DeepGOF-1: a pretrained goodness-of-fit test for logistic regression

Tests whether a fitted binomial glm is correctly specified, using a convolutional network that was trained once, offline, on simulated departures and is shipped frozen with this package. The analyst never trains anything: the network reads the model's residual map and the p-value is the rank of the observed score within the analyst's own parametric bootstrap, so the level does not depend on what the network learned.

Usage

deepgof1(fit, B = 199L, K = 6L)

Arguments

fit: a fitted glm with family = binomial() and at least two covariates.
B: number of parametric-bootstrap replicates. The p-value lies on a grid of 1/(B+1), so B = 199 makes the nominal .05 attainable exactly.
K: grid resolution. Leave at 6: the shipped weights were trained at K = 6 and are not valid at any other resolution.

Value

An object of class "deepgof1": a list with statistic (the observed score), p.value, B, K, axes (the two covariates the grid was built on), boot (the B bootstrap scores) and method.

Details

The residual map is a K by K grid over the empirical ranks of the two covariates with the largest \(|\hat\beta_j| \hat\sigma_j\); each cell holds a standardized residual sum, approximately standard normal under a correct model. Misfit therefore has a location on the map – an omitted quadratic paints a stripe, an omitted interaction a saddle – which is what the convolutional statistic reads.

The test is a small-sample instrument. Against the classical partition tests it gains most at \(n\) of 50 to 200 and the gain decays as \(n\) grows; because the grid uses only two covariates, misfit that lives off those axes is harder for it to see than for covariate-space or smoothing tests. It is not one of the tests run.all.gof selects: call it directly on the same fitted model and read its p-value beside the panel. See run.all.gof for the classical battery.

Reproducibility

A bootstrap refit that fails to converge is scored +Inf, so it counts against rejection – the conservative direction. Set a seed before calling for a reproducible p-value.

References

Ebrahim EK (2026). "DeepGOF-1: A Pretrained Convolutional Goodness-of-Fit Test for Logistic Regression with a Computable Consistency Certificate." Manuscript under review. Reproduction materials and frozen weights: doi:10.5281/zenodo.22113220

Besag, J. and Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika 76, 633–642. doi:10.1093/biomet/76.4.633

Examples

set.seed(1)
n  <- 150
x1 <- runif(n, -3, 3); x2 <- rnorm(n)
# a model with an omitted quadratic term
y  <- rbinom(n, 1, plogis(0.3 + 0.8 * x1 - 0.5 * x2 + 0.9 * (x1^2 - mean(x1^2))))
fit <- glm(y ~ x1 + x2, family = binomial())
deepgof1(fit, B = 49)   # B = 49 to keep the example fast; use the default in practice
#> 
#> 	DeepGOF-1: pretrained residual-map goodness-of-fit test
#> 
#> data:  fit
#> S = 6.8465, B = 49, p-value = 0.0200
#> grid: 6 x 6 over ranks of x1 and x2
#> alternative hypothesis: the logistic model is misspecified
#>