Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Simulation‐selection‐extrapolation: Estimation in high‐dimensional errors‐in‐variables models

Errors‐in‐variables models in high‐dimensional settings pose two challenges in application. First, the number of observed covariates is larger than the sample size, while only a small number of covariates are true predictors under an assumption of model sparsity. Second, the presence of measurement error can result in severely biased parameter estimates, and also affects the ability of penalized methods such as the lasso to recover the true sparsity pattern. A new estimation procedure called SIMulation‐SELection‐EXtrapolation (SIMSELEX) is proposed. This procedure makes double use of lasso methodology. First, the lasso is used to estimate sparse solutions in the simulation step, after which a group lasso is implemented to do variable selection. The SIMSELEX estimator is shown to perform well in variable selection, and has significantly lower estimation error than naive estimators that ignore measurement error. SIMSELEX can be applied in a variety of errors‐in‐variables settings, including linear models, generalized linear models, and Cox survival models. It is furthermore shown in the Supporting Information how SIMSELEX can be applied to spline‐based regression models. A simulation study is conducted to compare the SIMSELEX estimators to existing methods in the linear and logistic model settings, and to evaluate performance compared to naive methods in the Cox and spline models. Finally, the method is used to analyze a microarray dataset that contains gene expression measurements of favorable histology Wilms tumors.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

Robust best linear estimation for regression analysis using surrogate and instrumental variables

We investigate methods for regression analysis when covariates are measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies the classical measurement error model, but it may not have repeated measurements. In addition to the surrogate variables that are available among the subjects in the calibration sample, we assume that there is an instrumental variable (IV) that is available for all study subjects. An IV is correlated with the unobserved true exposure variable and hence can be useful in the estimation of the regression coefficients. We propose a robust best linear estimator that uses all the available data, which is the most efficient among a class of consistent estimators. The proposed estimator is shown to be consistent and asymptotically normal under very weak distributional assumptions. For Poisson or linear regression, the proposed estimator is consistent even if the measurement error from the surrogate or IV is heteroscedastic. Finite-sample performance of the proposed estimator is examined and compared with other estimators via intensive simulation studies. The proposed method and other methods are applied to a bladder cancer case-control study.     

Nonparametric correction for covariate measurement error in a stratified Cox model

Stratified Cox regression models with large number of strata and small stratum size are useful in many settings, including matched case-control family studies. In the presence of measurement error in covariates and a large number of strata, we show that extensions of existing methods fail either to reduce the bias or to correct the bias under nonsymmetric distributions of the true covariate or the error term. We propose a nonparametric correction method for the estimation of regression coefficients, and show that the estimators are asymptotically consistent for the true parameters. Small sample properties are evaluated in a simulation study. The method is illustrated with an analysis of Framingham data.     

Measurement error in a random walk model with applications to population dynamics

Population abundances are rarely, if ever, known. Instead, they are estimated with some amount of uncertainty. The resulting measurement error has its consequences on subsequent analyses that model population dynamics and estimate probabilities about abundances at future points in time. This article addresses some outstanding questions on the consequences of measurement error in one such dynamic model, the random walk with drift model, and proposes some new ways to correct for measurement error. We present a broad and realistic class of measurement error models that allows both heteroskedasticity and possible correlation in the measurement errors, and we provide analytical results about the biases of estimators that ignore the measurement error. Our new estimators include both method of moments estimators and "pseudo"-estimators that proceed from both observed estimates of population abundance and estimates of parameters in the measurement error model. We derive the asymptotic properties of our methods and existing methods, and we compare their finite-sample performance with a simulation experiment. We also examine the practical implications of the methods by using them to analyze two existing population dynamics data sets.     

Differential measurement errors in zero-truncated regression models for count data

Measurement errors in covariates may result in biased estimates in regression analysis. Most methods to correct this bias assume nondifferential measurement errors-i.e., that measurement errors are independent of the response variable. However, in regression models for zero-truncated count data, the number of error-prone covariate measurements for a given observational unit can equal its response count, implying a situation of differential measurement errors. To address this challenge, we develop a modified conditional score approach to achieve consistent estimation. The proposed method represents a novel technique, with efficiency gains achieved by augmenting random errors, and performs well in a simulation study. The method is demonstrated in an ecology application.     

Robust best linear estimator for Cox regression with instrumental variables in whole cohort and surrogates with additive measurement error in calibration sample

Biomedical researchers are often interested in estimating the effect of an environmental exposure in relation to a chronic disease endpoint. However, the exposure variable of interest may be measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies an additive measurement error model, but it may not have repeated measurements. The subset in which the surrogate variables are available is called a calibration sample. In addition to the surrogate variables that are available among the subjects in the calibration sample, we consider the situation when there is an instrumental variable available for all study subjects. An instrumental variable is correlated with the unobserved true exposure variable, and hence can be useful in the estimation of the regression coefficients. In this paper, we propose a nonparametric method for Cox regression using the observed data from the whole cohort. The nonparametric estimator is the best linear combination of a nonparametric correction estimator from the calibration sample and the difference of the naive estimators from the calibration sample and the whole cohort. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via intensive simulation studies. The methods are applied to the Nutritional Biomarkers Study of the Women's Health Initiative.     

Binary Regression Analysis with Pooled Exposure Measurements: A Regression Calibration Approach

Summary It has become increasingly common in epidemiological studies to pool specimens across subjects to achieve accurate quantitation of biomarkers and certain environmental chemicals. In this article, we consider the problem of fitting a binary regression model when an important exposure is subject to pooling. We take a regression calibration approach and derive several methods, including plug‐in methods that use a pooled measurement and other covariate information to predict the exposure level of an individual subject, and normality‐based methods that make further adjustments by assuming normality of calibration errors. Within each class we propose two ways to perform the calibration (covariate augmentation and imputation). These methods are shown in simulation experiments to effectively reduce the bias associated with the naive method that simply substitutes a pooled measurement for all individual measurements in the pool. In particular, the normality‐based imputation method performs reasonably well in a variety of settings, even under skewed distributions of calibration errors. The methods are illustrated using data from the Collaborative Perinatal Project.     

Use of baseline data for estimation of treatment effects in the presence of regression to the mean.

We consider a regression to the mean problem with a very large sample for the first measurement and relatively small subsample for the second measurement, selected on the basis of the initial measurement. This is a situation that often occurs in screening trials. We propose to estimate the unselected population mean and variance from the first measurement in the larger sample. Using these estimates, the correlation between the two measurements, as well as an effect of treatment, can be estimated in simple and explicit form. Under the condition that the size of the subsample is of a smaller order, the new estimators for all the four parameters are as asymptotically efficient as the usual maximum likelihood estimators. Tests based on this new approach are also discussed. An illustration from a cholesterol screening study is included.     

Robust inference on the average treatment effect using the outcome highly adaptive lasso

Many estimators of the average effect of a treatment on an outcome require estimation of the propensity score, the outcome regression, or both. It is often beneficial to utilize flexible techniques, such as semiparametric regression or machine learning, to estimate these quantities. However, optimal estimation of these regressions does not necessarily lead to optimal estimation of the average treatment effect, particularly in settings with strong instrumental variables. A recent proposal addressed these issues via the outcome-adaptive lasso, a penalized regression technique for estimating the propensity score that seeks to minimize the impact of instrumental variables on treatment effect estimators. However, a notable limitation of this approach is that its application is restricted to parametric models. We propose a more flexible alternative that we call the outcome highly adaptive lasso. We discuss the large sample theory for this estimator and propose closed-form confidence intervals based on the proposed estimator. We show via simulation that our method offers benefits over several popular approaches.     

No Longer Confidential: Estimating the Confidence of Individual Regression Predictions

Quantitative predictions in computational life sciences are often based on regression models. The advent of machine learning has led to highly accurate regression models that have gained widespread acceptance. While there are statistical methods available to estimate the global performance of regression models on a test or training dataset, it is often not clear how well this performance transfers to other datasets or how reliable an individual prediction is–a fact that often reduces a user’s trust into a computational method. In analogy to the concept of an experimental error, we sketch how estimators for individual prediction errors can be used to provide confidence intervals for individual predictions. Two novel statistical methods, named CONFINE and CONFIVE, can estimate the reliability of an individual prediction based on the local properties of nearby training data. The methods can be applied equally to linear and non-linear regression methods with very little computational overhead. We compare our confidence estimators with other existing confidence and applicability domain estimators on two biologically relevant problems (MHC–peptide binding prediction and quantitative structure-activity relationship (QSAR)). Our results suggest that the proposed confidence estimators perform comparable to or better than previously proposed estimation methods. Given a sufficient amount of training data, the estimators exhibit error estimates of high quality. In addition, we observed that the quality of estimated confidence intervals is predictable. We discuss how confidence estimation is influenced by noise, the number of features, and the dataset size. Estimating the confidence in individual prediction in terms of error intervals represents an important step from plain, non-informative predictions towards transparent and interpretable predictions that will help to improve the acceptance of computational methods in the biological community.     

Shrinkage Estimators for Covariance Matrices

Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.     

Recommendations for estimating mark rate of cetaceans in photo-ID research: A critique of field sampling protocols and variance estimation

Mark rate, or the proportion of the population with unique, identifiable marks, must be determined in order to estimate population size from photographic identification data. In this study we address field sampling protocols and estimation methods for robust estimation of mark rate and its uncertainty in cetacean populations. We present two alternatives for estimating the variance of mark rate: (1) a variance estimator for clusters of unequal sizes (SRCS) and (2) a hierarchical Bayesian model (SRCS-Bayes), and compare them to the simple random sampling (SRS) variance estimator. We tested these variance estimators using a simulation to see how they perform at varying mark rates, number of groups sampled, photos per group, and mean group sizes. The hierarchical Bayesian model outperformed the frequentist variance estimators, with the true mark rate of the population held in its 95% HDI 91.9% of the time (compared with coverage of 79% for the SRS method and 76.3% for the SRCS-Cochran method). The simulation results suggest that, ideally, mark rate and its precision should be quantified using hierarchical Bayesian modeling, and researchers should attempt to sample as many unique groups as possible to improve accuracy and precision.     

Random-Effects,Fixed-Effects and the within-between Specification for Clustered Data in Observational Health Studies: A Simulation Study

Background

When unaccounted-for group-level characteristics affect an outcome variable, traditional linear regression is inefficient and can be biased. The random- and fixed-effects estimators (RE and FE, respectively) are two competing methods that address these problems. While each estimator controls for otherwise unaccounted-for effects, the two estimators require different assumptions. Health researchers tend to favor RE estimation, while researchers from some other disciplines tend to favor FE estimation. In addition to RE and FE, an alternative method called within-between (WB) was suggested by Mundlak in 1978, although is utilized infrequently.

Methods

We conduct a simulation study to compare RE, FE, and WB estimation across 16,200 scenarios. The scenarios vary in the number of groups, the size of the groups, within-group variation, goodness-of-fit of the model, and the degree to which the model is correctly specified. Estimator preference is determined by lowest mean squared error of the estimated marginal effect and root mean squared error of fitted values.

Results

Although there are scenarios when each estimator is most appropriate, the cases in which traditional RE estimation is preferred are less common. In finite samples, the WB approach outperforms both traditional estimators. The Hausman test guides the practitioner to the estimator with the smallest absolute error only 61% of the time, and in many sample sizes simply applying the WB approach produces smaller absolute errors than following the suggestion of the test.

Conclusions

Specification and estimation should be carefully considered and ultimately guided by the objective of the analysis and characteristics of the data. The WB approach has been underutilized, particularly for inference on marginal effects in small samples. Blindly applying any estimator can lead to bias, inefficiency, and flawed inference.     

A simulation study of measurement error correction methods in logistic regression

Measurement error models in logistic regression have received considerable theoretical interest over the past 10-15 years. In this paper, we present the results of a simulation study that compares four estimation methods: the so-called regression calibration method, probit maximum likelihood as an approximation to the logistic maximum likelihood, the exact maximum likelihood method based on a logistic model, and the naive estimator, which is the result of simply ignoring the fact that some of the explanatory variables are measured with error. We have compared the behavior of these methods in a simple, additive measurement error model. We show that, in this situation, the regression calibration method is a very good alternative to more mathematically sophisticated methods.     

Influence functions and robust Bayes and empirical Bayes small area estimation

We introduce new robust small area estimation procedures basedon area-level models. We first find influence functions correspondingto each individual area-level observation by measuring the divergencebetween the posterior density functions of regression coefficientswith and without that observation. Next, based on these influencefunctions, properly standardized, we propose some new robustBayes and empirical Bayes small area estimators. The mean squarederrors and estimated mean squared errors of these estimatorsare also found. A small simulation study compares the performanceof the robust and the regular empirical Bayes estimators. Whenthe model variance is larger than the sample variance, the proposedrobust empirical Bayes estimators are superior.     

Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models

Line transect sampling is one of the most widely used methods for animal abundance assessment. Standard estimation methods assume certain detection on the transect, no animal movement, and no measurement errors. Failure of the assumptions can cause substantial bias. In this work, the effect of error measurement on line transect estimators is investigated. Based on considerations of the process generating the errors, a multiplicative error model is presented and a simple way of correcting estimates based on knowledge of the error distribution is proposed. Using beta models for the error distribution, the effect of errors and of the proposed correction is assessed by simulation. Adequate confidence intervals for the corrected estimates are obtained using a bootstrap variance estimate for the correction and the delta method. As noted by Chen (1998, Biometrics 54, 899-908), even unbiased estimators of the distances might lead to biased density estimators, depending on the actual error distribution. In contrast with the findings of Chen, who used an additive model, unbiased estimation of distances, given a multiplicative model, lead to overestimation of density. Some error distributions result in observed distance distributions that make efficient estimation impossible, by removing the shoulder present in the original detection function. This indicates the need to improve field methods to reduce measurement error. An application of the new methods to a real data set is presented.     

Comparison of estimators for measures of linkage disequilibrium

The measurement of biallelic pair-wise association called linkage disequilibrium (LD) is an important issue in order to understand the genomic architecture. A plethora of measures of association in two by two tables have been proposed in the literature. Beside the problem of choosing an appropriate measure, the problem of their estimation has been neglected in the literature. It needs to be emphasized that the definition of a measure and the choice of an estimator function for it are conceptually unrelated tasks. In this paper, we compare the performance of various estimators for the three popular LD measures D', r and Y in a simulation study for small to moderate samples sizes (N<=500). The usual frequency-plug-in estimators can lead to unreliable or undefined estimates. Estimators based on the computationally expensive volume measures have been proposed recently as a remedy to this well-known problem. We confirm that volume estimators have better expected mean square error than the naive plug-in estimators. But they are outperformed by estimators plugging-in easy to calculate non-informative Bayesian probability estimates into the theoretical formulae for the measures. Fully Bayesian estimators with non-informative Dirichlet priors have comparable accuracy but are computationally more expensive. We recommend the use of non-informative Bayesian plug-in estimators based on Jeffreys' prior, in particular when dealing with SNP array data where the occurrence of small table entries and table margins is likely.     

Sampling Considerations for Disease Surveillance in Wildlife Populations

Abstract Disease surveillance in wildlife populations involves detecting the presence of a disease, characterizing its prevalence and spread, and subsequent monitoring. A probability sample of animals selected from the population and corresponding estimators of disease prevalence and detection provide estimates with quantifiable statistical properties, but this approach is rarely used. Although wildlife scientists often assume probability sampling and random disease distributions to calculate sample sizes, convenience samples (i.e., samples of readily available animals) are typically used, and disease distributions are rarely random. We demonstrate how landscape-based simulation can be used to explore properties of estimators from convenience samples in relation to probability samples. We used simulation methods to model what is known about the habitat preferences of the wildlife population, the disease distribution, and the potential biases of the convenience-sample approach. Using chronic wasting disease in free-ranging deer (Odocoileus virginianus) as a simple illustration, we show that using probability sample designs with appropriate estimators provides unbiased surveillance parameter estimates but that the selection bias and coverage errors associated with convenience samples can lead to biased and misleading results. We also suggest practical alternatives to convenience samples that mix probability and convenience sampling. For example, a sample of land areas can be selected using a probability design that oversamples areas with larger animal populations, followed by harvesting of individual animals within sampled areas using a convenience sampling method.