Statistical & Econometric
Lin’s Concordance Correlation Coefficient (CCC, ρ_c) Formula
Calculate Lawrence Lin’s Concordance Correlation Coefficient ρ_c = (2 * r * s_x * s_y) / (s_x^2 + s_y^2 + (x̄ - ȳ)^2) and bias correction factor C_b = ρ_c / r to measure 45° identity-line assay agreement.
Penalizing BOTH scale shift and location shift off the 45° line of identity!
Pearson correlation r measures scatter around ANY straight line—so if Lab B reports biomarker values 20% higher or +4 units shifted above Lab A, Pearson r can still hit 0.96 while absolute reproducibility fails! Lawrence I-Kuei Lin (Biometrics, 1989) derived the Concordance Correlation Coefficient ρ_c = (2 * r * s_x * s_y) / (s_x^2 + s_y^2 + (x̄ - ȳ)^2), which factors cleanly into ρ_c = r * C_b: precision r (scatter around best-fit line) multiplied by bias correction factor C_b <= 1 (how close that line sits to the 45° y = x identity line)!
Watch two clinical chemistry analyzers with high Pearson precision r = 0.9600, standard deviations s_x = 12.0 and s_y = 12.5, and means x̄ = 100.0 vs. ȳ = 104.0 (+4.0 calibration offset): location shift pulls C_b down to 0.9486 and ρ_c down to 0.9107; recalibrating Lab B to eliminate the mean offset (ȳ = 100.0) restores ρ_c to 0.9592!
| Assay Calibration State (r=0.96, sx=12.0, sy=12.5) | Mean Shift |x̄ - ȳ| | Bias Factor C_b = ρ_c / r | Lin’s Concordance ρ_c |
|---|---|---|---|
| Uncalibrated Offset (x̄ = 100, ȳ = 104) | 4.0 units | 0.9486 | 0.9107 (Moderate CLSI Tier) |
| Zero-Offset Recalibrated (x̄ = 100, ȳ = 100) | 0.0 units | 0.9992 | 0.9592 (Substantial Agreement) |
| Invalid SD <= 0 or |r| > 1 | Error Guard | Domain Check | "Invalid CCC Summary Stats" |
Lin’s Concordance Correlation Coefficient ρ_c (-1 to +1)
Evaluates (2 * r * s_x * s_y) / (s_x^2 + s_y^2 + (x̄ - ȳ)^2) from Pearson r (B2), s_x (B3), s_y (B4), mean x̄ (B5), and mean ȳ (B6).
=IF(OR(ABS(B2) > 1, B3 <= 0, B4 <= 0), "Invalid CCC Summary Stats", ROUND((2 * B2 * B3 * B4) / ((B3 ^ 2) + (B4 ^ 2) + ((B5 - B6) ^ 2)), 4))Accuracy / Bias Correction Factor C_b (0 to 1)
Computes C_b = (2 * s_x * s_y) / (s_x^2 + s_y^2 + (x̄ - ȳ)^2), isolating systematic scale and location drift independent of random noise r.
=IF(OR(B3 <= 0, B4 <= 0), "Invalid CCC Summary Stats", ROUND((2 * B3 * B4) / ((B3 ^ 2) + (B4 ^ 2) + ((B5 - B6) ^ 2)), 4))Try the example
Use the original values or change them to test your understanding.