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The New England Journal of Statistics in Data Science


A Safe Hosmer-Lemeshow Test
Volume 2, Issue 2 (2024), pp. 175–189
Pub. online: 18 December 2023      Type: Methodology Article      Open accessOpen Access
Area: Statistical Methodology

1 The first two authors contributed equally to this work.
Accepted
2 December 2023
Published
18 December 2023

Abstract

This article proposes an alternative to the Hosmer-Lemeshow (HL) test for evaluating the calibration of probability forecasts for binary events. The approach is based on e-values, a new tool for hypothesis testing. An e-value is a random variable with expected value less or equal to one under a null hypothesis. Large e-values give evidence against the null hypothesis, and the multiplicative inverse of an e-value is a p-value. Our test uses online isotonic regression to estimate the calibration curve as a ‘betting strategy’ against the null hypothesis. We show that the test has power against essentially all alternatives, which makes it theoretically superior to the HL test and at the same time resolves the well-known instability problem of the latter. A simulation study shows that a feasible version of the proposed eHL test can detect slight miscalibrations in practically relevant sample sizes, but trades its universal validity and power guarantees against a reduced empirical power compared to the HL test in a classical simulation setup. We illustrate our test on recalibrated predictions for credit card defaults during the Taiwan credit card crisis, where the classical HL test delivers equivocal results.

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Copyright
© 2024 New England Statistical Society
Open access article under the CC BY license.

Keywords
E-value Probability forecast Calibration validation Goodness-of-fit Isotonic regression

Funding
A. Henzi and J. Ziegel gratefully acknowledge financial support from the Swiss National Science Foundation. T. Dimitriadis gratefully acknowledges financial support from the German Research Foundation (DFG) through grant number 502572912.

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