Lin's Concordance Correlation Coefficient

Computes Lin's concordance correlation coefficient (CCC) for assessing agreement between two continuous measurements.

Usage

ccc(
  x,
  y,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  method = c("z-transform", "boot", "asymptotic"),
  na.rm = FALSE,
  ...
)

Arguments

x: a numeric vector
y: a numeric vector of equal length to x
conf.level: confidence level of the interval. If set to NA (the default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().
method: a character string specifying the confidence interval method. One of "z-transform", "boot", or "asymptotic".
na.rm: logical; if TRUE, incomplete observation pairs are removed before computation
...: additional arguments controlling the bootstrap procedure. Currently R, parallel and ncpus are supported.

Value

a named numeric vector containing only est when conf.level = NA; otherwise a named numeric vector with elements:

est: point estimate.
lci: lower confidence interval bound.
uci: upper confidence interval bound.

Additional diagnostics are stored as attributes:

nObs: number of observations used
scaleShift: scale shift parameter
locationShift: location shift parameter
biasCorrection: bias correction factor
method: confidence interval method, if applicable
confLevel: confidence level, if applicable
sides: confidence interval type, if applicable

Details

The CCC combines measures of precision and accuracy and quantifies the deviation of the observed data from the line of perfect concordance. Values range from -1 to 1, where 1 indicates perfect agreement.

Confidence intervals can be computed using a Fisher z-transformation, a nonparametric bootstrap, or the asymptotic approximation of Lin (2000). The asymptotic variance implemented here is the corrected form given by Lin (2000), superseding the expression in Lin (1989). Internally it is held on the scale of \(\rho_c\) itself; the "z-transform" method rescales it to the z scale via the delta method, where \(d\,\mathrm{atanh}(\rho)/d\rho = 1/(1 - \rho^2)\).

sides names the side on which the finite bound lies: "left" yields an interval bounded below, with the upper limit fixed at 1, and "right" one bounded above, with the lower limit fixed at -1.

Missing values are handled according to package conventions: if na.rm = FALSE and either x or y contains missing values, NA is returned. If na.rm = TRUE, complete cases are used. Infinite values carry no comparable convention - they leave the moments undefined and are rejected with an error.

References

Lin, L. I.-K. (1989). A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45(1), 255-268.

Lin, L. I.-K. (2000). A note on the concordance correlation coefficient. Biometrics, 56(1), 324-325.

Examples

set.seed(123)

x <- rnorm(100)
y <- x + rnorm(100, sd = 0.2)

ccc(x, y)
#>       est 
#> 0.9775499 
#> attr(,"nObs")
#> [1] 100
#> attr(,"scaleShift")
#> [1] 1.011879
#> attr(,"locationShift")
#> [1] -0.02354303
#> attr(,"biasCorrection")
#> [1] 0.9996533

ccc(x, y, conf.level = 0.95)
#>       est       lci       uci 
#> 0.9775499 0.9668474 0.9848240 
#> attr(,"nObs")
#> [1] 100
#> attr(,"scaleShift")
#> [1] 1.011879
#> attr(,"locationShift")
#> [1] -0.02354303
#> attr(,"biasCorrection")
#> [1] 0.9996533
#> attr(,"method")
#> [1] "z-transform"
#> attr(,"confLevel")
#> [1] 0.95
#> attr(,"sides")
#> [1] "two.sided"

ccc(
  x, y,
  conf.level = 0.95,
  method = "boot",
  R = 999
)
#>       est       lci       uci 
#> 0.9775499 0.9666290 0.9851318 
#> attr(,"nObs")
#> [1] 100
#> attr(,"scaleShift")
#> [1] 1.011879
#> attr(,"locationShift")
#> [1] -0.02354303
#> attr(,"biasCorrection")
#> [1] 0.9996533
#> attr(,"method")
#> [1] "boot"
#> attr(,"confLevel")
#> [1] 0.95
#> attr(,"sides")
#> [1] "two.sided"