Regression Analysis with Response-biased Sampling

Response-biased sampling, in which samples are drawn from a popula- tion according to the values of the response variable, is common in biomedical, epidemiological, economic and social studies. In particular, the complete obser- vations in data with censoring, truncation or missing covariates can be regarded as response-biased sampling under certain conditions. This paper 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 re- gression 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 corre- lation 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 trans- formation 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 weight- ing 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.

[1]  Jing Ning,et al.  Non‐parametric tests for right‐censored data with biased sampling , 2010, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[2]  Steven R. Lerman,et al.  The Estimation of Choice Probabilities from Choice Based Samples , 1977 .

[3]  R. Pyke,et al.  Logistic disease incidence models and case-control studies , 1979 .

[4]  R. Sherman.  The Limiting Distribution of the Maximum Rank Correlation Estimator , 1993 .

[5]  J. Heckman.  Sample Selection Bias as a Specification Error (with an Application to the Estimation of Labor Supply Functions) , 1977 .

[6]  Rupert G. Miller,et al.  Regression with censored data , 1982 .

[7]  Charles F. Manski,et al.  3 The selection problem in econometrics and statistics , 1993 .

[8]  Mei-Cheng Wang,et al.  Hazards regression analysis for length-biased data , 1996 .

[9]  Jason Abrevaya,et al.  Computation of the maximum rank correlation estimator , 1999 .

[10]  Jerald F. Lawless,et al.  Semiparametric methods for response‐selective and missing data problems in regression , 1999 .

[11]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[12]  Aaron K. Han.  Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator , 1987 .

[13]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[14]  David A. Binder,et al.  Fitting Cox's proportional hazards models from survey data , 1992 .

[15]  Donglin Zeng,et al.  Maximum likelihood estimation in semiparametric regression models with censored data , 2007, Statistica Sinica.

[16]  J. Anderson.  Separate sample logistic discrimination , 1972 .

[17]  Peter J. Bickel,et al.  Large sample theory of estimation in biased sampling regression , 1991 .

[18]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[19]  S. Cosslett,et al.  Maximum likelihood estimator for choice-based samples , 1981 .

[20]  Kjell A. Doksum,et al.  Estimation and Testing in a Two-Sample Generalized Odds-Rate Model , 1988 .

[21]  Chiung-Yu Huang,et al.  Nonparametric estimation for length-biased and right-censored data. , 2011, Biometrika.

[22]  N. Breslow,et al.  The analysis of case-control studies , 1980 .

[23]  A. Scott,et al.  Fitting Logistic Models Under Case‐Control or Choice Based Sampling , 1986 .

[24]  W. Greene.  Sample Selection Bias as a Specification Error: Comment , 1981 .

[25]  Danyu Lin,et al.  On fitting Cox's proportional hazards models to survey data , 2000 .

[26]  Elie Tamer,et al.  Partial rank estimation of duration models with general forms of censoring , 2007 .

[27]  Jian Huang,et al.  A semiparametric approach for the nonparametric transformation survival model with multiple covariates. , 2007, Biostatistics.

[28]  W. Tsai,et al.  Pseudo-partial likelihood for proportional hazards models with biased-sampling data. , 2009, Biometrika.

[29]  Kani Chen,et al.  Parametric models for response‐biased sampling , 2001 .

[30]  Robert P. Sherman,et al.  Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators , 1994 .

[31]  Xiaodong Luo,et al.  Nonparametric estimation for right-censored length-biased data: a pseudo-partial likelihood approach , 2009 .

[32]  D. Horvitz,et al.  A Generalization of Sampling Without Replacement from a Finite Universe , 1952 .

[33]  M. Plummer,et al.  International agency for research on cancer. , 2020, Archives of pathology.

[34]  Jing Ning,et al.  Analyzing Length-Biased Data With Semiparametric Transformation and Accelerated Failure Time Models , 2009, Journal of the American Statistical Association.

[35]  J. Heckman.  Sample selection bias as a specification error , 1979 .

[36]  Z. Ying,et al.  Analysis of transformation models with censored data , 1995 .

[37]  Z. Ying,et al.  A simple resampling method by perturbing the minimand , 2001 .

[38]  Zhiliang Ying,et al.  Semiparametric analysis of transformation models with censored data , 2002 .

[39]  Xiaodong Luo,et al.  Pseudo-partial likelihood estimators for the Cox regression model with missing covariates. , 2009, Biometrika.