Journals Information
Mathematics and Statistics Vol. 13(6), pp. 425 - 438
DOI: 10.13189/ms.2025.130601
Reprint (PDF) (963Kb)
Summary Goodness-of-Fit Statistics for Binary Generalized Linear Models with Noncanonical Probit, Log-Log and Complementary Log-Log Links
Xuelu Sun 1, Stephen J. Quinn 1,*, Sunil Bhar 2
1 Department of Health Science and Biostatistics, Swinburne University of Technology, Australia
2 Department of Psychological Sciences, Swinburne University of Technology, Australia
ABSTRACT
Logistic regression is a popular and widely used method in applied research. However, other noncanonical link models can be more appropriate when the logistic model does not suit the characteristics of the response variable. Goodness-of-fit statistics play a crucial role in evaluating model adequacy by providing quantitative measures of how closely predicted probabilities align with observed outcomes. Despite their importance, limited research has focused on assessing goodness-of-fit under noncanonical links. In this paper, we conducted a simulation study to compare the performance of the Hosmer Lemeshow (
) statistic, the normalized unweighted sum of squares (
) statistic, and the Hjort-Hosmer (
) statistic in generalized linear models with noncanonical links, specifically probit, log-log and complementary log-log. The simulation results show that all three statistics remained the expected Type I error rate of 5% under correctly specified models. In scenarios with model misspecification, the (34.8%) and (33.0%) statistics generally achieved higher overall rejection rates than the statistic (27.8%). However, in some scenarios, the statistic outperformed both the and statistics, suggesting that all three statistics have a role in assessing model adequacy.
KEYWORDS
Goodness-of-Fit, Noncanonical Generalized Linear Models, Hosmer Lemeshow, Unweighted Sum of Squares, Hjort-Hosmer
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Xuelu Sun , Stephen J. Quinn , Sunil Bhar , "Summary Goodness-of-Fit Statistics for Binary Generalized Linear Models with Noncanonical Probit, Log-Log and Complementary Log-Log Links," Mathematics and Statistics, Vol. 13, No. 6, pp. 425 - 438, 2025. DOI: 10.13189/ms.2025.130601.
(b). APA Format:
Xuelu Sun , Stephen J. Quinn , Sunil Bhar (2025). Summary Goodness-of-Fit Statistics for Binary Generalized Linear Models with Noncanonical Probit, Log-Log and Complementary Log-Log Links. Mathematics and Statistics, 13(6), 425 - 438. DOI: 10.13189/ms.2025.130601.