Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

ABSTRACT

Due to cost-effectiveness and high efficiency, two-phase case-control sampling has been widely used in epidemiology studies. We develop a semi-parametric empirical likelihood approach to two-phase case-control data under the logistic regression model. We show that the maximum empirical likelihood estimator has an asymptotically normal distribution, and the empirical likelihood ratio follows an asymptotically central chi-square distribution. We find that the maximum empirical likelihood estimator is equal to Breslow and Holubkov (1997)'s maximum likelihood estimator. Even so, the limiting distribution of the likelihood ratio, likelihood-ratio-based interval, and test are all new. Furthermore, we construct new Kolmogorov–Smirnov type goodness-of-fit tests to test the validation of the underlying logistic regression model. Our simulation results and a real application show that the likelihood-ratio-based interval and test have certain merits over the Wald-type counterparts and that the proposed goodness-of-fit test is valid.

KEYWORDS:

• Bootstrap
• case-control data
• empirical likelihood
• goodness-of-fit test
• two-phase case-control sampling

Acknowledgments

The authors thank the Editor, the Associate Editor, and the two anonymous referees for helpful comments and suggestions that have led to significant improvements in the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the National Natural Science Foundation of China [grant number 11771144], the State Key Program of National Natural Science Foundation of China [grant number 71931004], [grant number 32030063], the development fund for Shanghai talents, and the 111 project (B14019).

Notes on contributors

Zhen Sheng

Zhen Sheng is a PhD candidate in School of Statistics, Faculty of Economic and Management, East China Normal University, China. She received her master degree in Statistics in 2018 from Qufu Normal University, China. Her research interests include case-control study, Genome-wide association study, and experimental design.

Yukun Liu

Yukun Liu is a Professor in School of Statistics, Faculty of Economic and Management, East China Normal University, China. He received his PhD in Statistics in 2009 from Nankai University, China. His research is focused on empirical likelihood and its applications to case-control data, capture-recapture data, selection biased data, and finite mixture models.

Jing Qin

Jing Qin is a Mathematical Statistician in the Biostatistics Research Branch of the National Institute of Allergy and Infectious Diseases (NIAID), USA. He received his PhD in Statistics from the University of Waterloo in 1992. His research interests include empirical likelihood, case-control study, length bias sampling, survival analysis, missing data, causal inference, and survey sampling.