Handbook of Measurement Error Models

The book belongs to the series Handbooks of Modern Statistical Methods and presents a compendium of 24 papers by 28 leading experts on the problems of modern statistical regression modeling with the Measurement Error (ME) in the covariates. These methods are important in medical and epidemiological, economics and environmental studies, survey research and other fields of error-prone data. The book is structured in 7 parts with 24 chapters given in multiple sections.

Part I of Introduction starts with the Chapter 1 “Measurement error models—A brief account of past developments and modern advancements” which describes linear and nonlinear regression models with ME in response and covariates. The STRATOS group provides guidance for design and analysis in observational studies describing types of ME, their effect on parameter estimation, and correcting methods (https://www.stratos-initiative.org/). Chapter 2 “The impact of unacknowledged measurement error” considers estimation of parameters in regressions with ME in the predictors and bias toward the null, confounding and differential misclassification in a predictor, specifics for the logit and survival models, polychotomous predictors and multiple covariates measured with error, Berkson error and other ways of errorpropagation.

Part II of Identifiability and Estimation contains three chapters. Chapter 3 “Identifiability in measurement error models” deals with statistical inference when ME in covariates produce so much uncertainty that it is hardly possible to provide sufficient amount of information to identify the unknown parameters, no matter how large the sample size is. Various ME types are formalized, conditional moments and surrogate predictors described, and examples given. Chapter 4 “Partial learning of misclassification parameters” focuses on incorporating the prior information into Bayesian analysis where the posterior distribution integrates several levels of uncertainty, and aggregating across different underlying values in the parameter space yields the posterior mean closer to the truth than the prior mean. Chapter 5 “Using instrumental variables to estimate models with mismeasured regressors” reviews classical approaches to regression analysis in presence of ME, including works by Frisch, Koopmans, Deming, Durbin, and modern instrumental variables (IV)developments.

Part III of General Methodology covers four chapters. Chapter 6 “Likelihood methods for measurement error and misclassification” draws an inference framework for handling ME in covariates, in response, and in them both. Induced regression and bias assessment for the probit and linear regression with interaction and quadratic terms, functional and structural schemes, conditional score, observed likelihood, validation data, likelihood with repeated surrogate measurements, expectation-maximization, and other problems and algorithms are described. Chapter 7 “Regression calibration for covariate measurement error” provides mathematical justification of this method for linear and high-degree of non-linearity models, with discussion and examples. Chapter 8 “Conditional and corrected score methods” demonstrates how to improve estimations with ME by employing the functional modeling approaches. The likelihood score function, conditional and corrected score approaches are described for the generalized linear model (GLM) and Cox hazard regression. Modified corrected score for discrete covariates in misclassification problem, complex variable simulation extrapolation for corrected score, regularization extrapolation corrected score (RECS), and trend constrained corrected score methods are considered. Chapter 9 “Semiparametric methods for measurement error and misclassification” describes the Maximum Likelihood Estimator (MLE), Method of Moment Estimator (MME), and Regular Asymptotically Linear (RAL) estimator. Feasibility and alteration questions, algorithm for ME model and implementation, and logistic binary response model are investigated.

Part IV of Nonparametric Inference includes five chapters. Chapter 10 “Deconvolution kernel density estimation” describes variables observed with an independent additive noise of known distribution, the classical ME model, the most popular nonparametric deconvolution kernel estimator and its theoretical properties, mean integrated squared error and its asymptotic, Fourier domain and supersmooth distribution of a variable. Problems of practical bandwidth estimation and computing the kernel estimator, cross-validation and bootstrap bandwidth, simulation extrapolation (SIMEX) procedure are described. Generalizations include analytic inversion formulae, vanishing of Fourier transform, heteroscedastic errors, and multivariate cases. Chapter 11 “Nonparametric deconvolution by Fourier transformation and other related approaches” continues with density estimation in series expansion and deconvolution by splines and log-splines with the Expectation-Maximization (EM) algorithm, wavelets and low-order approximations, and problems in image analysis related to inverse of the ill-posed structures. Chapter 12 “Deconvolution with unknown error distribution” considers more complicated cases of unknown errors incorporated into covariates when estimation is possible by error density found via additional observations with replicated measurements of homoscedastic or heteroscedastic noise. Boundary estimation with ME and misspecified error distributions are studied. Chapter 13 “Nonparametric inference methods for Berkson errors” is devoted to the specific error structures known as Berkson ME. When a possibly multidimensional random variable X cannot be directly measured but a proxy variable X* is observed, the classical ME model assumes the relation X*=X + U, while the Berkson modeling assumes the relation X = X*+U, where the random variable U represents ME. The stochastic structure of the classic model defines X and U as independent, but in Berkson model X* and U are independent. Berkson model finds applications in regression and designed experiments in agriculture, biology, and epidemiology. Its features are discussed, including consistent and kernel-based estimators, Mean Integrated Squared Error (MISE), regression with Berkson ME, and expansion by trigonometric functions. Chapter 14 “Nonparametric measurement errors models for regression” present theoretical description of estimator in a general form and modifications of the classic estimator, asymptotic properties and asymptotic normality, consistency and optimal convergence rates, unknown or partially known error distributions, nonparametric predictions, unbiased score and Monte Carlo methods, confidence interval and bands, and multivariateregression.

Part V of Applications presents five more chapters. Chapter 15 “Covariate measurement error in survival data” explores ME modeling in the time-to-event analysis. It describes censoring data, identifiability and auxiliary data, induced hazard function and ME in proportional hazard model, regression calibration and SIMEX, corrected likelihood and Bayesian methods. Chapter 16 “Mixed effects models with measurement errors in time-dependent covariates” describes the Generalized Linear Mixed Models (GLMM) and Nonlinear Mixed Effects (NLME) with covariate ME, applied in HIV/AIDS study. Chapter 17 “Estimation in mixed-effects models with measurement error” continues with longitudinal data analysis in GLMM with ME describing simulation-based numerical computations for mixed Poisson and logistic models, with mathematical proofs for theorems of asymptotic theory. Chapter 18 “Measurement error in dynamic models” focuses on parameter estimation for autoregressive (AR) and autoregressive moving average (ARMA) models with ME variances. General dynamic nonlinear autoregressive models and their estimators are presented in correcting methods, including SIMEX, bootstrapping, and regression calibration. Chapter 19 “Spatial exposure measurement error in environmental epidemiology” discusses evaluation of the air, water, land, and multipollutant exposures on humanhealth.

Part VI of Other Features presents three chapters. Chapter 20 “Measurement error as a missing data problem” considers the true variable as missing and applies statistical methods for handling missing data in accounting for ME in covariates. Regression calibration and conditional mean imputation, multiple imputation for ME correction, problems of validation and replication are considered, and example on risk factors for cardiovascular disease is discussed. Chapter 21 “Measurement error in causal inference” is devoted to synthesis of potential outcomes and casual modeling in application to ME estimations. Assumptions of consistency, exchangeability, and positivity in causal inference are described, together with the directed acyclic graphs (DAG), structural representation of ME in the mediator, estimation of casual effects in the ME presence, and ME correction in casual inference. Chapter 22 “Measurement error and misclassification in meta-analysis” concentrates on the random-effect models and control risk regression with structural and functional approaches for correcting techniques, diagnostic tests with sensitivity-specificity, conditional probability and ROC curve, bivariate approach and multiple thresholds. It describes SAS codes and R software packages, including CopulaREMADA, xmeta, Metatron, mada, bamdit, andmeta4diag.

Part VII of Bayesian Analysis presents two last chapters. Chapter 23 “Bayesian adjustment for misclassification” considers ME in binary variables that is a common problem known in public health, epidemiology, and quality control. Bayesian two-stage method used in validation data, multiple diagnostic tests, regression models, and alternative prior structures are described. Software is discussed, including OpenBUGS, JAGS, and Stan. Chapter 24 “Bayesian approaches for handling covariate measurement error” covers a broad spectrum of techniques handling uncertainties in data with ME. They include continuous and binary responses and time-to-event models, useful in genetic studies of diseases, and other medical, engineering, marketing, and social sciences.

Author and subject detail indices finalize the book. Each chapter is completed with the recent bibliography in the topics. Written by rigorous mathematical language, the papers in the book can be useful to professional statisticians and graduate students specializing in advanced regression modeling and analysis of data with measurement errors. Some resources for modeling with ME can also be found in Lipovetsky (2019, 2022).

Stan Lipovetsky
Minneapolis, MN