Summary Goodness-of-Fit Statistics for Binary Generalized Linear Models with Noncanonical Probit, Log-Log and Complementary Log-Log Links

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 (<img src=image/13442801_01.gif>) statistic, the normalized unweighted sum of squares (<img src=image/13442801_02.gif>) statistic, and the Hjort-Hosmer (<img src=image/13442801_03.gif>) 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 <img src=image/13442801_02.gif> (34.8%) and <img src=image/13442801_03.gif> (33.0%) statistics generally achieved higher overall rejection rates than the <img src=image/13442801_01.gif> statistic (27.8%). However, in some scenarios, the <img src=image/13442801_01.gif> statistic outperformed both the <img src=image/13442801_02.gif> and <img src=image/13442801_03.gif> statistics, suggesting that all three statistics have a role in assessing model adequacy.