Response-biased 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. In particular, the complete observations in data with censoring, truncation or missing covariates can be regarded as response-biased sampling under certain conditions. This work proposes to use transformation models, known as the generalized accelerated failure time model in econometrics, for regression analysis with response-biased sampling. With unknown error distribution, the transformation models are broad enough to cover linear regression models, the Cox's model and the proportional odds model as special cases. To the best of our knowledge, except for the case-control logistic regres...[ Read more ]

Response-biased 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. In particular, the complete observations in data with censoring, truncation or missing covariates can be regarded as response-biased sampling under certain conditions. This work proposes to use transformation models, known as the generalized accelerated failure time model in econometrics, for regression analysis with response-biased sampling. With unknown error distribution, the transformation models are broad enough to cover linear regression models, the 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 no report in the literature that a prospective estimation approach can work for biased sampling without any modification. We prove that the maximum rank correlation estimation is valid for response-biased sampling and establish its consistency and asymptotic normality. Unlike the inverse probability methods, the proposed method of estimation does not involve the 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, which 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. Applications are illustrated with the Forbes Global 2000 data and the Stanford heart transplant data. Inspired by the maximum rank estimation with response-biased data, a similar rank method for partial rank estimation with censored data is proposed. This method is also supported by presented simulation studies.