Directed Ebrahim-Farrington (DEF) Goodness-of-Fit Test

Performs the Directed Ebrahim-Farrington (DEF) goodness-of-fit test for a fitted binary logistic regression model. DEF concentrates its power on a small set of calibration-curve "shape" directions by projecting the grouped standardized residuals onto a low-dimensional basis and testing the squared length of that projection.

Naming note: this test is published under the name EDGE (Ebrahim Directed Goodness-of-fit Evaluation), and edge.gof is the primary interface going forward. def.gof() is retained, unchanged, as a fully supported legacy name.

Usage

def.gof(
  object,
  predicted_probs = NULL,
  X = NULL,
  G = 10,
  basis = c("poly3", "poly2", "stukel", "ensemble"),
  method = c("satterthwaite", "imhof")
)

Arguments

object

A fitted binary logistic glm, or a binary (0/1) response vector y (then supply predicted_probs).

predicted_probs

Numeric predicted probabilities; required when object is a y vector, ignored when it is a glm.

X

Optional design matrix, used only with the y/predicted_probs form: it enables the exact estimation-adjusted (\(\Omega\)) calibration (logit working weights assumed). Without it the conservative \(\chi^2_k\) reference is used and a warning is issued. Ignored when object is a glm.

G

Integer number of equal-frequency groups (default 10; must be >= 3).

basis

One of "poly3" (default), "poly2", "stukel", or "ensemble".

method

One of "satterthwaite" (default) or "imhof".

Value

A one-row data.frame with columns Test, Basis, Test_Statistic (the statistic \(S\)), df, Method, and p_value. When basis = "ensemble", the return is that of def.ensemble.gof.

Details

The observations are sorted by predicted probability and split into G equal-frequency groups; the standardized grouped residual vector \(r\) is projected onto a basis matrix \(Z\) of smooth shapes, giving \(S = (Z'r)'(Z'Z)^{-1}(Z'r)\). Its null distribution is a weighted sum of \(\chi^2_1\) variables with weights equal to the eigenvalues of \((Z'Z)^{-1}Z'\Omega Z\), where \(\Omega = I - U(X'WX)^{-1}U'\) is the estimation-adjusted covariance of the grouped residuals. The p-value uses a Satterthwaite scaled-\(\chi^2\) approximation (default) or Imhof's method (if the CompQuadForm package is installed). Bases: "poly2", "poly3" (default), "stukel"; "ensemble" runs all three and combines them via def.ensemble.gof.

References

Ebrahim EK, El-Kotory A (2026). "A Directional Hosmer-Lemeshow Goodness-of-Fit Test for Sparse Logistic Regression." arXiv:2607.15454 [stat.ME]. doi:10.48550/arXiv.2607.15454

Ebrahim EK, El-Kotory A (2026). "EDGE: A Closed-Form Directed Goodness-of-Fit Test for Sparse Logistic Regression." arXiv:2608.20511 [stat.ME]. doi:10.48550/arXiv.2608.20511

See also

ef.gof, def.ensemble.gof.

Author

Ebrahim Khaled Ebrahim ebrahimkhaled@alexu.edu.eg

Examples

## gof_demo carries a documented smooth calibration misfit: the risk bends in age,
## and a model linear in age misses it. The point of a directed test is to see that.
data("gof_demo", package = "ebrahim.gof")
wrong <- glm(outcome ~ age + bmi + sex + treatment,
             data = gof_demo, family = binomial())
def.gof(wrong)                       # default poly3 basis
#>                          Test Basis Test_Statistic       df        Method
#> 1 Directed Ebrahim-Farrington poly3       8.329512 2.072814 satterthwaite
#>      p_value
#> 1 0.01496412
def.gof(wrong, basis = "stukel")     # tail-shape basis
#>                          Test  Basis Test_Statistic       df        Method
#> 1 Directed Ebrahim-Farrington stukel       6.499717 1.564118 satterthwaite
#>     p_value
#> 1 0.0118151
def.gof(wrong, basis = "ensemble")   # combine all three (CCT)
#>           Test Combiner         Components k     p_value
#> 1 DEF ensemble      cct poly2+poly3+stukel 3 0.006959379

## give the model the term it was missing, and the same test stands down
right <- glm(outcome ~ poly(age, 2) + bmi + sex + treatment,
             data = gof_demo, family = binomial())
def.gof(right)
#>                          Test Basis Test_Statistic       df        Method
#> 1 Directed Ebrahim-Farrington poly3      0.4259731 2.046178 satterthwaite
#>     p_value
#> 1 0.7966492