Lin's Concordance Correlation Coefficient

Computes Lin's (1989) concordance correlation coefficient (CCC) for a pair of vectors of paired observations, together with a confidence interval built on Lin's z-transformed standard error (Lin, 1989; see also the note in Lin, 2000). The CCC measures agreement (not merely correlation) between two methods of measurement: a CCC of 1 means perfect agreement ($y_i = x_i$ for all $i$), while Pearson's $r$ would still be 1 for any straight-line relationship, even one with non-unit slope.

Usage

lin_ccc(x, y, conf_level = 0.95, method = "lin")

Arguments

x, y: Paired numeric vectors of equal length (e.g., the two measurement methods).
conf_level: Confidence level for the CI. Default 0.95.
method: The confidence interval method. Currently the only option is "lin", the Lin (1989) z-transformed standard error.

Value

A data.frame with rows for the CCC point estimate, the lower and upper CI limits, and decomposition components (Pearson $r$, accuracy $C_b$, location-shift $u$, scale-shift $v$).

Details

Definition. Lin (1989) defined the CCC as $$\rho_c \;=\; \frac{2 \rho\, \sigma_x \sigma_y} {\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2},$$ where $\rho = \mathrm{Cor}(X, Y)$ is the Pearson correlation and the denominator is inflated by the squared mean difference and by any inequality of the two variances, so disagreement in location or scale pulls $\rho_c$ below $\rho$. $\rho_c$ factors as $\rho_c = \rho \cdot C_b$, where $C_b \in [0, 1]$ is the "bias correction factor" that captures location and scale agreement, and $C_b = 1$ iff $\mu_x = \mu_y$ and $\sigma_x = \sigma_y$.

Confidence interval. The Fisher-style z-transform of the CCC, $z = \frac{1}{2} \log\{(1 + \rho_c)/(1 - \rho_c)\}$, has approximate variance (Lin, 1989, as corrected in Lin, 2000) $$\mathrm{Var}(z) \;\approx\; \frac{1}{n - 2} \left[ \frac{(1 - \rho^2) \rho_c^2}{(1 - \rho_c^2) \rho^2} + \frac{2 \rho_c^3 (1 - \rho_c) u^2}{\rho (1 - \rho_c^2)^2} - \frac{\rho_c^4 u^4}{2 \rho^2 (1 - \rho_c^2)^2}\right],$$ where $u = (\mu_x - \mu_y) / \sqrt{\sigma_x \sigma_y}$. The CI is built on the z-scale and back-transformed via $\tanh$.

This variance is derived under bivariate normality, and the interval inherits that assumption. Under normality its coverage is modestly below the nominal rate in small samples (roughly 0.92 to 0.94 at $n$ of 10 to 20 for a nominal 0.95) and approaches the nominal rate as $n$ grows (about 0.94 at $n = 50$ for a moderate CCC). With clearly skewed data the situation is worse and more data do not repair it: with heavy-tailed or log-normal style measurements the interval can cover far below the nominal rate at any sample size (Carrasco, Jover, King, & Chinchilli, 2007). With such data, transform toward symmetry before computing the CCC, or use a bootstrap interval on $\hat\rho_c$.

References

Carrasco, J. L., Jover, L., King, T. S., & Chinchilli, V. M. (2007). Comparison of concordance correlation coefficient estimating approaches with skewed data. Journal of Biopharmaceutical Statistics, 17(4), 673–684. doi:10.1080/10543400701329463

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. doi:10.1111/j.0006-341X.2000.00324.x

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Two methods of measuring the same quantity:
set.seed(113)
method_a <- rnorm(40, mean = 100, sd = 15)
method_b <- method_a + rnorm(40, mean = 2, sd = 5)
lin_ccc(method_a, method_b)
#>  term        value 
#>  ccc         0.928 
#>  lower_limit 0.871 
#>  upper_limit 0.96  
#>  pearson_r   0.941 
#>  C_b         0.986 
#>  u           -0.154
#>  v           0.933 
#> 
#> Confidence level: 95%

# 2. Compare CCC with Pearson r when there is a systematic offset:
lin_ccc(method_a, method_a + 5)$value[1:2]   # CCC < r
#> [1] 0.9492315 0.9224162
cor(method_a, method_a + 5)                  # Pearson r = 1
#> [1] 1