Methods: The analysis includes two simulation exercises and an empirical application. The two simulations differ by the distribution (Gamma or Normal) of the continuous covariate. Simulations are conducted for 1000 replications and for sample sizes ranging from 100 to 50000. The statistic of interest is the proportion of replicates for which the p-value on the Hosmer-Lemeshow (HL) test exceeds 0.05. An empirical application uses data on 48254 live discharges in Maryland following a hospitalization for stroke. The dependent variable is an indicator for a discharge to a medical care facility. Covariates include race, gender, age, marital status, insurance status, emergency room admission, and stroke type. The HL, modified deviance (MD) and modified Pearson (MP) tests assess model fit.
Results: Simulation results (proportion of well-calibrated replicates for LS; proportion of well-calibrated replicates for BRM) show that the relative advantage of the BRM increases with sample size. The models perform well at a sample size of 100 for the Gamma (0.948; 0.953) and the Normal (0.773; 0.913). However, their performance diverges as the sample size increases; the BRM outperforms the LS at a sample of 50000 for the Gamma (0.083; 0.957) and at a sample of 5000 for the Normal (0.000; 0.967). In the empirical application, diagnostic tests (p-values) indicate that the BRM outperforms the LS. The HL (p<0.0001), MD (p<0.0001) and MP (p<0.0001) tests indicate that the LS is misspecified and the HL (p= 0.25), MD (p= 0.10) and MP (p=0.06) tests indicate that the BRM is well-calibrated to the data.
Conclusion: The BRM outperforms the LS across reasonable sample sizes and utilizing continuous covariate distributions that are likely in retrospective analyses. While the specificity problem of Monte Carlo studies applies, these simulation results are suggestive of the advantage of BRM over LS. An empirical application shows the superiority of the BRM in a specific population-based dataset.