Measurement Error in Nonlinear Models

Transcription

1 Measurement Error in Nonlinear Models R.J. CARROLL Professor of Statistics Texas A&M University, USA D. RUPPERT Professor of Operations Research and Industrial Engineering Cornell University, USA and L.A. STEFANSKI Professor, Department of Statistics North Carolina State University, USA m CHAPMAN & HALL London Glasgow Weinheim New York Tokyo Melbourne Madras

2 Contents Preface Guide to Notation xvii xxiii 1 INTRODUCTION Measurement Error Examples Nutrition Studies Nurses' Health Study Bioassay in a Herbicide Study Lung Function in Children Coronary Heart Disease and Blood Pressure A-Bomb Survivor Data Blood Pressure and Urinary Sodium Chloride Functional and Structural Models Models for Measurement Error General Approaches Transportability of Models Potential Dangers of Transporting Models Sources of Data Is There an "Exact" Predictor? Differential and Nondifferential Error True and Approximate Replicates Measurement Error as a Missing Data Problem Prediction A Brief Tour 19 2 REGRESSION AND ATTENUATION Introduction 21

3 X CONTENTS 2.2 Bias Caused by Measurement Error Simple Linear Regression with Additive Error Simple Linear Regression, More Complex Error Structure Multiple Regression: Single Covariate Measured with Error Multiple Covariates Measured with Error Correcting for Bias Method of Moments Orthogonal Regression Bias Versus Variance Attenuation in General Problems An Illustration of Nondifferential Measurement Error Other References Appendix 38 REGRESSION CALIBRATION Overview The Regression Calibration Algorithm Correction for Attenuation NHANES Example Estimating the Calibration Function Parameters Overview and First Methods Best Linear Approximations Using Replicate Data Nonlinear Calibration Function Models Alternatives When Using Partial Replicates James-Stein Calibration Standard Errors Expanded Regression Calibration Models The Expanded Approximation Defined Implementation Models Without Severe Curvature Bioassay Data Heuristics and Accuracy of the Approximations Examples of the Approximations Linear Regression Logistic Regression Loglinear Mean Models 66

4 CONTENTS XI 3.10 Theoretical Examples Homoscedastic Regression Quadratic Regression with Homoscedastic Regression Calibration Loglinear Mean Model Small Curvature, Heteroscedastic Calibration Other References Appendix Error Variance Estimation in the CSFII Standard Errors and Replication Quadratic Regression: Details of The Expanded Calibration Model 78 SIMULATION EXTRAPOLATION Overview Simulation Extrapolation Heuristics The SIMEX Algorithm The Simulation and Extrapolation Steps Modifications of the Simulation Step Estimating the Measurement Error Variance Extrapolant Function Considerations Inference and Standard Errors Relation to the Jackknife Nonadditive Measurement Error Framingham Heart Study Füll Replication Partial Replication SIMEX in Some Important Special Cases Multiple Linear Regression Loglinear Mean Models Quadratic Mean Models Segmented Linear Regression Mean Models Theory and Variance Estimation Simulation Extrapolation Variance Estimation Estimating Equation Approach to Variance Estimation 101 INSTRUMENTAL VARIABLES Overview Approximate Instrumental Variable Estimation 108

5 CONTENTS First Regression Calibration Instrumental Variable Algorithm Second Regression Calibration Instrumental Variable Algorithm An Example Derivation of the Estimators First Regression Calibration Instrumental Variable Algorithm Second Regression Calibration Instrumental Variable Algorithm Asymptotic Distribution Approximations Two-Stage Estimation Computing Estimates and Standard Errors 120 FUNCTIONAL METHODS Overview Linear, Logistic and Gamma-Loglinear Models Framingham Data Unbiased Score Functions via Conditioning Linear and Logistic Regression Other Canonical Models Computation Inference Exact Corrected Estimating Equations Likelihoods With Exponentials and Powers Asymptotic Distribution Approximation Estimated Y, uu Infinite Series Corrected Estimating Equations Rare-Event Logistic Regression Extensions to Mean and Variance Function Models Comparison of Methods Appendix Technical Complements to Conditional Score Theory Technical Complements to Distribution Theory for Estimated 139 LIKELIHOOD AND QUASILIKELIHOOD Introduction 141

6 CONTENTS xin Identifiable Models Measurement Error Models and Missing Data Likelihood Methods when X is Unobserved Error Models Likelihood and External Second Measures The Berkson Model Error Model Choice Likelihood When X is Partly Observed Numerical Computation of Likelihoods Framingham Data Bronchitis Example Quasilikelihood and Variance Function Models Appendix Monte-Carlo Computation of Integrals Linear, Probit and Logistic Regression 162 BAYESIAN METHODS Overview The Gibbs Sampler Direct Sampling without Measurement Error The Weighted Bootstrap Forming Complete Data Importance Sampling Cervical Cancer Framingham Data Details of the Gibbs Sampler and Weighted Bootstrap 178 SEMIPARAMETRIC METHODS Using Only Complete Data Special Two-Stage Designs for Binary Responses Pseudolikelihood Mean Score Method General Unbiased Estimating Functions Using Polynomials Optimal Moment-Based Estimators Mean Based Moment-Based Estimators Semiparametric Regression Calibration Comparison of the Methods Appendix 194

7 XIV CONTENTS Use of Complete Data Only Theory for Complete Data Only Theory of Moment-Estimating Functions UNKNOWN LINK FUNCTIONS Overview Constants of Proportionality Estimation Methods Some Basic Facts Least Squares and Sliced Inverse Regression Details of Implementation Framingham Heart Study Appendix Basic Theory HYPOTHESIS TESTING Overview The Regression Calibration Approximation Testing H Testing H 0 :ß x =0 ßz= Testing H 0 (ßißtr = Hypotheses about Subvectors of ß x and ß z Efficient Score Tests of H 0 : ß x = Generalized Score Tests DENSITY ESTIMATION AND NONPARAMET- RIC REGRESSION Deconvolution Parametric Deconvolution via Moments Estimating Distribution Functions Optimal Score Tests Framingham Data NHANES Data Nonparametric Regression SIMEX Regression Calibration QVF and Likelihood Models Framingham Data Other Methods 228

8 13 RESPONSE VARIABLE ERROR Additive/Multiplicative Error and QVF Models Unbiased Measures of True Response Recommendations Biased Responses Calibration Likelihood Methods General Likelihood Theory and Surrogates Use of Complete Data Only Other Methods Semiparametric Methods Pseudolikelihood Simple Random Subsampling Modified Pseudolikelihood Other Types of Subsampling Example OTHER TOPICS Logistic Case-Control Studies The Case that X is Observed Measurement Error Normal Discriminant Model Differential Measurement Error Likelihood Formulation Functional Methods in Two-Stage Studies Comparison of Functional and Likelihood Approaches Mixture Methods as Functional Modeling Overview Nonparametric Mixture Likelihoods A Cholesterol Example Covariates Measured Without Error Design of Two-Stage Validation and Replication Studies Misclassification Survival Analysis General Considerations Rare Events Risk Set Calibration 255

9 CONTENTS FITTING METHODS AND MODELS 257 A.l Overview 257 A.2 Likelihood Methods 257 A.2.1 Notation 257 A.2.2 Maximum likelihood Estimation 258 A.2.3 Likelihood Ratio Tests 259 A.2.4 Profile Likelihood and Likelihood Ratio Confidence Intervals 259 A.2.5 Efficient Score Tests 260 A.3 Unbiased Estimating Equations 261 A.3.1 Introduction and Basic Large Sample Theory 261 A.3.2 Sandwich Formula Example: Linear Regression Without Measurement Error 264 A.3.3 Sandwich Method and Likelihood-type Inference 265 A.3.4 Unbiased, But Conditionally Biased, Estimating Equations 266 A.3.5 Biased Estimating Equations 267 A.3.6 Stacking Estimating Equations: Using Prior Estimates of Some Parameters 267 A.4 Quasilikelihood and Variance Function (QVF) Models269 A.4.1 General Ideas 269 A.4.2 Estimation and Inference for QVF Models 270 A.5 Generalized Linear Models 273 A.6 Bootstrap Methods 273 A.6.1 Introduction 273 A.6.2 Nonlinear Regression Without Measurement Error 274 A.6.3 Bootstrapping Heteroscedastic Regression Models 276 A.6.4 Bootstrapping Logistic Regression Models 277 A.6.5 Bootstrapping Measurement Error Models 277 A.6.6 Bootstrap Confidence Intervals 278 References 280 Author index 298 Subject index 301