CCC

- Developed by Lin L(1989) to measure the agreement between two sets of measurements of the same variable, usually between the new measurements and the gold standard.

- Anova assumptions don't need to be satisfied, however needs to apply one of transformation method (for example: Fisher Z-Transformation) in order to get asymptotic normal distribution to compute confidence interval.

Interpretation:

The degree of agreement among single measurements rated by 2 raters. McBride(2005) gives the following suggestion for interpreting CCC:
below 0.90: Poor
between 0.90 to 0.95: Moderate
between 0.95 to 0.99: Substantial
above 0.99 almost Perfect.

Formula:

\(\frac{2\rho \sigma_x \sigma_y}{(\mu_x - \mu_y)^2 + \sigma_x^2 + \sigma_y^2}\)

\(\rho\) = correlation coefficient between variables x and y;

\(\sigma_x^2\) = variance of x;

\(\sigma_y^2\) = variance of y;

\(\mu_x\) = mean of x;

\(\mu_y\) = mean of y;

Example in R

Untitled.utf8 
library("epiR")
## Loading required package: survival
## Package epiR 1.0-15 is loaded
## Type help(epi.about) for summary information
## Type browseVignettes(package = 'epiR') to learn how to use epiR for applied epidemiological analyses
## 
set.seed(seed = 1234)
method1 <- rnorm(n = 25, mean = 0, sd = 1)
method2 <- method1 + runif(n = 25, min = -0.5, max = 0.5)
tmp <- data.frame(method1, method2)
head(tmp)
##      method1     method2
## 1 -1.2070657 -1.63328587
## 2  0.2774292  0.08711584
## 3  1.0844412  1.30171292
## 4 -2.3456977 -2.34115179
## 5  0.4291247  0.08212365
## 6  0.5060559  0.50998938

Apply z-transformation, compute 95% confidence interval, CCC = 0.9461401, 95% confidence interval (0.8857455,0.9750329)

tmp.ccc <- epi.ccc(method1, method2, ci = "z-transform", conf.level = 0.95, 
   rep.measure = FALSE)
tmp.ccc
## $rho.c
##         est     lower     upper
## 1 0.9461401 0.8857455 0.9750329
## 
## $s.shift
## [1] 1.08911
## 
## $l.shift
## [1] -0.1038458
## 
## $C.b
## [1] 0.9910435
## 
## $blalt
##            mean        delta
## 1  -1.420175809  0.426220120
## 2   0.182272543  0.190313398
## 3   1.193077048 -0.217271743
## 4  -2.343424747 -0.004545912
## 5   0.255624168  0.347001041
## 6   0.508022636 -0.003933488
## 7  -0.577759499  0.006039077
## 8  -0.421031757 -0.251200197
## 9  -0.727127087  0.325350176
## 10 -0.715841624 -0.348392410
## 11 -0.294775784 -0.364833832
## 12 -1.227457807  0.458142725
## 13 -0.867662817  0.182817845
## 14 -0.178666213  0.486250061
## 15  0.829006922  0.260974273
## 16 -0.007038186 -0.206494617
## 17 -0.606962127  0.191905243
## 18 -0.906921634 -0.008547566
## 19 -1.061348371  0.448353381
## 20  2.448120098 -0.064569840
## 21 -0.055171686  0.378519813
## 22 -0.294267706 -0.392836382
## 23 -0.683234244  0.485372744
## 24  0.601149993 -0.283121104
## 25 -0.898739580  0.410038667
## 
## $sblalt
##          est  delta.sd      lower    upper
## 1 0.09806206 0.3015703 -0.4930049 0.689129
## 
## $nmissing
## [1] 0

Live Example: Try this yourself!

References:

Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
Lin L.I-K (1989) A [CCC] to evaluate reproducibility. Biometrics 45:255-268
McBride GB (2005) A proposal for strength-of-agreement criteria for Lin’s concordance… NIWA Client Report: HAM2005-062.
Lin L (2000). A note on the concordance correlation coefficient. Biometrics 56: 324 – 325.