1
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Hernandez-Velasco LL, Abanto-Valle CA, Dey DK, Castro LM. A Bayesian approach for mixed effects state-space models under skewness and heavy tails. Biom J 2023; 65:e2100302. [PMID: 37853834 DOI: 10.1002/bimj.202100302] [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: 09/25/2021] [Revised: 05/29/2023] [Accepted: 06/15/2023] [Indexed: 10/20/2023]
Abstract
Human immunodeficiency virus (HIV) dynamics have been the focus of epidemiological and biostatistical research during the past decades to understand the progression of acquired immunodeficiency syndrome (AIDS) in the population. Although there are several approaches for modeling HIV dynamics, one of the most popular is based on Gaussian mixed-effects models because of its simplicity from the implementation and interpretation viewpoints. However, in some situations, Gaussian mixed-effects models cannot (a) capture serial correlation existing in longitudinal data, (b) deal with missing observations properly, and (c) accommodate skewness and heavy tails frequently presented in patients' profiles. For those cases, mixed-effects state-space models (MESSM) become a powerful tool for modeling correlated observations, including HIV dynamics, because of their flexibility in modeling the unobserved states and the observations in a simple way. Consequently, our proposal considers an MESSM where the observations' error distribution is a skew-t. This new approach is more flexible and can accommodate data sets exhibiting skewness and heavy tails. Under the Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is implemented. To evaluate the properties of the proposed models, we carried out some exciting simulation studies, including missing data in the generated data sets. Finally, we illustrate our approach with an application in the AIDS Clinical Trial Group Study 315 (ACTG-315) clinical trial data set.
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Affiliation(s)
- Lina L Hernandez-Velasco
- Facultad de Ciencias Básicas, Universidad Santiago de Cali, Calle 5 62-00, Santiago de Cali, Colombia
| | - Carlos A Abanto-Valle
- Department of Statistics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
| | - Dipak K Dey
- Department of Statistics, University of Connecticut, Storrs, Connecticut, USA
| | - Luis M Castro
- Department of Statistics, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
- Center for the Discovery of Structures in Complex Data, Casilla 306, Correo 22, Santiago, Chile
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2
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Gong M, Mao Z, Zhang D, Ren J, Zuo S. Study on Bayesian Skew-Normal Linear Mixed Model and Its Application in Fire Insurance. FIRE TECHNOLOGY 2023:1-26. [PMID: 37360677 PMCID: PMC10245366 DOI: 10.1007/s10694-023-01436-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/22/2022] [Accepted: 05/20/2023] [Indexed: 06/28/2023]
Abstract
Fire insurance is a crucial component of property insurance, and its rating depends on the forecast of insurance loss claim data. Fire insurance loss claim data have complicated characteristics such as skewness and heavy tail. The traditional linear mixed model is commonly difficult to accurately describe the distribution of loss. Therefore, it is crucial to establish a scientific and reasonable distribution model of fire insurance loss claim data. In this study, the random effects and random errors in the linear mixed model are firstly assumed to obey the skew-normal distribution. Then, a skew-normal linear mixed model is established using the Bayesian MCMC method based on a set of U.S. property insurance loss claims data. Comparative analysis is conducted with the linear mixed model of logarithmic transformation. Afterward, a Bayesian skew-normal linear mixed model for Chinese fire insurance loss claims data is designed. The posterior distribution of claim data parameters and related parameter estimation are employed with the R language JAGS package to obtain the predicted and simulated loss claim values. Finally, the optimization model in this study is used to determine the insurance rate. The results demonstrate that the model established by the Bayesian MCMC method can overcome data skewness, and the fitting and correlation with the sample data are better than the log-normal linear mixed model. Hence, it can be concluded that the distribution model proposed in this paper is reasonable for describing insurance claims. This study innovates a new approach for calculating the insurance premium rate and expands the application of the Bayesian method in the fire insurance field.
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Affiliation(s)
- Meiling Gong
- School of Fire Protection Engineering, China People’s Police University, Langfang, China
| | - Zhanli Mao
- School of Fire Protection Engineering, China People’s Police University, Langfang, China
| | - Di Zhang
- School of Fire Protection Engineering, China People’s Police University, Langfang, China
| | - Jianxing Ren
- School of Fire Protection Engineering, China People’s Police University, Langfang, China
| | - Songtao Zuo
- School of Fire Protection Engineering, China People’s Police University, Langfang, China
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3
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Bandyopadhyay D, Prates MO, Zhao X, Lachos VH. Spatial skew-normal/independent models for nonrandomly missing clustered data. Stat Med 2021; 40:3085-3105. [PMID: 33782991 DOI: 10.1002/sim.8960] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2018] [Revised: 03/02/2021] [Accepted: 03/05/2021] [Indexed: 11/06/2022]
Abstract
Clinical studies on periodontal disease (PD) often lead to data collected which are clustered in nature (viz. clinical attachment level, or CAL, measured at tooth-sites and clustered within subjects) that are routinely analyzed under a linear mixed model framework, with underlying normality assumptions of the random effects and random errors. However, a careful look reveals that these data might exhibit skewness and tail behavior, and hence the usual normality assumptions might be questionable. Besides, PD progression is often hypothesized to be spatially associated, that is, a diseased tooth-site may influence the disease status of a set of neighboring sites. Also, the presence/absence of a tooth is informative, as the number and location of missing teeth informs about the periodontal health in that region. In this paper, we develop a (shared) random effects model for site-level CAL and binary presence/absence status of a tooth under a Bayesian paradigm. The random effects are modeled using a spatial skew-normal/independent (S-SNI) distribution, whose dependence structure is conditionally autoregressive (CAR). Our S-SNI density presents an attractive parametric tool to model spatially referenced asymmetric thick-tailed structures. Both simulation studies and application to a clinical dataset recording PD status reveal the advantages of our proposition in providing a significantly improved fit, over models that do not consider these features in a unified way.
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Affiliation(s)
| | - Marcos O Prates
- Department of Statistics, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
| | | | - Victor H Lachos
- Departament of Statistics, University of Connecticut, Storrs, Connecticut, USA
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4
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Wang W. Bayesian analysis of multivariate linear mixed models with censored and intermittent missing responses. Stat Med 2020; 39:2518-2535. [DOI: 10.1002/sim.8554] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2019] [Revised: 02/01/2020] [Accepted: 03/31/2020] [Indexed: 11/05/2022]
Affiliation(s)
- Wan‐Lun Wang
- Department of Statistics, Graduate Institute of Statistics and Actuarial ScienceFeng Chia University Taichung Taiwan
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5
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A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications. Symmetry (Basel) 2020. [DOI: 10.3390/sym12010118] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Within the context of flexible parametric families of distributions, much work has been dedicated in recent years to the theme of skew-symmetric distributions, or symmetry-modulated distributions, as we prefer to call them. The present contribution constitutes a review of this area, with special emphasis on multivariate skew-elliptical families, which represent the subset with more immediate impact on applications. After providing background information of the distribution theory aspects, we focus on the aspects more relevant for applied work. The exposition is targeted to non-specialists in this domain, although some general knowledge of probability and multivariate statistics is assumed. Given this aim, the mathematical profile is kept to the minimum required.
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6
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Burger DA, Schall R, Jacobs R, Chen D. A generalized Bayesian nonlinear mixed‐effects regression model for zero‐inflated longitudinal count data in tuberculosis trials. Pharm Stat 2019; 18:420-432. [DOI: 10.1002/pst.1933] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2018] [Revised: 01/05/2019] [Accepted: 01/24/2019] [Indexed: 12/25/2022]
Affiliation(s)
| | - Robert Schall
- Department of Mathematical Statistics and Actuarial ScienceUniversity of the Free State Bloemfontein South Africa
- IQVIABiostatistics Bloemfontein South Africa
| | - Rianne Jacobs
- Bernoulli Institute for Mathematics, Computer Science and Artificial IntelligenceUniversity Groningen Groningen The Netherlands
| | - Ding‐Geng Chen
- Department of StatisticsUniversity of Pretoria Pretoria South Africa
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7
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Bhingare A, Sinha D, Pati D, Bandyopadhyay D, Lipsitz SR. Semiparametric Bayesian latent variable regression for skewed multivariate data. Biometrics 2019; 75:528-538. [PMID: 30365158 DOI: 10.1111/biom.12989] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2017] [Accepted: 10/05/2018] [Indexed: 11/26/2022]
Abstract
For many real-life studies with skewed multivariate responses, the level of skewness and association structure assumptions are essential for evaluating the covariate effects on the response and its predictive distribution. We present a novel semiparametric multivariate model and associated Bayesian analysis for multivariate skewed responses. Similar to multivariate Gaussian densities, this multivariate model is closed under marginalization, allows a wide class of multivariate associations, and has meaningful physical interpretations of skewness levels and covariate effects on the marginal density. Other desirable properties of our model include the Markov Chain Monte Carlo computation through available statistical software, and the assurance of consistent Bayesian estimates of the parameters and the nonparametric error density under a set of plausible prior assumptions. We illustrate the practical advantages of our methods over existing alternatives via simulation studies and the analysis of a clinical study on periodontal disease.
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Affiliation(s)
- Apurva Bhingare
- Eunice Kennedy Shriver National Institute of Child Health & Human Development, Bethesda, Maryland
| | - Debajyoti Sinha
- Department of Statistics, Florida State University, Tallahassee, Florida
| | - Debdeep Pati
- Department of Statistics, Texas A & M University, College Station, Texas
| | | | - Stuart R Lipsitz
- Department of Medicine, Brigham and Women's Hospital, Boston, Massachusetts
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8
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Lachos VH, A Matos L, Castro LM, Chen MH. Flexible longitudinal linear mixed models for multiple censored responses data. Stat Med 2018; 38:1074-1102. [PMID: 30421470 DOI: 10.1002/sim.8017] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2018] [Revised: 09/27/2018] [Accepted: 10/01/2018] [Indexed: 11/06/2022]
Abstract
In biomedical studies and clinical trials, repeated measures are often subject to some upper and/or lower limits of detection. Hence, the responses are either left or right censored. A complication arises when more than one series of responses is repeatedly collected on each subject at irregular intervals over a period of time and the data exhibit tails heavier than the normal distribution. The multivariate censored linear mixed effect (MLMEC) model is a frequently used tool for a joint analysis of more than one series of longitudinal data. In this context, we develop a robust generalization of the MLMEC based on the scale mixtures of normal distributions. To take into account the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is considered. For this complex longitudinal structure, we propose an exact estimation procedure to obtain the maximum-likelihood estimates of the fixed effects and variance components using a stochastic approximation of the EM algorithm. This approach allows us to estimate the parameters of interest easily and quickly as well as to obtain the standard errors of the fixed effects, the predictions of unobservable values of the responses, and the log-likelihood function as a byproduct. The proposed method is applied to analyze a set of AIDS data and is examined via a simulation study.
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Affiliation(s)
- Victor H Lachos
- Department of Statistics, University of Connecticut, Storrs, Connecticut
| | - Larissa A Matos
- Departamento de Estatística, Universidade Estadual de Campinas, Campinas, Brazil
| | - Luis M Castro
- Departamento de Estadística, Pontificia Universidad Católica de Chile, Santiago, Chile
| | - Ming-Hui Chen
- Department of Statistics, University of Connecticut, Storrs, Connecticut
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9
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Castro LM, Wang WL, Lachos VH, Inácio de Carvalho V, Bayes CL. Bayesian semiparametric modeling for HIV longitudinal data with censoring and skewness. Stat Methods Med Res 2018; 28:1457-1476. [PMID: 29551086 DOI: 10.1177/0962280218760360] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In biomedical studies, the analysis of longitudinal data based on Gaussian assumptions is common practice. Nevertheless, more often than not, the observed responses are naturally skewed, rendering the use of symmetric mixed effects models inadequate. In addition, it is also common in clinical assays that the patient's responses are subject to some upper and/or lower quantification limit, depending on the diagnostic assays used for their detection. Furthermore, responses may also often present a nonlinear relation with some covariates, such as time. To address the aforementioned three issues, we consider a Bayesian semiparametric longitudinal censored model based on a combination of splines, wavelets, and the skew-normal distribution. Specifically, we focus on the use of splines to approximate the general mean, wavelets for modeling the individual subject trajectories, and on the skew-normal distribution for modeling the random effects. The newly developed method is illustrated through simulated data and real data concerning AIDS/HIV viral loads.
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Affiliation(s)
- Luis M Castro
- 1 Department of Statistics, Pontificia Universidad Católica de Chile, Chile
| | - Wan-Lun Wang
- 2 Department of Statistics, Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taichung, Taiwan
| | - Victor H Lachos
- 3 Department of Statistics, University of Connecticut, Storrs, CT, USA
| | | | - Cristian L Bayes
- 5 Department of Sciences, Pontificia Universidad Católica del Perú, Lima, Perú
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10
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Rubio FJ, Steel MFJ. Flexible linear mixed models with improper priors for longitudinal and survival data. Electron J Stat 2018. [DOI: 10.1214/18-ejs1401] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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11
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Lu T, Lu M, Wang M, Zhang J, Dong GH, Xu Y. Partially linear mixed-effects joint models for skewed and missing longitudinal competing risks outcomes. J Biopharm Stat 2017; 29:971-989. [PMID: 29252088 DOI: 10.1080/10543406.2017.1378663] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Abstract
Longitudinal competing risks data frequently arise in clinical studies. Skewness and missingness are commonly observed for these data in practice. However, most joint models do not account for these data features. In this article, we propose partially linear mixed-effects joint models to analyze skew longitudinal competing risks data with missingness. In particular, to account for skewness, we replace the commonly assumed symmetric distributions by asymmetric distribution for model errors. To deal with missingness, we employ an informative missing data model. The joint models that couple the partially linear mixed-effects model for the longitudinal process, the cause-specific proportional hazard model for competing risks process and missing data process are developed. To estimate the parameters in the joint models, we propose a fully Bayesian approach based on the joint likelihood. To illustrate the proposed model and method, we implement them to an AIDS clinical study. Some interesting findings are reported. We also conduct simulation studies to validate the proposed method.
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Affiliation(s)
- Tao Lu
- Department of Mathematics and Statistics, University of Nevada, Reno, NV, USA
| | - Minggen Lu
- School of Community Health Sciences, University of Nevada, Reno, NV, USA
| | - Min Wang
- Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan, USA
| | - Jun Zhang
- College of Mathematics and Statistics, Shenzhen University, Shenzhen, China
| | - Guang-Hui Dong
- Department of Preventive Medicine, Sun Yat-sen University, Guangzhou, China
| | - Yong Xu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, China
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12
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Chen G, Luo S. Bayesian Hierarchical Joint Modeling Using Skew-Normal/Independent Distributions. COMMUN STAT-SIMUL C 2017; 47:1420-1438. [PMID: 30174369 DOI: 10.1080/03610918.2017.1315730] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Abstract
The multiple longitudinal outcomes collected in many clinical trials are often analyzed by multilevel item response theory (MLIRT) models. The normality assumption for the continuous outcomes in the MLIRT models can be violated due to skewness and/or outliers. Moreover, patients' follow-up may be stopped by some terminal events (e.g., death or dropout) which are dependent on the multiple longitudinal outcomes. We proposed a joint modeling framework based on the MLIRT model to account for three data features: skewness, outliers, and dependent censoring. Our method development was motivated by a clinical study for Parkinson's disease.
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Affiliation(s)
- Geng Chen
- Clinical Statistics, GlaxoSmithKline, 1250 S Collegeville Rd., Collegeville, Pennsylvania 19426, USA
| | - Sheng Luo
- Department of Biostatistics, School of Public Health, The University of Texas Health Science Center at Houston, 1200 Pressler St., Houston, Texas 77030, USA
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13
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Huang Y, Yan C. Piecewise Mixture Modeling for Longitudinal Virologic Data With Heterogeneity, Nonnormality, and Missingness. Stat Biopharm Res 2017. [DOI: 10.1080/19466315.2016.1215347] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
Affiliation(s)
- Yangxin Huang
- Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL
| | - Chunning Yan
- School of Management, Shanghai University, Shanghai, China
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14
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Huang Y, Dagne GA, Park JG. Mixture Joint Models for Event Time and Longitudinal Data With Multiple Features. Stat Biopharm Res 2016. [DOI: 10.1080/19466315.2016.1142891] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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15
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Su X, Luo S. Analysis of Censored Longitudinal Data with Skewness and a Terminal Event. COMMUN STAT-SIMUL C 2016; 46:5378-5391. [PMID: 29056818 DOI: 10.1080/03610918.2016.1157181] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
In HIV/AIDS study, the measurements viral load are often highly skewed and left-censored because of a lower detection limit. Furthermore, a terminal event (e.g., death) stops the follow-up process. The time to terminal event may be dependent on the viral load measurements. In this article, we present a joint analysis framework to model the censored longitudinal data with skewness and a terminal event process. The estimation is carried out by adaptive Gaussian quadrature techniques in SAS procedure NLMIXED. The proposed model is evaluated by a simulation study and is applied to the motivating Multicenter AIDS Cohort Study (MACS).
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Affiliation(s)
- Xiao Su
- Department of Biostatistics, School of Public Health, The University of Texas Health Science Center at Houston, 1200 Pressler St., Houston, Texas 77030, USA
| | - Sheng Luo
- Department of Biostatistics, School of Public Health, The University of Texas Health Science Center at Houston, 1200 Pressler St., Houston, Texas 77030, USA
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16
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Khoundabi B, Kazemnejad A, Mansourian M, Faghihimani E. Factors Associated With Serum Albumin in Diabetes Mellitus Type 2 With Microalbuminuria Using Non-Normal Mixed Models: A Prospective Cohort Study. IRANIAN RED CRESCENT MEDICAL JOURNAL 2016; 18:e20671. [PMID: 26889385 PMCID: PMC4752729 DOI: 10.5812/ircmj.20671] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/26/2014] [Revised: 07/06/2014] [Accepted: 08/05/2014] [Indexed: 01/08/2023]
Abstract
Background: The globally increasing epidemic of diabetes will lead to serious problems including diabetic nephropathy and kidney diseases in near future. The first clinical diagnosable stage in a diabetic kidney disease is microalbuminuria (urinary albumin excretion of 30 - 300 g/24 hours). Objectives: This prospective cohort study investigated the risk factors of microalbuminuria in patients with type 2 diabetes who had been registered in endocrine and metabolism research center in Isfahan city, Iran. Patients and Methods: This prospective cohort study was performed on 90 diabetic type 2 patients with microalbuminuria, who were selected according to the consecutive sample selection method during 6 years. Data were collected through regular and systematic measurements of serum albumin as the response variable and body mass index, systolic and diastolic blood pressure, the duration of diabetes, glycosylated hemoglobin (HbA1c), total cholesterol, triglyceride (TG), fasting blood sugar (FBS), low-density lipoprotein (LDL), and high-density lipoprotein (HDL) as the related factors. Non-normal mixed models were used to investigate the impact of effective factors on the amount of excreted serum albumin. Results: According to the deviance information criterion (DIC = 56.2), the non-normal mixed effects model with the skewed t distribution had a best fit and indicated that HbA1c, HDL and total cholesterol had a significant effect on the amount of albumin in urine (P < 0.05). Conclusions: Using nonnormal mixed models may lead to the best results as compared to common normality assumption.
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Affiliation(s)
- Batoul Khoundabi
- Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, IR Iran
| | - Anoshirvan Kazemnejad
- Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, IR Iran
- Corresponding Author: Anoshirvan Kazemnejad, Department of Biostatistics, Faculty of Medical Sciences, Tarbiat Modares University, Tehran, IR Iran. Tel: +98-2182883875, Fax: +98-2182884524, E-mail:
| | - Marjan Mansourian
- Department of Epidemiology and Biostatistics, School of Public Health, Isfahan University of Medical Sciences, Isfahan, IR Iran
| | - Elham Faghihimani
- Department of Internal Medicine, Isfahan Endocrine and Metabolism Research Center, Isfahan University of Medical Sciences, Isfahan, IR Iran
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17
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Influence assessment in censored mixed-effects models using the multivariate Student's- t distribution. J MULTIVARIATE ANAL 2015; 141:104-117. [PMID: 26190871 DOI: 10.1016/j.jmva.2015.06.014] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
In biomedical studies on HIV RNA dynamics, viral loads generate repeated measures that are often subjected to upper and lower detection limits, and hence these responses are either left- or right-censored. Linear and non-linear mixed-effects censored (LMEC/NLMEC) models are routinely used to analyse these longitudinal data, with normality assumptions for the random effects and residual errors. However, the derived inference may not be robust when these underlying normality assumptions are questionable, especially the presence of outliers and thick-tails. Motivated by this, Matos et al. (2013b) recently proposed an exact EM-type algorithm for LMEC/NLMEC models using a multivariate Student's-t distribution, with closed-form expressions at the E-step. In this paper, we develop influence diagnostics for LMEC/NLMEC models using the multivariate Student's-t density, based on the conditional expectation of the complete data log-likelihood. This partially eliminates the complexity associated with the approach of Cook (1977, 1986) for censored mixed-effects models. The new methodology is illustrated via an application to a longitudinal HIV dataset. In addition, a simulation study explores the accuracy of the proposed measures in detecting possible influential observations for heavy-tailed censored data under different perturbation and censoring schemes.
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18
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Lu T. Bayesian nonparametric mixed-effects joint model for longitudinal-competing risks data analysis in presence of multiple data features. Stat Methods Med Res 2015; 26:2407-2423. [PMID: 26265770 DOI: 10.1177/0962280215597939] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Recently, the joint analysis of longitudinal and survival data has been an active research area. Most joint models focus on survival data with only one type of failure. The research on joint modeling of longitudinal and competing risks survival data is sparse. Even so, many joint models for this type of data assume parametric function forms for both longitudinal and survival sub-models, thus limits their use. Further, the common data features that are usually observed in practice, such as asymmetric distribution and missingness in response, measurement errors in covariate, need to be taken into account for reliable parameter estimation. The statistical inference is complicated when all these factors are considered simultaneously. In the article, driven by a motivating example, we assume nonparametric function forms for the varying coefficients in both longitudinal and competing risks survival sub-models. We propose a Bayesian nonparametric mixed-effects joint model for the analysis of longitudinal-competing risks data with asymmetry, missingness, and measurement errors. Simulation studies are conducted to assess the performance of the proposed method. We apply the proposed method to an AIDS dataset and compare a few candidate models under various settings. Some interesting results are reported.
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Affiliation(s)
- Tao Lu
- Department of Epidemiology and Biostatistics, State University of New York, Albany, USA
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19
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Rodrigues-Motta M, Galvis Soto DM, Lachos VH, Vilca F, Baltar VT, Junior EV, Fisberg RM, Lobo Marchioni DM. A mixed-effect model for positive responses augmented by zeros. Stat Med 2015; 34:1761-78. [DOI: 10.1002/sim.6450] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2013] [Revised: 12/16/2014] [Accepted: 01/16/2015] [Indexed: 12/30/2022]
Affiliation(s)
| | | | - Victor H. Lachos
- Department of Statistics; State University of Campinas; São Paulo Brazil
| | - Filidor Vilca
- Department of Statistics; State University of Campinas; São Paulo Brazil
| | - Valéria Troncoso Baltar
- Department of Epidemiology and Biostatistics, Institute of Community Health; Fluminense Federal University; Niterói, Rio de Janeiro Brazil
| | - Eliseu Verly Junior
- Department of Epidemiology, Institute of Social Medicine; Rio de Janeiro State University; Rio de Janeiro Brazil
| | - Regina Mara Fisberg
- Department of Nutrition, School of Public Health; University of São Paulo; São Paulo Brazil
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20
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Lu T, Huang Y. Bayesian inference on mixed-effects varying-coefficient joint models with skew- t distribution for longitudinal data with multiple features. Stat Methods Med Res 2015; 26:1146-1164. [PMID: 25670749 DOI: 10.1177/0962280215569294] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
In AIDS clinical study, two biomarkers, HIV viral load and CD4 cell counts, play important roles. It is well known that there is inverse relationship between the two. Nevertheless, the relationship is not constant but time varying. The mixed-effects varying-coefficient model is capable of capturing the time varying nature of such relationship from both population and individual perspective. In practice, the nucleic acid sequence-based amplification assay is used to measure plasma HIV-1 RNA with a limit of detection (LOD) and the CD4 cell counts are usually measured with much noise and missing data often occur during the treatment. Furthermore, most of the statistical models assume symmetric distribution, such as normal, for the response variables. Often time, normality assumption does not hold in practice. Therefore, it is important to explore all these factors when modeling the real data. In this article, we establish a joint model that accounts for asymmetric and LOD data for the response variable, and covariate measurement error and missingness simultaneously in the mixed-effects varying-coefficient modeling framework. A Bayesian inference procedure is developed to estimate the parameters in the joint model. The proposed model and method are applied to a real AIDS clinical study and various comparisons of a few models are performed.
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Affiliation(s)
- Tao Lu
- 1 Department of Epidemiology and Biostatistics, State University of New York, Albany, NY, USA
| | - Yangxin Huang
- 2 Department of Epidemiology and Biostatistics, College of Public Health University of South Florida, Tampa, FL, USA
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Zhao YY, Tang NS. Maximum-likelihood estimation and influence analysis in multivariate skew-normal reproductive dispersion mixed models for longitudinal data. STATISTICS-ABINGDON 2015. [DOI: 10.1080/02331888.2014.993638] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Bandyopadhyay D, Castro LM, Lachos VH, Pinheiro HP. Robust Joint Non-linear Mixed-Effects Models and Diagnostics for Censored HIV Viral Loads with CD4 Measurement Error. JOURNAL OF AGRICULTURAL BIOLOGICAL AND ENVIRONMENTAL STATISTICS 2015. [DOI: 10.1007/s13253-014-0195-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Garay AM, Castro LM, Leskow J, Lachos VH. Censored linear regression models for irregularly observed longitudinal data using the multivariate- t distribution. Stat Methods Med Res 2014; 26:542-566. [PMID: 25296865 DOI: 10.1177/0962280214551191] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
In acquired immunodeficiency syndrome (AIDS) studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student's t-distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed. An efficient expectation maximization type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step that rely on formulas for the mean and variance of a truncated multivariate Student's t-distribution. The methodology is illustrated through an application to an Human Immunodeficiency Virus-AIDS (HIV-AIDS) study and several simulation studies.
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Affiliation(s)
- Aldo M Garay
- 1 Departamento de Estatística, Universidade Estadual de Campinas, Campinas, Brazil
| | - Luis M Castro
- 2 Departamento de Estadística, Universidad de Concepción, Concepción, Chile
| | - Jacek Leskow
- 3 Institute of Mathematics, Cracow Technical University, Cracow, Poland
| | - Victor H Lachos
- 1 Departamento de Estatística, Universidade Estadual de Campinas, Campinas, Brazil
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Lu X, Huang Y. Bayesian analysis of nonlinear mixed-effects mixture models for longitudinal data with heterogeneity and skewness. Stat Med 2014; 33:2830-49. [PMID: 24623529 DOI: 10.1002/sim.6136] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2013] [Revised: 01/19/2014] [Accepted: 02/16/2014] [Indexed: 11/05/2022]
Abstract
It is a common practice to analyze complex longitudinal data using nonlinear mixed-effects (NLME) models with normality assumption. The NLME models with normal distributions provide the most popular framework for modeling continuous longitudinal outcomes, assuming individuals are from a homogeneous population and relying on random-effects to accommodate inter-individual variation. However, the following two issues may standout: (i) normality assumption for model errors may cause lack of robustness and subsequently lead to invalid inference and unreasonable estimates, particularly, if the data exhibit skewness and (ii) a homogeneous population assumption may be unrealistically obscuring important features of between-subject and within-subject variations, which may result in unreliable modeling results. There has been relatively few studies concerning longitudinal data with both heterogeneity and skewness features. In the last two decades, the skew distributions have shown beneficial in dealing with asymmetric data in various applications. In this article, our objective is to address the simultaneous impact of both features arisen from longitudinal data by developing a flexible finite mixture of NLME models with skew distributions under Bayesian framework that allows estimates of both model parameters and class membership probabilities for longitudinal data. Simulation studies are conducted to assess the performance of the proposed models and methods, and a real example from an AIDS clinical trial illustrates the methodology by modeling the viral dynamics to compare potential models with different distribution specifications; the analysis results are reported.
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Affiliation(s)
- Xiaosun Lu
- Department of Epidemiology and Biostatistics, College of Public Health, University of South Florida, Tampa, FL 33612, U.S.A
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Vock DM, Davidian M, Tsiatis AA. SNP_NLMM: A SAS Macro to Implement a Flexible Random Effects Density for Generalized Linear and Nonlinear Mixed Models. J Stat Softw 2014; 56:2. [PMID: 24688453 DOI: 10.18637/jss.v056.c02] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022] Open
Abstract
Generalized linear and nonlinear mixed models (GMMMs and NLMMs) are commonly used to represent non-Gaussian or nonlinear longitudinal or clustered data. A common assumption is that the random effects are Gaussian. However, this assumption may be unrealistic in some applications, and misspecification of the random effects density may lead to maximum likelihood parameter estimators that are inconsistent, biased, and inefficient. Because testing if the random effects are Gaussian is difficult, previous research has recommended using a flexible random effects density. However, computational limitations have precluded widespread use of flexible random effects densities for GLMMs and NLMMs. We develop a SAS macro, SNP_NLMM, that overcomes the computational challenges to fit GLMMs and NLMMs where the random effects are assumed to follow a smooth density that can be represented by the seminonparametric formulation proposed by Gallant and Nychka (1987). The macro is flexible enough to allow for any density of the response conditional on the random effects and any nonlinear mean trajectory. We demonstrate the SNP_NLMM macro on a GLMM of the disease progression of toenail infection and on a NLMM of intravenous drug concentration over time.
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Sun W, Larsen MD, Lachin JM. Methods for a longitudinal quantitative outcome with a multivariate Gaussian distribution multi-dimensionally censored by therapeutic intervention. Stat Med 2013; 33:1288-306. [PMID: 24258796 DOI: 10.1002/sim.6037] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2013] [Revised: 10/05/2013] [Accepted: 10/21/2013] [Indexed: 11/07/2022]
Abstract
In longitudinal studies, a quantitative outcome (such as blood pressure) may be altered during follow-up by the administration of a non-randomized, non-trial intervention (such as anti-hypertensive medication) that may seriously bias the study results. Current methods mainly address this issue for cross-sectional studies. For longitudinal data, the current methods are either restricted to a specific longitudinal data structure or are valid only under special circumstances. We propose two new methods for estimation of covariate effects on the underlying (untreated) general longitudinal outcomes: a single imputation method employing a modified expectation-maximization (EM)-type algorithm and a multiple imputation (MI) method utilizing a modified Monte Carlo EM-MI algorithm. Each method can be implemented as one-step, two-step, and full-iteration algorithms. They combine the advantages of the current statistical methods while reducing their restrictive assumptions and generalizing them to realistic scenarios. The proposed methods replace intractable numerical integration of a multi-dimensionally censored MVN posterior distribution with a simplified, sufficiently accurate approximation. It is particularly attractive when outcomes reach a plateau after intervention due to various reasons. Methods are studied via simulation and applied to data from the Diabetes Control and Complications Trial/Epidemiology of Diabetes Interventions and Complications study of treatment for type 1 diabetes. Methods proved to be robust to high dimensions, large amounts of censored data, low within-subject correlation, and when subjects receive non-trial intervention to treat the underlying condition only (with high Y), or for treatment in the majority of subjects (with high Y) in combination with prevention for a small fraction of subjects (with normal Y).
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Affiliation(s)
- Wanjie Sun
- Division of Biometrics VI, Center for Drug Evaluation and Research, Food and Drug Administration, Rockville, MD 20852, U.S.A
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