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

Abstract

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. A remarkable 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 chi-squared 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.