt= "I" option is related to the log-rank
statistic.
When the number of strata is very large, such as in a conditional
logistic regression for instance (clogit function), a much
faster computation is available when the individual strata results
are not retained; use keepstrata=FALSE or keepstrata=0
to do so. In the general case the keepstrata = 10
default simply keeps the printout managable: it retains and prints
per-strata counts if the number of strata is <= 10.
F Harrell, R Califf, D Pryor, K Lee and R Rosati, Evaluating the yield of medical tests, J Am Medical Assoc, 1982.
R Peto and J Peto, Asymptotically efficient rank invariant test procedures (with discussion), J Royal Stat Soc A, 1972.
M Schemper, Cox analysis of survival data with non-proportional hazard functions, The Statistician, 1992.
H Uno, T Cai, M Pencina, R D'Agnostino and Lj Wei, On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data, Statistics in Medicine, 2011.
coxph
fit1 <- coxph(Surv(ptime, pstat) ~ age + sex + mspike, mgus2)
concordance(fit1, timewt="n/G2") # Uno's weighting
# logistic regression
fit2 <- glm(I(sex=='M') ~ age + log(creatinine), binomial, data= flchain)
concordance(fit2) # equal to the AUC
# compare multiple models
options(na.action = na.exclude) # predict all 1384 obs, including missing
fit3 <- glm(pstat ~ age + sex + mspike + offset(log(ptime)),
poisson, data= mgus2)
fit4 <- coxph(Surv(ptime, pstat) ~ age + sex + mspike, mgus2)
fit5 <- coxph(Surv(ptime, pstat) ~ age + sex + hgb + creat, mgus2)
tdata <- mgus2; tdata$ptime <- 60 # prediction at 60 months
p3 <- -predict(fit3, newdata=tdata)
p4 <- -predict(fit4) # high risk scores predict shorter survival
p5 <- -predict(fit5)
options(na.action = na.omit) # return to the R default
cfit <- concordance(Surv(ptime, pstat) ~p3 + p4 + p5, mgus2)
cfit
round(coef(cfit), 3)
round(cov2cor(vcov(cfit)), 3) # high correlation
test <- c(1, -1, 0) # contrast vector for model 1 - model 2
round(c(difference = test %*% coef(cfit),
sd= sqrt(test %*% vcov(cfit) %*% test)), 3)
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