Bias, variance components and expected error along the severity range

Functional part of the one-stage SAD validation. Describes, for each condition, how the bias, the variation among raters, the image variation, the residual variation and the expected absolute error change along the severity range, and compares the bias curves between conditions with simultaneous confidence bands, so that the severity ranges where a SAD helps or hurts are identified.

Usage

sad_curve(
  fit,
  type = c("linear", "smooth"),
  reference = NULL,
  pairs = NULL,
  n = 100,
  level = 0.95,
  k = 5,
  nsim = 5000,
  seed = 1
)

Arguments

fit: An object returned by sad_fit().
type: "linear" (default) uses the fitted sad_fit model, so bias is linear in severity within each condition. "smooth" fits a functional generalised additive mixed model with mgcv::bam(): a smooth error curve per condition, a random smooth curve per rater and condition and a random image effect; bias can then take any shape.
reference: Reference for the differences: a condition (e.g. "UN_Old") or a method (e.g. "UN"). With a method, every other condition is compared with the reference method in the same group of raters. Defaults to the first condition.
pairs: Optional data frame with columns a and b; differences a - b are computed for each row. Overrides reference.
n: Number of grid points along the severity range.
level: Confidence level of the bands.
k: Basis dimension of the condition smooths (type = "smooth").
nsim: Number of draws used for the simultaneous bands.
seed: Random seed for the simultaneous bands. The user's random number stream is restored afterwards.

Value

An object of class sad_curve: a list with curves (one row per condition and grid point; for "linear" also sd_rater, sd_image, sd_resid, sd_total and expected_abs_error), differences (with significant), type and level. For "smooth" also gam and edf.

Details

With type = "linear", the variance among raters at severity \(z\) is \(\sigma^2_{b0} + 2z\sigma_{b0b1} + z^2\sigma^2_{b1}\), from the rater intercept and slope covariance of the method. The total standard deviation \(\sigma_T\) adds image, image x condition and residual variances, and the expected absolute error is the mean of a folded normal distribution, \(\sigma_T\sqrt{2/\pi}\exp(-b^2/2\sigma_T^2) + b(1 - 2\Phi(-b/\sigma_T))\), where \(b\) is the bias. Differences are evaluated only where the severity ranges of the two conditions overlap. Simultaneous bands use the distribution of the maximum absolute t statistic over the grid, from draws of the fixed effects.

Examples

# \donttest{
if (requireNamespace("glmmTMB", quietly = TRUE)) {
  x <- sad_data(SADRustSoybean, wide = c(UN = "unaided", SAD = "aided"), group = "sad")
  fit <- sad_fit(x)
  cv <- sad_curve(fit, reference = "UN")
  sad_regions(cv)
  plot(cv, "difference")
}
#> Structure 'full' did not converge; trying the next one.
#> Structure 'noim' did not converge; trying the next one.

# }