Bayesian Inference for Semiparametric Binary Regression

Abstract We propose a regression model for binary response data that places no structural restrictions on the link function except monotonicity and known location and scale. Predictors enter linearly. We demonstrate Bayesian inference calculations in this model. By modifying the Dirichlet process, we obtain a natural prior measure over this semiparametric model, and we use Polya sequence theory to formulate this measure in terms of a finite number of unobserved variables. We design a Markov chain Monte Carlo algorithm for posterior simulation and apply the methodology to data on radiotherapy treatments for cancer.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  E. C. Fieller.  SOME PROBLEMS IN INTERVAL ESTIMATION , 1954 .

[4]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[5]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[6]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[7]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[8]  D. Blackwell.  Discreteness of Ferguson Selections , 1973 .

[9]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[10]  C. Manski.  MAXIMUM SCORE ESTIMATION OF THE STOCHASTIC UTILITY MODEL OF CHOICE , 1975 .

[11]  R. Prentice,et al.  A generalization of the probit and logit methods for dose response curves. , 1976, Biometrics.

[12]  J. Kingman.  The population structure associated with the Ewens sampling formula. , 1977, Theoretical population biology.

[13]  D. Pregibon.  Goodness of Link Tests for Generalized Linear Models , 1980 .

[14]  J. Sethuraman,et al.  Convergence of Dirichlet Measures and the Interpretation of Their Parameter. , 1981 .

[15]  Francisco J. Aranda-Ordaz,et al.  On Two Families of Transformations to Additivity for Binary Response Data , 1981 .

[16]  Víctor M. Guerrero,et al.  Use of the Box-Cox transformation with binary response models , 1982 .

[17]  S. Cosslett.  DISTRIBUTION-FREE MAXIMUM LIKELIHOOD ESTIMATOR OF THE BINARY CHOICE MODEL1 , 1983 .

[18]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  Hani Doss.  Bayesian Nonparametric Estimation of the Median; Part I: Computation of the Estimates , 1985 .

[20]  S. Weisberg.  Plots, transformations, and regression , 1985 .

[21]  Nils Lid Hjort.  Discussion: On the Consistency of Bayes Estimates , 1986 .

[22]  D. Freedman,et al.  On the consistency of Bayes estimates , 1986 .

[23]  Trevor Hastie,et al.  Non‐Parametric Logistic and Proportional Odds Regression , 1987 .

[24]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[25]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[26]  Jeremy M. G. Taylor.  The Cost of Generalizing Logistic Regression , 1988 .

[27]  T. Stukel.  Generalized Logistic Models , 1988 .

[28]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[29]  Diane Lambert,et al.  Generalizing Logistic Regression by Nonparametric Mixing , 1989 .

[30]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[31]  A. Gelfand,et al.  Nonparametric Bayesian bioassay including ordered polytomous response , 1991 .

[32]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[33]  R. Arratia,et al.  Poisson Process Approximations for the Ewens Sampling Formula , 1992 .

[34]  T. Santner,et al.  Orthogonalizing parametric link transformation families in binary regression analysis , 1992 .

[35]  Claudia Czado,et al.  On Link Selection in Generalized Linear Models , 1992 .

[36]  Claudia Czado,et al.  The effect of link misspecification on binary regression inference , 1992 .

[37]  J. Fowler,et al.  Sensitivity of human squamous cell carcinoma of the larynx to fractionated radiotherapy. , 1993, The British journal of radiology.

[38]  R. Spady,et al.  AN EFFICIENT SEMIPARAMETRIC ESTIMATOR FOR BINARY RESPONSE MODELS , 1993 .

[39]  D. Spiegelhalter,et al.  Modelling Complexity: Applications of Gibbs Sampling in Medicine , 1993 .

[40]  J. Besag,et al.  Spatial Statistics and Bayesian Computation , 1993 .

[41]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[42]  Peter Green,et al.  Spatial statistics and Bayesian computation (with discussion) , 1993 .

[43]  Hani Doss.  Bayesian Nonparametric Estimation for Incomplete Data Via Successive Substitution Sampling , 1994 .

[44]  Parametric link modification of both tails in binary regression , 1994 .

[45]  Bayesian inference of binary regression models with parametric link , 1994 .

[46]  K. Cheng,et al.  Testing Goodness of Fit for a Parametric Family of Link Functions , 1994 .

[47]  Bani K. Mallick,et al.  Generalized linear models with unknown link functions , 1994 .

[48]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[49]  M. Escobar.  Estimating Normal Means with a Dirichlet Process Prior , 1994 .

[50]  S. MacEachern.  Estimating normal means with a conjugate style dirichlet process prior , 1994 .

[51]  R. Chappell,et al.  Modeling Dose and Local Control in Radiotherapy , 1995 .

[52]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[53]  J. Skilling,et al.  Bayesian Density Estimation , 1996 .

[54]  Jun S. Liu.  Nonparametric hierarchical Bayes via sequential imputations , 1996 .