Skip to main content
SpringerLink
Log in

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

  • Published:
Statistics in Biosciences Aims and scope Submit manuscript

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abarin T, Wu Y, Warrington N, Pennell C, Briollais L (2011) The impact of breastfeeding on FTO-related BMI growth trajectories: an application to the RAINE birth and child cohort study. Int J Epidemiol (to appear)

  2. Abarin T, Wang L (2010) Instrumental variable approach to covariate measurement error in generalized linear models. Ann Inst Stat Math. doi:10.1007/s10463-010-0319-0

    Google Scholar 

  3. Abarin T, Wang L (2009) Second-order least squares estimation of censored regression models. J Stat Plan Inference 139:125–135

    Article  MATH  MathSciNet  Google Scholar 

  4. Abarin T, Wang L (2006) Comparison of GMM with second-order least squares estimation in nonlinear models. Far East J Theor Stat 20(2):179–196

    MATH  MathSciNet  Google Scholar 

  5. Berkson J (1950) Are there two regressions? J Am Stat Assoc 45:164–180

    Article  MATH  Google Scholar 

  6. Biggs R, et al. (2009) Spurious certainty: how ignoring measurement error and environmental heterogeneity may contribute to environmental controversies. Bioscience 59(1):65–76

    Article  Google Scholar 

  7. Bland RM, et al. (2003) Maternal recall of exclusive breast feeding duration. Arch Dis Child 88(9):778–783

    Article  Google Scholar 

  8. Buonaccorsi J (2010) Measurement error: models, methods, and applications. Chapman & Hall, London

    Book  Google Scholar 

  9. Buonaccorsi J (1996) Measurement error in the response in the general linear model. J Am Stat Assoc 91:633–642

    Article  MATH  MathSciNet  Google Scholar 

  10. Buonaccorsi J, Tosteson T (1993) Correcting for nonlinear measurement errors in the dependent variable in the general linear model. Commun Stat, Theory Methods 22:2687–2702

    Article  MATH  MathSciNet  Google Scholar 

  11. Buonaccorsi J, Lin Ch (2002) Berkson measurement error in designed repeated measures studies with random coefficients. J Stat Plan Inference 104:53–72

    Article  MATH  MathSciNet  Google Scholar 

  12. Buonaccorsi J, et al. (2000) Estimation in longitudinal random effects models with measurement error. Stat Sin 10:885–903

    MATH  MathSciNet  Google Scholar 

  13. Burton PR, et al. (2009) Size matters: just how big is BIG? Quantifying realistic sample size requirements for human genome epidemiology. Int J Epidemiol 38:263–273

    Article  Google Scholar 

  14. Carroll RJ, et al. (2006) Measurement error in nonlinear models: a modern perspective, 2nd edn. Chapman & Hall, London

    Book  Google Scholar 

  15. Carroll RJ, Wand MP (1991) Semiparametric estimation in logistic measurement error models. J R Stat Soc B 53:573–585

    MATH  MathSciNet  Google Scholar 

  16. Faith MS, et al. (2004) Parental feeding attitudes and styles and child body mass index: prospective analysis of a gene-environment interaction. Pediatrics 114(4):e429–e436

    Article  Google Scholar 

  17. Fuller AW (1987) Measurement error models. Wiley, New York

    Book  MATH  Google Scholar 

  18. Joseph ML, Carriquiry A (2010) A measurement error approach to assess the association between dietary diversity, nutrient intake, and mean probability of adequacy. J Nutr 140(11):2094S–2101S

    Article  Google Scholar 

  19. Kamarainen AM, et al. (2008) Zooplankton and the total phosphorus—chlorophyll a relationship: hierarchical Bayesian analysis of measurement error. Can J Fish Aquat Sci 65(12):2644–2655

    Article  Google Scholar 

  20. Kipnis V, et al. (2003) Structure of dietary measurement error: results of the OPEN biomarker study. Am J Epidemiol 158(1):14–21

    Article  Google Scholar 

  21. Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38:963–974

    Article  MATH  Google Scholar 

  22. Li H (2005) A simulation study of the second-order least squares estimators for nonlinear mixed effects models. Master’s thesis, University of Manitoba

  23. Li H, Wang L (2011) Second-order least squares estimation in linear mixed models. University of Manitoba. Comm Statist Theory Methods doi:10.1080/s03610926.2011.601837

  24. Marchand L, Wilkens L (2008) Design considerations for genomic association studies: importance of gene-environment interactions. Cancer Epidemiol Biomark Prev 17:263–267

    Article  Google Scholar 

  25. Pan W, et al. (2009) Semiparametric transition measurement error models for longitudinal data. Biometrics 65:728–736

    Article  MATH  MathSciNet  Google Scholar 

  26. Parsons TJ, et al. (2001) Fetal and early life growth and body mass index from birth to early adulthood in 1958 British cohort: longitudinal study. Br Med J 323:1331–1335

    Article  Google Scholar 

  27. Prentice RL, et al. (2002) Research strategies and the use of nutriet biomarkers in studies of diet and chronic disease. Public Health Nutr 5:977–984

    Article  Google Scholar 

  28. Rampersaud E, et al. (2008) Physical activity and the association of common FTO gene variants with body mass index and obesity. Arch Intern Med 168(16):1791–1797

    Article  Google Scholar 

  29. Rios E, et al. (1992) Accuracy of mothers’ responses to questions about breast-feeding practices. Food Nutr Bull 14(2):115–118

    Google Scholar 

  30. Schennach MS (2007) Instrumental variable estimation of nonlinear errors-in-variables models. Econometrica 75:201–239

    Article  MATH  MathSciNet  Google Scholar 

  31. Thomas D (2010) Methods for investigating gene-environment interactions in candidate pathway and genome-wide association studies. Annu Rev Public Health 31:21–36

    Article  Google Scholar 

  32. Tosteson TD, et al. (1998) Covariate measurement error and the estimation of random effect parameters in a mixed model for longitudinal data. Stat Med 8:1139–1147

    Article  Google Scholar 

  33. Tsiatis T, Davidian M (2001) A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika 88(2):447–458

    Article  MATH  MathSciNet  Google Scholar 

  34. Vimaleswaran KS, et al. (2009) Physical activity attenuates the body mass index–increasing influence of genetic variation in the FTO gene. Am J Clin Nutr 90(2):425–428

    Article  Google Scholar 

  35. Wang L (2004) Estimation of nonlinear models with Berkson measurement errors. Ann Stat 32:2559–2579

    Article  MATH  Google Scholar 

  36. Wang L, Hsiao C (2010) Method of moments estimation and identifiability of nonlinear semiparametric errors-in-variables models. J Econom 165:30–44. doi:10.1016/j.jeconom.2011.05.004

    Article  MathSciNet  Google Scholar 

  37. Wang N, et al. (1998) Bias analysis and SIMEX approach in generalized mixed measurement error models. J Am Stat Assoc 93:249–261

    Article  MATH  Google Scholar 

  38. Wang N, et al. (1999) A bias correction regression calibration approach in generalized linear mixed measurement error model. Commun Stat, Theory Methods 28:217–232

    Article  MATH  Google Scholar 

  39. Wong MY, et al. (2003) The detection of gene-environment interaction for continuous traits: should we deal with measurement error by bigger studies or better measurement? Int J Epidemiol 32:51–57

    Article  Google Scholar 

Download references

Acknowledgements

We thank the referees for their comments and helpful suggestions for manuscript changes. These have improved both the content and clarity of the manuscript. We would also like to thank Wei Ang, Nicole Warrington, Julie Marsh and Louise Mckenzie, for their help in preparing the data set and choosing the variables of the model in the application, and the RAINE study (http://www.rainestudy.org.au/) for providing the data for analysis. Financial support from “The Alva Foundation” and “Samuel Lunenfeld Research Institute—New Opportunities Funds” and MITACS are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University, St. John’s, NL, Canada

    Taraneh Abarin

  2. Department of Statistics, University of Manitoba, Winnipeg, Canada

    He Li & Liqun Wang

  3. Samuel Lunenfeld Research Institute, Mount Sinai Hospital, Toronto, Canada

    Laurent Briollais

Authors
  1. Taraneh Abarin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. He Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Laurent Briollais
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taraneh Abarin.

Additional information

This study has been partially supported by NSERC.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(PDF 176 kB)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abarin, T., Li, H., Wang, L. et al. On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction. Stat Biosci 6, 1–18 (2014). https://doi.org/10.1007/s12561-012-9074-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12561-012-9074-5

Keywords

Search

Navigation