Covariate-Space Directed Ebrahim-Farrington (CDEF) Goodness-of-Fit Test

A directed goodness-of-fit test for binary logistic regression whose direction lives in covariate space (functions of the predictors) rather than in fitted-probability space like def.gof. It projects the standardized residuals onto a covariate-space basis (polynomials and pairwise products, natural splines, or a combination that also includes fitted-probability bends) and calibrates the quadratic form with the Farrington estimation-adjusted projection, exactly as in def.gof. This makes it sensitive to omitted interactions and to local / oscillatory departures that fitted-probability grouping can miss.

Usage

cdef.gof(
  object,
  predicted_probs = NULL,
  X = NULL,
  basis = c("poly", "spline", "combined"),
  method = c("satterthwaite", "imhof")
)

Arguments

object: A fitted binary logistic glm, or a binary (0/1) response vector y (then supply predicted_probs and X).
predicted_probs: Numeric predicted probabilities; required when object is a y vector.
X: Design/covariate matrix (with or without an intercept column); required when object is a y vector. Ignored when object is a glm.
basis: One of "poly" (squares, cubes, pairwise products), "spline" (natural cubic splines per covariate plus a pairwise term; needs splines), or "combined" (covariate polynomials plus fitted-probability bends).
method: One of "satterthwaite" (default) or "imhof".

Value

A one-row data.frame with Test, Basis, Test_Statistic, df, Method, and p_value.

Details

Let \(\tilde r_i=(y_i-\hat p_i)/\sqrt{\hat p_i(1-\hat p_i)}\) be the standardized residuals and \(Z\) a covariate-space basis matrix. The statistic is \(S=(Z'\tilde r)'(Z'Z)^{-1}(Z'\tilde r)\), whose null distribution is a weighted sum of \(\chi^2_1\) variables with weights the eigenvalues of \((Z'Z)^{-1}Z'\Omega Z\), where \(\Omega=I-V^{1/2}X(X'VX)^{-1}X'V^{1/2}\) adjusts for estimating \(\hat\beta\). The p-value uses a Satterthwaite scaled-\(\chi^2\) approximation (default) or Imhof's method (CompQuadForm). Rank-deficient bases are reduced automatically.

References

Farrington, C. P. (1996). On Assessing Goodness of Fit of Generalized Linear Models to Sparse Data. JRSS-B 58(2), 349-360.

Examples

set.seed(1)
n <- 600; x1 <- runif(n, -3, 3); x2 <- rnorm(n)
# truth has an omitted interaction; fit the additive model
y <- rbinom(n, 1, plogis(0.3 + 0.8 * x1 - 0.5 * x2 + 0.4 * x1 * x2))
fit <- glm(y ~ x1 + x2, family = binomial())
cdef.gof(fit)                    # covariate-space directed test (poly basis)
#>                          Test Basis Test_Statistic       df        Method
#> 1 Covariate-space Directed EF  poly       86.40403 4.271854 satterthwaite
#>        p_value
#> 1 2.805972e-20
cdef.gof(fit, basis = "spline")  # for local / oscillatory misfit
#>                          Test  Basis Test_Statistic       df        Method
#> 1 Covariate-space Directed EF spline       89.49195 7.099734 satterthwaite
#>        p_value
#> 1 1.112775e-16