1
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Cai J, Zhang N, Zhou X, Spiegelman D, Wang M. Correcting for bias due to mismeasured exposure history in longitudinal studies with continuous outcomes. Biometrics 2023; 79:3739-3751. [PMID: 37222518 PMCID: PMC11214728 DOI: 10.1111/biom.13877] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2022] [Accepted: 04/28/2023] [Indexed: 05/25/2023]
Abstract
Epidemiologists are often interested in estimating the effect of functions of time-varying exposure histories in relation to continuous outcomes, for example, cognitive function. However, the individual exposure measurements that constitute the history upon which an exposure history function is constructed are usually mismeasured. To obtain unbiased estimates of the effects for mismeasured functions in longitudinal studies, a method incorporating main and validation studies was developed. Simulation studies under several realistic assumptions were conducted to assess its performance compared to standard analysis, and we found that the proposed method has good performance in terms of finite sample bias reduction and nominal confidence interval coverage. We applied it to a study of long-term exposure toPM 2.5 $\text{PM}_{2.5}$ , in relation to cognitive decline in the Nurses' Health Study Previously, it was found that the 2-year decline in the standard measure of cognition was 0.018 (95% CI, -0.034 to -0.001) units worse per 10μ g/m 3 $\mu \text{g/m}^3$ increase inPM 2.5 $\text{PM}_{2.5}$ exposure. After correction, the estimated impact ofPM 2.5 $\text{PM}_{2.5}$ on cognitive decline increased to 0.027 (95% CI, -0.059 to 0.005) units lower per 10μ g/m 3 $\mu \text{g/m}^3$ increase. To put this into perspective, effects of this magnitude are about 2/3 of those found in our data associated with each additional year of aging: 0.044 (95% CI, -0.047 to -0.040) units per 1 year older after applying our correction method.
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Affiliation(s)
- Jiachen Cai
- Department of Biostatistics, Yale School of Public Health, New Haven, Connecticut, USA
| | - Ning Zhang
- Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, Massachusetts, USA
| | - Xin Zhou
- Department of Biostatistics, Yale School of Public Health, New Haven, Connecticut, USA
- Center for Methods in Implementation and Prevention Science, Yale School of Public Health, New Haven, Connecticut, USA
| | - Donna Spiegelman
- Department of Biostatistics, Yale School of Public Health, New Haven, Connecticut, USA
- Center for Methods in Implementation and Prevention Science, Yale School of Public Health, New Haven, Connecticut, USA
| | - Molin Wang
- Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, Massachusetts, USA
- Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, Massachusetts, USA
- Channing Division of Network Medicine, Harvard Medical School, Brigham and Women’s Hospital, Boston, Massachusetts, USA
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2
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He PF, Duan XD. Bayesian variable selection and estimation in multivariate skew-normal generalized partial linear mixed models for longitudinal data. COMMUN STAT-SIMUL C 2022. [DOI: 10.1080/03610918.2022.2154796] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Peng-Fei He
- School of Mathematics Science, Guizhou Normal University, Guiyang, P.R. China
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Xind-De Duan
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
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3
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Ding H, Zhang R, Zhu H. New estimation for heteroscedastic single-index measurement error models. J Nonparametr Stat 2022. [DOI: 10.1080/10485252.2021.2025238] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Hui Ding
- School of Economics, Nanjing University of Finance and Economics, Nanjing, People's Republic of China
| | - Riquan Zhang
- School of Statistics, Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, East China Normal University, Shanghai, People's Republic of China
| | - Hanbing Zhu
- School of Statistics, Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, East China Normal University, Shanghai, People's Republic of China
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4
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Huang Z, Jiang Z, Li J. Statistical inference for a single-index varying coefficient model with measurement errors in all covariates. J STAT COMPUT SIM 2021. [DOI: 10.1080/00949655.2021.1984486] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Zhensheng Huang
- School of Science, Nanjing University of Science and Technology, Nanjing, People's Republic of China
| | - Zhiqiang Jiang
- School of Science, Nanjing University of Science and Technology, Nanjing, People's Republic of China
| | - Jing Li
- School of Science, Nanjing University of Science and Technology, Nanjing, People's Republic of China
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5
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Zhang J, Cui X. Logarithmic calibration for nonparametric multiplicative distortion measurement errors models. J STAT COMPUT SIM 2021. [DOI: 10.1080/00949655.2021.1904240] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
Affiliation(s)
- Jun Zhang
- College of Mathematics and Statistics, Shenzhen University, Shenzhen, People's Republic of China
| | - Xia Cui
- School of Economics and Statistics, Guangzhou University, Guangzhou, People's Republic of China
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6
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Zhang H, Huang Y. Bayesian joint modeling for partially linear mixed-effects quantile regression of longitudinal and time-to-event data with limit of detection, covariate measurement errors and skewness. J Biopharm Stat 2020; 31:295-316. [PMID: 33284096 DOI: 10.1080/10543406.2020.1852248] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
Joint modeling analysis of longitudinal and time-to-event data has been an active area of statistical methodological study and biomedical research, but the majority of them are based on mean-regression. Quantile regression (QR) can characterize the entire conditional distribution of the outcome variable, and may be more robust to outliers/heavy tails and misspecification of error distribution. Additionally, a parametric specification may be insufficient and inflexible to capture the complicated longitudinal pattern of biomarkers. Thus, this study proposes novel QR-based partially linear mixed-effects joint models with three components (QR-based longitudinal response, longitudinal covariate, and time-to-event processes), and applies to Multicenter AIDS Cohort Study (MACS). Many common data features, including left-censoring due to a limit of detection, covariate measurement error, and asymmetric distribution, are considered to obtain reliable parameter estimates. Many interesting findings are discovered by the complicated joint models under Bayesian inference framework. Simulation studies are also implemented to assess the performance of the proposed joint models under different scenarios.
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Affiliation(s)
- Hanze Zhang
- Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, Florida, USA
| | - Yangxin Huang
- Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, Florida, USA.,Department of Statistics, Yunnan University, Kunming, PR China
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7
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Yalaz S, Kuran Ö. Kernel estimator and predictor of partially linear mixed-effect errors-in-variables model. J STAT COMPUT SIM 2020. [DOI: 10.1080/00949655.2020.1836642] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Seçil Yalaz
- Department of Statistics, Dicle University Science Faculty, Diyarbakır, Turkey
| | - Özge Kuran
- Department of Statistics, Dicle University Science Faculty, Diyarbakır, Turkey
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8
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Parast L, Garcia TP, Prentice RL, Carroll RJ. Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics 2020; 78:9-23. [PMID: 33021738 DOI: 10.1111/biom.13386] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2019] [Revised: 09/16/2020] [Accepted: 09/30/2020] [Indexed: 11/27/2022]
Abstract
The identification of valid surrogate markers of disease or disease progression has the potential to decrease the length and costs of future studies. Most available methods that assess the value of a surrogate marker ignore the fact that surrogates are often measured with error. Failing to adjust for measurement error can erroneously identify a useful surrogate marker as not useful or vice versa. We investigate and propose robust methods to correct for the effect of measurement error when evaluating a surrogate marker using multiple estimators developed for parametric and nonparametric estimates of the proportion of treatment effect explained by the surrogate marker. In addition, we quantify the attenuation bias induced by measurement error and develop inference procedures to allow for variance and confidence interval estimation. Through a simulation study, we show that our proposed estimators correct for measurement error in the surrogate marker and that our inference procedures perform well in finite samples. We illustrate these methods by examining a potential surrogate marker that is measured with error, hemoglobin A1c, using data from the Diabetes Prevention Program clinical trial.
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Affiliation(s)
- Layla Parast
- RAND Corporation, Statistics Group, Santa Monica, California
| | - Tanya P Garcia
- Department of Biostatistics, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina
| | - Ross L Prentice
- Public Health Sciences Division, Fred Hutchinson Cancer Research Center, Seattle, Washington
| | - Raymond J Carroll
- Department of Statistics, Texas A&M University, College Station, Texas.,School of Mathematical and Physical Sciences, University of Technology Sydney, Broadway, NSW, Australia
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9
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Taavoni M, Arashi M, Wang WL, Lin TI. Multivariate t semiparametric mixed-effects model for longitudinal data with multiple characteristics. J STAT COMPUT SIM 2020. [DOI: 10.1080/00949655.2020.1812608] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
Affiliation(s)
- M. Taavoni
- Department of Statistic, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
| | - M. Arashi
- Department of Statistic, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
| | - Wan-Lun Wang
- Department of Statistics, Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taichung, Taiwan
| | - Tsung-I Lin
- Institute of Statistics, National Chung Hsing University, Taichung, Taiwan
- Department of Public Health, China Medical University, Taichung, Taiwan
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10
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McGee G, Kioumourtzoglou M, Weisskopf MG, Haneuse S, Coull BA. On the interplay between exposure misclassification and informative cluster size. J R Stat Soc Ser C Appl Stat 2020. [DOI: 10.1111/rssc.12430] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Affiliation(s)
- Glen McGee
- Harvard T.H. Chan School of Public Health Boston USA
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11
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Zhu H, Zhang R, Li H. Estimation on semi-functional linear errors-in-variables models. COMMUN STAT-THEOR M 2019. [DOI: 10.1080/03610926.2018.1494836] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Hanbing Zhu
- School of Statistics, East China Normal University, Shanghai, P. R. China
| | - Riquan Zhang
- School of Statistics, East China Normal University, Shanghai, P. R. China
| | - Huiying Li
- School of Statistics, East China Normal University, Shanghai, P. R. China
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12
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Taavoni M, Arashi M. Kernel estimation in semiparametric mixed effect longitudinal modeling. Stat Pap (Berl) 2019. [DOI: 10.1007/s00362-019-01125-8] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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13
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Zhu H, Zhang R, Yu Z, Lian H, Liu Y. Estimation and testing for partially functional linear errors-in-variables models. J MULTIVARIATE ANAL 2019. [DOI: 10.1016/j.jmva.2018.11.005] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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14
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Feng Z, Gai Y, Zhang J. Correlation curve estimation for multiplicative distortion measurement errors data. J Nonparametr Stat 2019. [DOI: 10.1080/10485252.2019.1580708] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Zhenghui Feng
- School of Economics and the Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China
| | - Yujie Gai
- School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, China
- Department of Biostatistics, School of Public Health, University of Texas at Houston, Houston, TX, USA
| | - Jun Zhang
- College of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, Shenzhen-Hong Kong Joint Research Center for Applied Statistical Sciences, Shenzhen, China
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15
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SIMEX estimation for single-index model with covariate measurement error. ASTA ADVANCES IN STATISTICAL ANALYSIS 2018. [DOI: 10.1007/s10182-018-0327-6] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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16
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Garcia TP, Ma Y. Simultaneous treatment of unspecified heteroskedastic model error distribution and mismeasured covariates for restricted moment models. JOURNAL OF ECONOMETRICS 2017; 200:194-206. [PMID: 29200600 PMCID: PMC5708600 DOI: 10.1016/j.jeconom.2017.06.005] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We develop consistent and efficient estimation of parameters in general regression models with mismeasured covariates. We assume the model error and covariate distributions are unspecified, and the measurement error distribution is a general parametric distribution with unknown variance-covariance. We construct root-n consistent, asymptotically normal and locally efficient estimators using the semiparametric efficient score. We do not estimate any unknown distribution or model error heteroskedasticity. Instead, we form the estimator under possibly incorrect working distribution models for the model error, error-prone covariate, or both. Empirical results demonstrate robustness to different incorrect working models in homoscedastic and heteroskedastic models with error-prone covariates.
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Affiliation(s)
- Tanya P. Garcia
- Department of Epidemiology and Biostatistics, Texas A&M University
| | - Yanyuan Ma
- Department of Statistics, Pennsylvania State University
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17
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Zhang H, Huang Y, Wang W, Chen H, Langland-Orban B. Bayesian quantile regression-based partially linear mixed-effects joint models for longitudinal data with multiple features. Stat Methods Med Res 2017; 28:569-588. [PMID: 28936916 DOI: 10.1177/0962280217730852] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
In longitudinal AIDS studies, it is of interest to investigate the relationship between HIV viral load and CD4 cell counts, as well as the complicated time effect. Most of common models to analyze such complex longitudinal data are based on mean-regression, which fails to provide efficient estimates due to outliers and/or heavy tails. Quantile regression-based partially linear mixed-effects models, a special case of semiparametric models enjoying benefits of both parametric and nonparametric models, have the flexibility to monitor the viral dynamics nonparametrically and detect the varying CD4 effects parametrically at different quantiles of viral load. Meanwhile, it is critical to consider various data features of repeated measurements, including left-censoring due to a limit of detection, covariate measurement error, and asymmetric distribution. In this research, we first establish a Bayesian joint models that accounts for all these data features simultaneously in the framework of quantile regression-based partially linear mixed-effects models. The proposed models are applied to analyze the Multicenter AIDS Cohort Study (MACS) data. Simulation studies are also conducted to assess the performance of the proposed methods under different scenarios.
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Affiliation(s)
- Hanze Zhang
- 1 Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL, USA
| | - Yangxin Huang
- 1 Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL, USA
| | - Wei Wang
- 1 Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL, USA
| | - Henian Chen
- 1 Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL, USA
| | - Barbara Langland-Orban
- 2 Department of Health Policy and Management, College of Public Health, University of South Florida, Tampa, FL, USA
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18
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Duan X, Kam Fung W, Tang N. Bayesian semiparametric reproductive dispersion mixed models for non-normal longitudinal data: estimation and case influence analysis. J STAT COMPUT SIM 2017. [DOI: 10.1080/00949655.2017.1298766] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Xingde Duan
- Institute of Applied Statistics, Chuxiong Normal School, People's Republic of China
| | - Wing Kam Fung
- Department of Statistics & Actuarial Science, The University of Hong Kong, Hongkong, People's Republic of China
| | - Niansheng Tang
- Department of Statistics, Yunnan University, Kunming, People's Republic of China
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19
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Liu L, Sun Z. Kernel-based global MLE of partial linear random effects models for longitudinal data. J Nonparametr Stat 2017. [DOI: 10.1080/10485252.2017.1339308] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Lei Liu
- Department of Preventive Medicine and Robert H. Lurie Comprehensive Cancer Center, Northwestern University, Chicago, IL, USA
| | - Zhihua Sun
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, People's Republic of China
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20
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Hu X, Yang W. Semi-parametric small area inference in generalized semi-varying coefficient mixed effects models. Stat Pap (Berl) 2016. [DOI: 10.1007/s00362-016-0862-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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21
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Duan XD, Tang NS. Bayesian estimation and influence diagnostics of generalized partially linear mixed-effects models for longitudinal data. STATISTICS-ABINGDON 2015. [DOI: 10.1080/02331888.2015.1078332] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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22
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Yi GY, Ma Y, Spiegelman D, Carroll RJ. Functional and Structural Methods with Mixed Measurement Error and Misclassification in Covariates. J Am Stat Assoc 2015; 110:681-696. [PMID: 26190876 DOI: 10.1080/01621459.2014.922777] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Abstract
Covariate measurement imprecision or errors arise frequently in many areas. It is well known that ignoring such errors can substantially degrade the quality of inference or even yield erroneous results. Although in practice both covariates subject to measurement error and covariates subject to misclassification can occur, research attention in the literature has mainly focused on addressing either one of these problems separately. To fill this gap, we develop estimation and inference methods that accommodate both characteristics simultaneously. Specifically, we consider measurement error and misclassification in generalized linear models under the scenario that an external validation study is available, and systematically develop a number of effective functional and structural methods. Our methods can be applied to different situations to meet various objectives.
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Affiliation(s)
- Grace Y Yi
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Yanyuan Ma
- Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143,
| | - Donna Spiegelman
- Departments of Epidemiology and Biostatistics, Harvard School of Public Health, 677 Huntington Ave, Boston, MA 02115,
| | - Raymond J Carroll
- Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143, and School of Mathematical Sciences, University of Technology, Sydney, Broadway NSW 2007,
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23
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Li W, Xue L. Efficient Inference in a Generalized Partially Linear Model with Random Effect for Longitudinal Data. COMMUN STAT-THEOR M 2014. [DOI: 10.1080/03610926.2012.740126] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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24
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Tang NS, Duan XD. Bayesian influence analysis of generalized partial linear mixed models for longitudinal data. J MULTIVARIATE ANAL 2014. [DOI: 10.1016/j.jmva.2013.12.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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25
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A semiparametric Bayesian approach to generalized partial linear mixed models for longitudinal data. Comput Stat Data Anal 2012. [DOI: 10.1016/j.csda.2012.03.018] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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26
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Yi GY. A functional generalized method of moments approach for longitudinal studies with missing responses and covariate measurement error. Biometrika 2012; 99:151-165. [PMID: 28781377 PMCID: PMC5541954 DOI: 10.1093/biomet/asr076] [Citation(s) in RCA: 36] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
Covariate measurement error and missing responses are typical features in longitudinal data analysis. There has been extensive research on either covariate measurement error or missing responses, but relatively little work has been done to address both simultaneously. In this paper, we propose a simple method for the marginal analysis of longitudinal data with time-varying covariates, some of which are measured with error, while the response is subject to missingness. Our method has a number of appealing properties: assumptions on the model are minimal, with none needed about the distribution of the mismeasured covariate; implementation is straightforward and its applicability is broad. We provide both theoretical justification and numerical results.
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Affiliation(s)
- Grace Y. Yi
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo,
Ontario, Canada N2L 3G1,
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