Calibrated Bayes, for Statistics in General, and Missing Data in Particular

It is argued that the Calibrated Bayesian (CB) approach to statistical inference capitalizes on the strength of Bayesian and frequen- tist approaches to statistical inference. In the CB approach, inferences under a particular model are Bayesian, but frequentist methods are useful for model development and model checking. In this article the CB approach is outlined. Bayesian methods for missing data are then reviewed from a CB perspective. The basic theory of the Bayesian ap- proach, and the closely related technique of multiple imputation, is described. Then applications of the Bayesian approach to normal mod- els are described, both for monotone and nonmonotone missing data patterns. Sequential Regression Multivariate Imputation and Penalized Spline of Propensity Models are presented as two useful approaches for relaxing distributional assumptions.

[1]  George E. P. Box,et al.  Sampling and Bayes' inference in scientific modelling and robustness , 1980 .

[2]  Xiao-Li Meng,et al.  The EM Algorithm—an Old Folk‐song Sung to a Fast New Tune , 1997 .

[3]  R. Little,et al.  Robust Likelihood-based Analysis of Multivariate Data with Missing Values , 2003 .

[4]  D. Rubin,et al.  The central role of the propensity score in observational studies for causal effects , 1983 .

[5]  D. Rubin.  Multiple Imputation After 18+ Years , 1996 .

[6]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[7]  Russell V. Lenth,et al.  Statistical Analysis With Missing Data (2nd ed.) (Book) , 2004 .

[8]  Geert Verbeke,et al.  Multiple Imputation for Model Checking: Completed‐Data Plots with Missing and Latent Data , 2005, Biometrics.

[9]  Allan R. Sampson,et al.  A multivariate correlation ratio , 1984 .

[10]  Chiu-Hsieh Hsu,et al.  Survival analysis using auxiliary variables via non‐parametric multiple imputation , 2006, Statistics in medicine.

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

[12]  D. Rubin,et al.  Multiple Imputation for Interval Estimation from Simple Random Samples with Ignorable Nonresponse , 1986 .

[13]  T. W. Anderson.  Maximum Likelihood Estimates for a Multivariate Normal Distribution when Some Observations are Missing , 1957 .

[14]  John Van Hoewyk,et al.  A multivariate technique for multiply imputing missing values using a sequence of regression models , 2001 .

[15]  D. Rubin.  Formalizing Subjective Notions about the Effect of Nonrespondents in Sample Surveys , 1977 .

[16]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .

[17]  Joseph Kang,et al.  Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data , 2007, 0804.2958.

[18]  Joseph L Schafer,et al.  Analysis of Incomplete Multivariate Data , 1997 .

[19]  R. Little.  Survey Nonresponse Adjustments for Estimates of Means , 1986 .

[20]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[21]  Donald B. Rubin,et al.  Multiple imputations in sample surveys , 1978 .

[22]  D. Rubin,et al.  Ignorability and Coarse Data , 1991 .

[23]  Trevillore E. Raghunathan,et al.  IVEware: Imputation and Variance Estimation Software User Guide , 2002 .

[24]  R. Little,et al.  Maximum likelihood inference for multiple regression with missing values , 1979 .

[25]  Andrew Gelman,et al.  Diagnostics for multivariate imputations , 2007 .

[26]  J. Robins,et al.  Estimation of Regression Coefficients When Some Regressors are not Always Observed , 1994 .

[27]  Kendrick,et al.  Applications of Mathematics to Medical Problems , 1925, Proceedings of the Edinburgh Mathematical Society.

[28]  Joseph G. Ibrahim,et al.  Missing data methods in longitudinal studies: a review , 2009 .

[29]  G. Tian,et al.  Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation , 2009 .

[30]  A. Dawid.  The Well-Calibrated Bayesian , 1982 .

[31]  Guangyu Zhang,et al.  Extensions of the Penalized Spline of Propensity Prediction Method of Imputation , 2009, Biometrics.

[32]  Central role , 2005, Veterinary Record.

[33]  R. R. Hocking,et al.  The analysis of incomplete data. , 1971 .

[34]  D. Rubin,et al.  Fully conditional specification in multivariate imputation , 2006 .

[35]  D. Rubin,et al.  Small-sample degrees of freedom with multiple imputation , 1999 .

[36]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[37]  R. Okafor.  Maximum likelihood estimation from incomplete data , 1987 .

[38]  R. Little.  SURVEY NONRESPONSE ADJUSTMENTS , 2002 .

[39]  Ingram Olkin,et al.  Multivariate Correlation Models with Mixed Discrete and Continuous Variables , 1961 .

[40]  Malay Ghosh,et al.  Small Area Estimation: An Appraisal , 1994 .

[41]  L. Baum,et al.  A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains , 1970 .

[42]  A. Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[43]  David Draper,et al.  Assessment and Propagation of Model Uncertainty , 2011 .

[44]  A. Rotnitzky,et al.  Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis by DANIELS, M. J. and HOGAN, J. W , 2009 .

[45]  J. Robins,et al.  Doubly Robust Estimation in Missing Data and Causal Inference Models , 2005, Biometrics.

[46]  M. Wand,et al.  Smoothing with Mixed Model Software , 2004 .

[47]  H. W. Peers.  On Confidence Points and Bayesian Probability Points in the Case of Several Parameters , 1965 .

[48]  B. L. Welch.  On Comparisons between Confidence Point Procedures in the Case of a Single Parameter , 1965 .

[49]  M. Woodbury.  A missing information principle: theory and applications , 1972 .

[50]  W. Gilks,et al.  Random-effects models, for longitudinal data using Gibbs sampling. , 1993, Biometrics.

[51]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[52]  D. Rubin.  Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician , 1984 .

[53]  A. M'Kendrick.  Applications of Mathematics to Medical Problems , 1925, Proceedings of the Edinburgh Mathematical Society.

[54]  James M. Robins,et al.  Semiparametric Regression for Repeated Outcomes With Nonignorable Nonresponse , 1998 .

[55]  Roderick J. A. Little,et al.  Approximately calibrated small sample inference about means from bivariate normal data with missing values , 1988 .