Regression analysis with response-selective sampling

Response-selective sampling, in which samples are drawn from a population according to the values of the response variable, is common in biomedical, epidemiological, economic and social studies. This paper proposes to use transformation models, the generalized accelerated failure time models in econometrics, for regression analysis with response-selective sampling. With unknown error distribution, the transformation models are broad enough to cover linear regression models, Cox's model, and the proportional odds model as special cases. To the best of our knowledge, except for the case-control logistic regression, there is presently no prospective estimation approach that can work for biased sampling without modification. We prove that the maximum rank correlation estimation is valid for response-selective sampling and establish its consistency and asymptotic normality. Unlike inverse probability methods, the proposed method of estimation does not involve sampling probabilities, which are often difficult to obtain in practice. Without the need of estimating the unknown transformation function or the error distribution, the proposed method is numerically easy to implement with the Nelder-Mead simplex algorithm that does not require convexity or continuity. We propose an inference procedure using random weighting to avoid the complication of density estimation when using the plug-in rule for variance estimation. Numerical studies with supportive evidence are presented. Application is illustrated with the Forbes Global 2000 data.