Quasi‐likelihood ratio tests and the Bartlett‐type correction for improved inferences of the modified Poisson and least‐squares regressions for binary outcomes

Logistic regression has been a standard multivariate analysis method for binary outcomes in clinical and epidemiological studies; however, the odds ratios cannot be interpreted as effect measures directly. The modified Poisson and least‐squares regressions are alternative effective methods to provide risk ratio and risk difference estimates. However, their ordinary Wald‐type inference methods using the sandwich variance estimator seriously underestimate the statistical errors under small or moderate sample settings. In this article, we develop alternative likelihood‐ratio‐type inference methods for these regression analyses based on Wedderburn's quasi‐likelihood theory. An advantage of the proposed methods is that we have correct information for the true models (i.e., the binomial log‐linear and linear models). Using this modeling information, we develop an effective parametric bootstrap algorithm for accurate inferences. In particular, we propose the Bartlett‐type mean calibration approach and bootstrap test‐based approach for the quasi‐likelihood ratio statistic. In addition, we propose another computationally efficient modified approximate quasi‐likelihood ratio statistic whose large sample distribution can be approximated by the χ2$$ {\chi}^2 $$ distribution and its bootstrap inference method. In numerical studies by simulations, the new bootstrap‐based methods outperformed the current standard Wald‐type confidence interval. We applied these methods to a clinical study of epilepsy.