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J Am Stat Assoc. 2013 Jan 1;108(501):217-227.

A Unified Approach to Semiparametric Transformation Models under General Biased Sampling Schemes.

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  • 1Department of Medicine, Stanford University, Stanford CA 94305.


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 data sets are presented.


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

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