Maximum likelihood estimation in generalized linear models with multiple covariates subject to detection limits
The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Mont...
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| Published in | Statistics in medicine Vol. 30; no. 20; pp. 2551 - 2561 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
10.09.2011
Wiley Subscription Services, Inc |
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| Online Access | Get full text |
| ISSN | 0277-6715 1097-0258 1097-0258 |
| DOI | 10.1002/sim.4280 |
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| Abstract | The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation–maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one‐dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M‐step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. Copyright © 2011 John Wiley & Sons, Ltd. |
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| AbstractList | The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation–maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one‐dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M‐step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. Copyright © 2011 John Wiley & Sons, Ltd. The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation-maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one-dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M-step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation-maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one-dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M-step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies.The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation-maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one-dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M-step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. The analysis of data subject to detection limits is becoming increasingly necessary in many environmental and laboratory studies. Covariates subject to detection limits are often left censored because of a measurement device having a minimal lower limit of detection. In this paper, we propose a Monte Carlo version of the expectation-maximization algorithm to handle large number of covariates subject to detection limits in generalized linear models. We model the covariate distribution via a sequence of one-dimensional conditional distributions, and sample the covariate values using an adaptive rejection metropolis algorithm. Parameter estimation is obtained by maximization via the Monte Carlo M-step. This procedure is applied to a real dataset from the National Health and Nutrition Examination Survey, in which values of urinary heavy metals are subject to a limit of detection. Through simulation studies, we show that the proposed approach can lead to a significant reduction in variance for parameter estimates in these models, improving the power of such studies. [PUBLICATION ABSTRACT] |
| Author | Chu, Haitao May, Ryan C. Ibrahim, Joseph G. |
| Author_xml | – sequence: 1 givenname: Ryan C. surname: May fullname: May, Ryan C. email: ryanmay@unc.edu, Ryan C. May, Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill NC 27599, USA, ryanmay@unc.edu organization: Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA – sequence: 2 givenname: Joseph G. surname: Ibrahim fullname: Ibrahim, Joseph G. organization: Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA – sequence: 3 givenname: Haitao surname: Chu fullname: Chu, Haitao organization: Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA |
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| Cites_doi | 10.2307/1907382 10.1093/aje/kwj039 10.2307/2347565 10.1080/01621459.1990.10474930 10.1289/ehp.7199 10.1111/j.2517-6161.1982.tb01203.x 10.1097/EDE.0b013e3181ce97d8 10.1111/j.1468-2354.2007.00470.x 10.1093/aje/kwf217 10.1002/sim.3285 10.1093/biomet/83.4.916 10.1063/1.1699114 10.1023/A:1023630425376 10.1016/j.chemosphere.2006.04.051 10.1080/01621459.1990.10474938 10.2307/2986138 10.1111/1467-9868.00170 10.1002/1097-0258(20010115)20:1<33::AID-SIM640>3.0.CO;2-O 10.1093/ije/dyp269 |
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| Keywords | NHANES Gibbs sampling logistic regression EM algorithm maximum likelihood estimation Monte Carlo EM |
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| References | D'Angelo G, Weissfeld L. An index approach for the Cox model with left censored covariates. Statistics in Medicine 2008; 27:4502-4514. Lipsitz SR, Ibrahim JG. A conditional model for incomplete covariates in parametric regression models. Biometrika 1996; 72:916-922. Nie L, Chu H, Liu C, Cole SR, Vexler A, Schisterman EF. Linear regression with an independent variable subject to a detection limit. Epidemiology 2010; 21(4):S17-S24. Wei GC, Tanner MA. A Monte Carlo implementation of the EM algorithm and the poor dan's data augmentation algorithms. Journal of the American Statistical Association 1990; 85:699-704. Schisterman E, Vexler A, Whitcomb B, Liu A. The limitations due to exposure detection limits for regression models. American Journal of Epidemiology 2006; 163(4):374-383. Cole SR, Chu H, Nie L, Schisterman EF. Estimating the odds ratio when exposure has a limit of detection.International Journal of Epidemiology 2009; 38:1674-1680. Gilks WR, Best NG, Tan KKC. Adaptive rejection metropolis sampling within Gibbs sampling. Applied Statistics 1995; 44:455-472. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equations of state calculations by fast computing machines. Journal of Chemical Physics 1953; 21:1087-1092. Lynn H. Maximum likelihood inference for left-censored HIV RNA data. Statistics in Medicine 2001; 20:33-45. Rigobon R, Stoker TM. Estimation with censored regressors: basic issues. International Economic Review 2007; 48(4):1441-1467. Ibrahim JG. Incomplete data in generalized linear models. Journal of the American Statistical Association 1990; 85:765-769. Thompson M, Nelson KP. Linear regression with Type I interval- and left-censored response data. Environmental and Ecological Statistics 2003; 10:221-230. Gilks WR, Wild P. Adaptive rejection sampling for Gibbs sampling. Applied Statistics 1992; 41:337-348. Helsel DR. Fabricating data: how substituting values for nondetects can ruin results, and what can be done about it. Chemosphere 2006; 65:2434-2439. Richardson DB, Ciampi A. Effects of exposure measurement error when an exposure variable is constrained by a lower limit. American Journal of Epidemiology 2003; 157(4):355-363. Singh A, Nocerino J. Chemometrics and Intelligent Laboratory Systems, Vol. 60, 2002. Ibrahim JG, Lipsitz SR, Chen M. Missing covariates in generalized linear models when the missing data mechanism is non-ignorable. Journal of the Royal Statistical Society, Series B 1999; 61:173-190. Louis T. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B 1982; 44:226-233. Lubin JH, Colt JS, Camann D, Davis S, Cerhan JR, Severson RK, Bernstein L, Hartge P. Epidemiologic evaluation of measurement data in the presence of detection limits. Environmental Health Perspectives 2004; 112(17):1691-1696. Helsel DR. Nondetects and Data Analysis: Statistics for Censored Environmental Data: New York, 2004. Tobin J. Estimation of relationships for limited dependent variables. Econometrica 1958; 26:24-36. 2004; 112 1990; 85 2010; 21 2006; 65 1958; 26 2008; 27 1995; 44 1982; 44 2006; 163 1996; 72 2004 2002; 60 1999; 61 2003; 157 2009; 38 2003; 10 1953; 21 2007; 48 2001; 20 1992; 41 e_1_2_8_17_1 e_1_2_8_18_1 Helsel DR (e_1_2_8_6_1) 2004 e_1_2_8_19_1 Louis T (e_1_2_8_23_1) 1982; 44 e_1_2_8_13_1 e_1_2_8_14_1 e_1_2_8_15_1 e_1_2_8_16_1 Singh A (e_1_2_8_10_1) 2002 e_1_2_8_3_1 e_1_2_8_2_1 e_1_2_8_5_1 e_1_2_8_4_1 e_1_2_8_7_1 e_1_2_8_9_1 e_1_2_8_8_1 e_1_2_8_20_1 e_1_2_8_21_1 e_1_2_8_11_1 e_1_2_8_22_1 e_1_2_8_12_1 |
| References_xml | – reference: Ibrahim JG. Incomplete data in generalized linear models. Journal of the American Statistical Association 1990; 85:765-769. – reference: D'Angelo G, Weissfeld L. An index approach for the Cox model with left censored covariates. Statistics in Medicine 2008; 27:4502-4514. – reference: Helsel DR. Nondetects and Data Analysis: Statistics for Censored Environmental Data: New York, 2004. – reference: Schisterman E, Vexler A, Whitcomb B, Liu A. The limitations due to exposure detection limits for regression models. American Journal of Epidemiology 2006; 163(4):374-383. – reference: Nie L, Chu H, Liu C, Cole SR, Vexler A, Schisterman EF. Linear regression with an independent variable subject to a detection limit. Epidemiology 2010; 21(4):S17-S24. – reference: Lubin JH, Colt JS, Camann D, Davis S, Cerhan JR, Severson RK, Bernstein L, Hartge P. Epidemiologic evaluation of measurement data in the presence of detection limits. Environmental Health Perspectives 2004; 112(17):1691-1696. – reference: Gilks WR, Best NG, Tan KKC. Adaptive rejection metropolis sampling within Gibbs sampling. Applied Statistics 1995; 44:455-472. – reference: Helsel DR. Fabricating data: how substituting values for nondetects can ruin results, and what can be done about it. Chemosphere 2006; 65:2434-2439. – reference: Gilks WR, Wild P. Adaptive rejection sampling for Gibbs sampling. Applied Statistics 1992; 41:337-348. – reference: Richardson DB, Ciampi A. Effects of exposure measurement error when an exposure variable is constrained by a lower limit. American Journal of Epidemiology 2003; 157(4):355-363. – reference: Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equations of state calculations by fast computing machines. Journal of Chemical Physics 1953; 21:1087-1092. – reference: Cole SR, Chu H, Nie L, Schisterman EF. Estimating the odds ratio when exposure has a limit of detection.International Journal of Epidemiology 2009; 38:1674-1680. – reference: Tobin J. Estimation of relationships for limited dependent variables. Econometrica 1958; 26:24-36. – reference: Thompson M, Nelson KP. Linear regression with Type I interval- and left-censored response data. Environmental and Ecological Statistics 2003; 10:221-230. – reference: Wei GC, Tanner MA. A Monte Carlo implementation of the EM algorithm and the poor dan's data augmentation algorithms. Journal of the American Statistical Association 1990; 85:699-704. – reference: Rigobon R, Stoker TM. Estimation with censored regressors: basic issues. International Economic Review 2007; 48(4):1441-1467. – reference: Ibrahim JG, Lipsitz SR, Chen M. Missing covariates in generalized linear models when the missing data mechanism is non-ignorable. Journal of the Royal Statistical Society, Series B 1999; 61:173-190. – reference: Lynn H. Maximum likelihood inference for left-censored HIV RNA data. Statistics in Medicine 2001; 20:33-45. – reference: Louis T. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B 1982; 44:226-233. – reference: Lipsitz SR, Ibrahim JG. A conditional model for incomplete covariates in parametric regression models. Biometrika 1996; 72:916-922. – reference: Singh A, Nocerino J. 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| SubjectTerms | Algorithms Computer Simulation Data analysis EM algorithm Gibbs sampling Humans Likelihood Functions Limit of Detection Linear Models logistic regression Male maximum likelihood estimation Maximum likelihood method Medical statistics Metals, Heavy - urine Monte Carlo EM Monte Carlo Method Monte Carlo simulation Neoplasms - urine NHANES Urology |
| Title | Maximum likelihood estimation in generalized linear models with multiple covariates subject to detection limits |
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