Lin's concordance correlation coefficient with its 95% interval and
decomposition, for any pair of vectors: subject estimates against
their generating values, or recovered correlations and effects
against the ones that were simulated. summary() of a recovery
object reports the same columns per parameter.
Value
A one-row tibble with ccc, ccc_low, ccc_high,
ccc_accuracy, ccc_scale_shift, ccc_location_shift,
calibration_slope, calibration_intercept and n, the number of
complete pairs. The coefficient and its components are NA with
fewer than three pairs or when either vector has no spread.
Details
The coefficient is 1 - E[(estimate - truth)^2] / (sd_e^2 + sd_t^2 + (mean_e - mean_t)^2), a rescaled mean squared deviation, and factors
into Pearson's r (precision) times ccc_accuracy (Lin, 1989, p. 258).
Accuracy depends on the scale shift v = sd_e / sd_t and the location
shift u = (mean_e - mean_t) / sqrt(sd_e sd_t); all moments divide by
n.
Reading the scale shift. Estimates from a hierarchical model are
shrunk towards the population mean, and a well-calibrated posterior
mean has ccc_scale_shift equal to r, not 1. calibration_slope
(r divided by v, the slope of truth regressed on the estimate) is 1 in that
case, above 1 when estimates are shrunk too much and below 1 when they
are shrunk too little. Because accuracy treats v and 1/v alike,
ccc rewards too little shrinkage: read it together with the RMSE and
the calibration slope, and prefer r when only the rank order of
subjects matters.
The calibration line. calibration_intercept is the intercept of
the same regression, mean_t - calibration_slope * mean_e: the truth
the line predicts at an estimate of 0. It is read on the scale of the
input, so it depends on where 0 lies on that scale, and with the slope
it gives the whole line. It is not calibration-in-the-large
(mean_t - mean_e, the slope fixed at 1), which is minus the bias; the
two are related by calibration_intercept = (mean_t - mean_e) - (calibration_slope - 1) * mean_e, so at a slope of 1 they agree.
The interval uses Lin's Z transformation, atanh(ccc), with the
asymptotic variance for random bivariate normal pairs, and is NA
with three or fewer pairs, at ccc = +/-1 and when the Pearson
correlation is 0. When the
generating values are fixed by design rather than drawn, it is an
approximation. It is always a 95% interval.
References
Lin, L. I.-K. (1989). A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45(1), 255–268. doi:10.2307/2532051
Lin, L. I.-K. (2000). A note on the concordance correlation coefficient. Biometrics, 56(1), 324–325. https://www.jstor.org/stable/2677159
Examples
truth <- c(0.2, 0.5, 0.9, 1.4, 1.8, 2.3)
# shrunk towards the mean and shifted up
estimate <- 0.7 * truth + 0.5
recovery_ccc(estimate, truth)
#> # A tibble: 1 × 9
#> ccc ccc_low ccc_high ccc_accuracy ccc_scale_shift ccc_location_shift
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 0.915 0.794 0.966 0.915 0.7 0.238
#> # ℹ 3 more variables: calibration_slope <dbl>, calibration_intercept <dbl>,
#> # n <int>