A directional Hosmer-Lemeshow goodness-of-fit test for sparse logistic regression

Goodness-of-fit assessment for the binary logistic regression model is difficult when covariates are continuous: the data are effectively sparse, the classical Pearson and deviance tests fail, and practitioners rely on partition-based tests, such as the Hosmer-Lemeshow test, that group observations before comparing observed and expected counts. We study a partition test that modifies the Hosmer-Lemeshow statistic with a single directional correction term, weighted by $(1-2\bar\pi_g)$ and referred to a $\chi^2_{G-2}$ distribution. The correction is the grouped form of the Osius-Rojek/Farrington standardization; grouping makes it well defined in the sparse regime, and it targets the asymmetric over- and under-prediction that a misspecified link induces. A single alignment functional captures its effect, predicting where the test gains power (asymmetric-link misspecification) and where it does not (symmetric departures, and covariate-space structure that no probability-grouping test can see). In simulations the test holds its size; no well-calibrated partition test is more sensitive to asymmetric-link misfit, and it clearly exceeds Hosmer-Lemeshow there, most so for the complementary log-log link -- a modest gain that fades as $n$ grows; it ties Hosmer-Lemeshow on an omitted interaction and is less powerful on an omitted quadratic (by about ten percentage points at $n=1000$). A real-data application illustrates its use, and the test is implemented in the R package ebrahim.gof.