A Unified Approach to Semiparametric Transformation Models Under General Biased Sampling Schemes

Abstract

We propose a unified estimation method for semiparametric linear transformation models under general biased sampling schemes. The new estimator is obtained from a set of counting process-based unbiased estimating equations, developed through introducing a general weighting scheme that offsets the sampling bias. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. A closed-form formula is derived for the limiting variance and the plug-in estimator is shown to be consistent. We demonstrate the unified approach through the special cases of left truncation, length bias, the case-cohort design, and variants thereof. Simulation studies and applications to real datasets are presented.

Keywords:

• Case-cohort design
• Counting process
• Cox model
• Estimating equations
• Importance sampling
• Length bias
• Proportional odds model
• Regression
• Survival data
• Truncation

Acknowledgments

The authors thank the associate editor and three anonymous referees for their constructive comments that led to substantial improvements. This research was supported in part by grants from the National Science Foundation and the National Institutes of Health and by a fellowship from Sir Edward Youde Memorial Fund.

Notes

NOTE: Bias, var, , and ECP are defined as the difference between the estimated and the true parameter values, the asymptotic variance estimated, the variance of the simulated estimated parameter values as well as the empirical coverage probability, respectively.

NOTE: Bias, var, , and ECP are defined as the difference between the estimated and the true parameter values, the asymptotic variance estimated, the variance of the simulated estimated parameter values as well as the empirical coverage probability, respectively.

NOTE: Bias, var, , and ECP are defined as the difference between the estimated and the true parameter values, the asymptotic variance estimated, the variance of the simulated estimated parameter values as well as the empirical coverage probability, respectively.

NOTE: Bias, var, , and ECP are defined as the difference between the estimated and the true parameter values, the asymptotic variance estimated, the variance of the simulated estimated parameter values as well as the empirical coverage probability, respectively.

NOTE: Bias, var, , and ECP are defined as the difference between the estimated and the true parameter values, the asymptotic variance estimated, the variance of the simulated estimated parameter values as well as the empirical coverage probability, respectively.