edge.belt: EDGE-Belt: the grouped calibration picture behind the EDGE...

In ebrahim.gof: Goodness-of-Fit and Calibration Tests for Logistic Regression

EDGE-Belt: the grouped calibration picture behind the EDGE test

Description

Returns, and draws, the group-level quantities that edge.gof (def.gof) computes and then discards: the standardized grouped residuals, their estimation-adjusted null standard deviations, the shape the test projects onto, and the bands that say which groups are out of line. The p-value reported with the belt is the one def.gof() itself returns, so the picture and the test cannot disagree.

Usage

edge.belt(
  object,
  predicted_probs = NULL,
  X = NULL,
  G = "auto",
  basis = c("poly3", "poly2", "stukel", "sym"),
  weights = c("unit", "score"),
  method = c("satterthwaite", "imhof"),
  external = FALSE,
  level = 0.95,
  nsim = 20000L,
  seed = NULL
)

## S3 method for class 'edge_belt'
print(x, ...)

## S3 method for class 'edge_belt'
plot(x, which = c(1, 2), main = NULL, palette = NULL, ...)

Arguments

object	A fitted binary logistic glm, or a binary (0/1) response vector (then supply predicted_probs).
predicted_probs	Numeric predicted probabilities; required when object is not a glm.
X	Optional design matrix, used for the estimation adjustment when object is not a glm. Without it the belt falls back to \Omega = I, which gives conservative (too wide) bands.
G	Number of equal-frequency groups, or "auto" for max(10, ceiling(n/25)). Default "auto".
basis	Shape basis: "poly3" (default), "poly2", "stukel" or "sym".
weights	"unit" (default, the statistic as published) or "score".
method	Reference for the p-value, passed to def.gof().
external	TRUE when the predicted risks were frozen before these outcomes were seen, as in external validation of a model fitted elsewhere. Nothing was estimated from these data, so \Omega = I is then the exact reference rather than a conservative fallback, and the belt says so instead of warning. The test is then def.gof(..., external = TRUE): a column of ones joins the basis, so the curve also carries the overall level. Only weights = "unit" is available. Default FALSE.
level	Band level. Default 0.95.
nsim	Draws used for the simultaneous band. Default 20000.
seed	Optional seed for that simulation, for a reproducible band.
x	An "edge_belt" object.
...	Ignored.
which	1 for the residual panel, 2 for the risk panel, c(1, 2) (default) for both.
main	Optional title; the default names the basis, the grouping and the p-value.
palette	Optional named character vector overriding the colours (band, point_band, pt, flag, curve, axis, grid).

Details

The observations are sorted by predicted risk and split into G equal-frequency groups, exactly as in def.gof(). For group g with O_g events, E_g = \sum \hat p_i and V_g = \sum \hat p_i(1-\hat p_i), the standardized residual is r_g = (O_g - E_g)/\sqrt{V_g}.

Because \beta was estimated from the same data, r is not a vector of independent standard normals: under the fitted null its covariance is \Omega = I - U(X'WX)^{-1}U', whose diagonal is below one. The belt divides each residual by its own \sqrt{\Omega_{gg}}, so a band drawn at \pm 1.96 is the correct pointwise band and not the too-wide naive one.

Two bands, and why the wider one matters. With G groups, a pointwise 95% band is expected to exclude about 0.05 * G groups when the model is perfect: at G = 40 that is two groups flagged by chance alone. The belt therefore also carries a simultaneous band, the level-level quantile of \max_g |r_g|/\sqrt{\Omega_{gg}} under r \sim N(0, \Omega), obtained by simulation from the known \Omega. Read the simultaneous band when asking "is this model miscalibrated anywhere"; read the pointwise band only for a group singled out in advance.

The curve. The orange curve is B(B'B)^{-1}B'r, the component of the residual vector that the EDGE statistic squares (B = Z for the unit form, B = Z\sqrt{V_g} for the score form). It is the shape the test looked for, drawn on the same axes as the residuals it was extracted from.

What this is not. The belt is G discrete groups, not a continuous curve over the risk range: it is coarser than the calibration belt of Nattino et al. (2014), implemented in givitiR, and it inherits whatever the equal-frequency grouping hides. It is a diagnostic companion to the test, not a replacement for a calibration curve.

Value

An object of class "edge_belt": a list with test (the def.gof() result), groups (one row per group: size, mean predicted risk, observed and expected events, variance, residual, null SD, standardized residual, the fitted direction, and the two flags), critical (the two band half-widths and the level), and the settings.

References

Nattino, G., Finazzi, S. and Bertolini, G. (2014). A new calibration test and a reappraisal of the calibration belt for the assessment of prediction models based on dichotomous outcomes. Statistics in Medicine 33, 2390-2407.

See Also

edge.gof, def.gof.

Examples

set.seed(1)
n  <- 800
x  <- runif(n, -3, 3)
y  <- rbinom(n, 1, 1 - exp(-exp(0.8 * x)))   # log-log truth, fitted as logit
fit <- glm(y ~ x, family = binomial())
b <- edge.belt(fit)
b
plot(b)

ebrahim.gof documentation built on Oct. 11, 2026, 5:07 p.m.