Covariate Measurement Error Research Articles

Spatial modelling of infectious diseases with covariate measurement error

AbstractIn spatial infectious disease models, it is typical to assume that only the distance between susceptible and infectious individuals is important for modelling, but not the actual spatial locations of the individuals. Recently introduced geographically-dependent individual level models (GD-ILMs) can be used to also consider the effect of spatial locations of individuals and the distance between susceptible and infectious individuals for determining the risk of infection. In these models, it is assumed that the covariates used to predict the occurrence of disease are measured accurately. However, there are many applications in which covariates are prone to measurement error. For instance, to study risk factors for influenza, people with low socio-economic status (SES) are known to be more at risk compared to the rest of the population. However, SES is prone to measurement error. In this paper, we propose a GD-ILM which accounts for measurement error in both individual-level and area-level covariates. A Monte Carlo expectation conditional maximisation algorithm is used for inference. We use models fitted to data to predict areas with high average infectivity rates. We evaluate the performance of the proposed approach through simulation studies and by a real-data application on influenza data in Manitoba, Canada.

Practical Considerations for Sandwich Variance Estimation in 2-Stage Regression Settings.

In this paper, we present a practical approach for computing the sandwich variance estimator in 2-stage regression model settings. As a motivating example for 2-stage regression, we consider regression calibration, a popular approach for addressing covariate measurement error. The sandwich variance approach has rarely been applied in regression calibration, despite its requiring less computation time than popular resampling approaches for variance estimation, specifically the bootstrap. This is probably because it requires specialized statistical coding. Here we first outline the steps needed to compute the sandwich variance estimator. We then develop a convenient method of computation in R for sandwich variance estimation, which leverages standard regression model outputs and existing R functions and can be applied in the case of a simple random sample or complex survey design. We use a simulation study to compare the sandwich estimator to a resampling variance approach for both settings. Finally, we further compare these 2 variance estimation approaches in data examples from the Women's Health Initiative (1993-2005) and the Hispanic Community Health Study/Study of Latinos (2008-2011). In our simulations, the sandwich variance estimator typically had good numerical performance, but simple Wald bootstrap confidence intervals were unstable or overcovered in certain settings, particularly when there was high correlation between covariates or large measurement error.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

A joint Bayesian framework for missing data and measurement error using integrated nested Laplace approximations.

Measurement error (ME) and missing values in covariates are often unavoidable in disciplines that deal with data, and both problems have separately received considerable attention during the past decades. However, while most researchers are familiar with methods for treating missing data, accounting for ME in covariates of regression models is less common. In addition, ME and missing data are typically treated as two separate problems, despite practical and theoretical similarities. Here, we exploit the fact that missing data in a continuous covariate is an extreme case of classical ME, allowing us to use existing methodology that accounts for ME via a Bayesian framework that employs integrated nested Laplace approximations (INLA) and thus to simultaneously account for both ME and missing data in the same covariate. As a useful by-product, we present an approach to handle missing data in INLA since this corresponds to the special case when no ME is present. In addition, we show how to account for Berkson ME in the same framework. In its broadest generality, the proposed joint Bayesian framework can thus account for Berkson ME, classical ME, and missing data, or any combination of these in the same or different continuous covariates of the family of regression models that are feasible with INLA. The approach is exemplified using both simulated and real data. We provide extensive and fully reproducible Supporting Information with thoroughly documented examples using R-INLA and inlabru.

Semiparametric M-quantile regression with measurement error in spatial covariates: an application to housing price modelling

Abstract Spatial data are becoming increasingly accessible to urban scientists, but these data are often prone to measurement error. Motivated by the analysis of the Milan (Italy) apartment market heterogeneity, we propose a semiparametric approach to adjust for the presence of measurement error in the covariates when estimating M-quantile regression. The M-quantile approach helps explain the heterogeneity across individual units, preserving robustness and efficiency in the estimates. The model’s parameters are estimated within a penalised likelihood framework and an analytical expression is proposed to estimate standard errors. Asymptotic properties of estimates are also provided.

Integrated partially linear model for multi-centre studies with heterogeneity and batch effect in covariates

Multi-centre study is increasingly used for borrowing strength from multiple research groups to obtain reproducible study findings. Regression analysis is widely used for analysing multi-group studies, however, some of the regression predictors are nonlinear and/or often measured with batch effects. Also, the group compositions are potentially heterogeneous across different centres. The conventional pooled data analysis can cause biased regression estimates. This paper proposes an integrated partially linear regression model (IPLM) to account for predictor's nonlinearity, general batch effect, group composition heterogeneity, and potential measurement-error in covariates simultaneously. A local linear regression-based approach is employed to estimate the nonlinear component and a regularization procedure is introduced to identify the predictors' effects. The IPLM-based method has estimation consistency and variable-selection consistency. Moreover, it has a fast computing algorithm and its effectiveness is supported by simulation studies. A multi-centre Alzheimer's disease research project is provided to illustrate the proposed IPLM-based analysis.

Approximate Gaussian variance inference for state‐space models

SummaryState‐space models require an accurate knowledge of the process error () and measurement error () covariance matrices for exact state estimation. Even though the matrix can be, in many situations, considered to be known from the measuring instrument specifications, it is still a challenge to infer the matrix online while providing reliable estimates along with a low computational cost. In this article, we propose an analytically tractable online Bayesian inference method for inferring the matrix in state‐space models. We refer to this method as approximate Gaussian variance inference (AGVI) using which we are able to treat the error variance and covariance terms in the full matrix as Gaussian hidden states and infer them simultaneously with the other hidden states in a closed‐form manner. The two case studies show that the method is able to provide statistically consistent estimates for the mean and uncertainties of the error variance terms for univariate and multivariate cases. The method also exceeds the performance of the existing adaptive Kalman filter methods both in terms of accuracy and computational efficiency.

Analysis of length-biased and partly interval-censored survival data with mismeasured covariates.

In this paper, we analyze the length-biased and partly interval-censored data, whose challenges primarily come from biased sampling and interfere induced by interval censoring. Unlike existing methods that focus on low-dimensional data and assume the covariates to be precisely measured, sometimes researchers may encounter high-dimensional data subject to measurement error, which are ubiquitous in applications and make estimation unreliable. To address those challenges, we explore a valid inference method for handling high-dimensional length-biased and interval-censored survival data with measurement error in covariates under the accelerated failure time model. We primarily employ the SIMEX method to correct for measurement error effects and propose the boosting procedure to do variable selection and estimation. The proposed method is able to handle the case that the dimension of covariates is larger than the sample size and enjoys appealing features that the distributions of the covariates are leftunspecified.

An augmented likelihood approach for the Cox proportional hazards model with interval-censored auxiliary and validated outcome data-with application to the Hispanic Community Health Study/Study of Latinos.

In large epidemiologic studies, it is typical for an inexpensive, non-invasive procedure to be used to record disease status during regular follow-up visits, with less frequent assessment by a gold standard test. Inexpensive outcome measures like self-reported disease status are practical to obtain, but can be error-prone. Association analysis reliant on error-prone outcomes may lead to biased results; however, restricting analyses to only data from the less frequently observed error-free outcome could be inefficient. We have developed an augmented likelihood that incorporates data from both error-prone outcomes and a gold standard assessment. We conduct a numerical study to show how we can improve statistical efficiency by using the proposed method over standard approaches for interval-censored survival data that do not leverage auxiliary data. We extend this method for the complex survey design setting so that it can be applied in our motivating data example. Our method is applied to data from the Hispanic Community Health Study/Study of Latinos to assess the association between energy and protein intake and the risk of incident diabetes. In our application, we demonstrate how our method can be used in combination with regression calibration to additionally address the covariate measurement error in self-reported diet.

The Tobit-Unscented-Kalman-Filter-Based Attitude Estimation Algorithm Using the Star Sensor and Inertial Gyro Combination.

For the orbit operation of spacecraft, due to environmental factors, a star sensor installed on the spacecraft must have data censoring, which greatly reduces the attitude determination ability of the traditional combined-attitude-determination algorithm. To address this problem, this paper proposes an algorithm for high-precision attitude estimation based on a Tobit unscented Kalman filter. This is on the basis of establishing the nonlinear state equation of the integrated star sensor and gyroscope navigation system. The measurement update process of the unscented Kalman filter is improved. The Tobit model is used to describe the gyroscope drift when the star sensor fails. The latent measurement values are calculated using the probability statistics, and the measurement error covariance expression is derived. The proposed design is verified via computer simulations. When the star sensor fails for 15 min, the accuracy of the Tobit unscented Kalman filter based on the Tobit model is improved by approximately 90% compared to the unscented Kalman filter. Based on the results, the proposed filter can effectively estimate the error caused by the gyro drift, and the method is effective and feasible, provided there is theoretical support for the engineering practice.

An adaptive max-type multivariate control chart by considering measurement errors and autocorrelation

The combined effect of two real-world-occurring phenomena: ‘measurement errors’ and ‘autocorrelation between observations’ has rarely been investigated. In this paper, it will be investigated for the first time on ‘adaptive’ and/or ’simultaneous monitoring’ charts and also for the first time by using the multivariate linearly covariate measurement errors and VARMA (vector mixed autoregressive and moving average) autocorrelation models, and Markov chains-based performance measures. In addition, this paper for the first time proposes a skip-sampling strategy in an ARMA/VARMA model for alleviating the autocorrelation effect. To do so, we add the above-mentioned measurement errors and autocorrelation models to a recently developed adaptive max-type chart. Then, we develop a Markov chain model to compute the performance measures. After that, extensive numerical analyses will be performed to investigate their combined effect as well as some methods to alleviate their negative effects. Finally, an illustrative example involving a real industrial case will be presented.

Particle filter algorithms for on-line prediction of transient heat flux of participating medium

The particle filter (PF) and corresponding improved algorithm (unscented particle filter (UPF)) are introduced to solve the on-line prediction of transient heat flux (q(t)) on the boundary of participating medium for the first time. The temperature-dependent thermophysical properties of the medium result in a highly nonlinear heat transfer system. The graded index medium (caused by the temperature-varying refractive index) is also taken into account, in which the radiation transfer is more complex than that in the uniform refractive index medium. The finite volume method is employed to address the transient coupled radiative and conductive heat transfer in medium. The impacts of particle number, process error covariance, measurement error covariance, sampling time interval, forms of heat flux and measurement errors on prediction results are researched in detail. The UPF algorithm, in which unscented Kalman filter technique is applied to pick an appropriate proposal density function to ameliorate the degradation phenomenon in original PF algorithm, outperforms the PF algorithm in terms of stability and accuracy of estimation results. All the prediction results demonstrate that UPF method is effective and reliable to solve the real-time reconstruction of q(t) even with measurement errors.

Measurement error models for spatial network lattice data: Analysis of car crashes in Leeds

Abstract Road casualties represent an alarming concern for modern societies. During the last years, several authors proposed sophisticated approaches to help authorities implement new policies. These models were usually developed considering a set of socio-economic variables and ignoring the measurement error, which can bias the statistical inference. This paper presents a Bayesian model to analyse car crash occurrences at the network-lattice level, taking into account measurement error in the spatial covariate. The suggested methodology is exemplified by considering the collisions in the road network of Leeds (UK) during 2011–2019. Traffic volumes are approximated using an extensive set of counts obtained from mobile devices and the estimates are adjusted using a spatial measurement error correction.

The mean residual life model for the right-censored data in the presence of covariate measurement errors.

In this article, we consider the mean residual life regression model in the presence of covariate measurement errors. In the whole cohort, the surrogate variable of the error-prone covariate is available for each subject, while the instrumental variable (IV), which is related to the underlying true covariates, is measured only for some subjects, the calibration sample. Without specifying distributions of measurement errors but assuming that the IV is missing at random, we develop two estimation methods, the IV calibration and cohort estimators, for the regression parameters by solving estimation equations (EEs) based on the calibration sample and cohort sample, respectively. To improve estimation efficiency, a synthetic estimator is derived by applying the generalized method of moment for all EEs. The large sample properties of the proposed estimators are established and their finite sample performance are evaluated via simulation studies. Simulation results show that the cohort and synthetic estimators outperform the IV calibration estimator and the relative efficiency of the cohort and synthetic estimators mainly depends on the missing rate of IV. In the case of low missing rate, the synthetic estimator is more efficient than the cohort estimator, while the result can be reversed when the missing rate is high. We illustrate the proposed method by application to data from the patients with stage 5 chronic kidney disease in Taiwan.

Leveraging neural network uncertainty in adaptive unscented Kalman Filter for spacecraft pose estimation

This paper introduces an adaptive Convolutional Neural Network (CNN)-based Unscented Kalman Filter for the pose estimation of uncooperative spacecraft. The validation is carried out at Stanford’s robotic Testbed for Rendezvous and Optical Navigation on the Satellite Hardware-In-the-loop Rendezvous Trajectories (SHIRT) dataset, which simulates vision-based rendezvous trajectories of a servicer spacecraft to PRISMA’s Tango spacecraft. The proposed navigation system is stress-tested on synthetic as well as realistic lab imagery by simulating space-like illumination conditions on-ground. The validation is performed at different levels of the navigation system by first training and testing the adopted CNN on SPEED+, Stanford’s spacecraft pose estimation dataset with specific emphasis on domain shift between a synthetic domain and an Hardware-In-the-Loop domain. A novel data augmentation scheme based on light randomization is proposed to improve the CNN robustness under adverse viewing conditions, reaching centimeter-level and 10 degree-level pose errors in 80% of the SPEED+ lab images. Next, the entire navigation system is tested on the SHIRT dataset. Results indicate that the inclusion of a new scheme to adaptively scale the heatmaps-based measurement error covariance based on filter innovations improves filter robustness by returning centimeter-level position errors and moderate attitude accuracies, suggesting that a proper representation of the measurements uncertainty combined with an adaptive measurement error covariance is key in improving the navigation robustness.

Near real-time prediction of thermophysical parameters and time-dependent boundary heat flux of participating medium

The scattering coefficient (κs), absorption coefficient (κa), thermal conductivity (λ) and time-varying boundary heat flux (q(t)) of 1D participating medium are simultaneously determined in near real time. The forward problem of coupled conductive and radiative heat transfer is solved by finite volume method (FVM). The unscented Kalman filter (UKF) and corresponding smoothing technique (UKS) are employed to estimate the q(t), λ, κa and κs. To achieve non-intrusive detection, the measured signals on boundaries of the medium (such as radiative intensity on right boundary and temperatures on boundaries) are employed as inputs in the inverse problem. The effects of measurement error covariance, process error covariance, future measured signals, sampling interval and thermophysical properties on estimation results are thoroughly investigated. The UKS algorithm, when compared to the UKF algorithm, can improve the stability and accuracy of estimated q(t), λ, κa and κs. It is found that the UKS algorithm is efficient and accurate to solve the near real-time prediction of thermophysical parameters and boundary heat flux of participating medium even with measurement errors.

Spatial Linear Regression with Covariate Measurement Errors: Inference and Scalable Computation in a Functional Modeling Approach

For spatial linear models, the classical maximum-likelihood estimators of both regression coefficients and variance components can be biased when the covariates are measured with errors. This work introduces a theoretically backed-up estimation framework for the spatial linear errors-in-variables model in a functional approach. Compared with the structural models, the functional approach treats the unobserved true covariates as fixed unknown parameters without imposing additional structures, thus leading to more robust parameter inference. Our model parameters are estimated simultaneously based on a set of unbiased estimating equations. Under some regularity conditions, we prove the consistency of the proposed estimating-equation estimators and derive their asymptotic distribution. In addition, a consistent variance estimator is developed for the estimating-equation estimators. To handle large spatial datasets, we provide two approaches to obtain scalable estimations based on our proposed estimating equations, where the required computational time and storage are reduced to be linear with sample size for each estimating-function evaluation. Simulation studies under different settings show that our estimators are consistent and the scalable algorithms work well. Finally, the proposed method is applied to studying the relationship between Arctic sea ice and related geophysical variables. Supplementary materials for this article are available online.

Two Sample Approaches to Regression Calibration for Measurement Error Correction

The goal of this work is to create methods for enhancing measurement error using regression calibration as a strategy by combining two samples, thereby increasing the relative efficiency of linear regression models. Because two or more samples are more likely to provide an accurate representation of the population than a single sample under inquiry, utilizing two samples in regression calibration is likely to produce a realistic depiction of what the actual population is when error-free. This study has generated independent estimates from two samples and combined them with weights equal to the inverse of their estimated probabilities of sample inclusion. It has also integrated two data sets into a single data set and suitably adjusted the weights on each sampled unit. The regression calibration method is most commonly used to correct predictor-response bias caused by variable measurement imperfections. Because of its simplicity, this method is often used. The fundamental principle behind regression calibration is to estimate the conditional expectation of a genuine response, given predictors measured with error and other covariates supposed to be measured without error. The predicted values are then estimated and used to assess the relationship between the response and an outcome in place of the unknown genuine response. Further information on the unobservable true predictors is required by the regression calibration program. This data is frequently obtained from a validation study that employs unbiased measurements for genuine predictors. This study has employed and compared the results obtained from the two sample approaches. Measuring errors can be produced by a variety of sources, including instrument error, laboratory error, human error, problems in documenting or executing measurements, self-reporting errors, and natural oscillations in the underlying amount. Covariate measurement error has three effects: In addition to hiding the properties of the data, which makes graphical model analysis difficult, it produces bias in parameter estimates for statistical models, resulting in a sometimes significant loss of power for detecting fascinating correlations between variables. The two sample approaches employed by the study have yielded acceptable results.

Logistic regression with correlated measurement error and misclassification in covariates.

Many areas of research, such as nutritional epidemiology, may encounter measurement errors of continuous covariates and misclassification of categorical variables when modeling. It is well known that ignoring measurement errors or misclassification can lead to biased results. But most research has focused on solving these two problems separately. Addressing both measurement error and misclassification simultaneously in a single analysis is less actively studied. In this article, we propose a new correction method for a logistic regression to handle correlated error variables involved in multivariate continuous covariates and misclassification in a categorical variable simultaneously. It is not computationally intensive since a closed-form of the approximate likelihood function conditional on observed covariates is derived. The asymptotic normality of this proposed estimator is established under regularity conditions and its finite-sample performance is empirically examined by simulation studies. We apply this new estimation method to handle measurement error in some nutrients of interest and misclassification of a categorical variable named physical activity in the European Prospective Investigation into Cancer and Nutrition-InterAct Study data. Analyses show that fruit is negatively associated with type 2 diabetes for a group of women doing active physical activity, protein has positive association with type 2 diabetes for the group of less active physical activity, and actual physical activity has a greater effect on reducing the risk of type 2 diabetes than observed physical activity.

Poisson-Lognormal model with measurement error in covariate for small area estimation of count data

Small Area Estimation (SAE) is a statistical technique to estimate parameters of subpopulation containing small-size of samples. SAE is an indirect estimation method by utilizing the strength of the neighbor area and data sources outside the area so that the sample becomes more effective and reduces the variance of parameter estimators. This paper deals with non-symmetrical count data in SAE which can be modelled based on Poisson-Lognormal distribution using hierarchical Bayesian (HB) approach and information on covariate contains measurement error. We develop the Poisson-Lognormal model for predicting small area counts with structural measurement error in the area-specific covariate. To get the HB Bayes estimator for Poisson-Lognormal model with measurement error in its covariates, the Metropolis-Hastings (MH) algorithm from the Markov Chain Monte Carlo (MCMC) technique is used. The HB estimator performance is studied through simulation and implementation of real data to predict the illiteracy rate at the sub-district level in Kepulauan Riau Province Indonesia based on The National Socio-Economic Survey (Susenas) data in March 2020.