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:

- concordance
the estimated concordance value or values

- count
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

- n
the number of observations

- var
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.

- cvar
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.)- dfbeta
optional, the vector of leverage estimates for the concordance

- influence
optional, the matrix of leverage values for each of the counts, one row per observation

- ranks
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.

- object
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.- data
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.- weights
optional vector of case weights. Only applicable if

`object`

is a formula.- subset
expression indicating which subset of the rows of data should be used in the fit. Only applicable if

`object`

is a formula.- na.action
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.- newdata
optional, a new data frame in which to evaluate (but not refit) the models

- cluster
optional grouping vector for calculating the robust variance

- ymin, ymax
compute the concordance over the restricted range ymin <= y <= ymax. (For survival data this is a time range.)

- timewt
the weighting to be applied. The overall statistic is a weighted mean over event times.

- influence
1= return the dfbeta vector, 2= return the full influence matrix, 3 = return both

- ranks
if TRUE, return a data frame containing the scaled ranks that make up the overall score.

- reverse
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.- timefix
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.

- keepstrata
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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