Abstract
A common goal in environmental epidemiologic studies is to undertake logistic regression modeling to associate a continuous measure of exposure with binary disease status, adjusting for covariates. A frequent complication is that exposure may only be measurable indirectly, through a collection of subject-specific variables assumed associated with it. Motivated by a specific study to investigate the association between lung function and exposure to metal working fluids, we focus on a multiplicative-lognormal structural measurement error scenario and approaches to address it when external validation data are available. Conceptually, we emphasize the case in which true untransformed exposure is of interest in modeling disease status, but measurement error is additive on the log scale and thus multiplicative on the raw scale. Methodologically, we favor a pseudo-likelihood (PL) approach that exhibits fewer computational problems than direct full maximum likelihood (ML) yet maintains consistency under the assumed models without necessitating small exposure effects and/or small measurement error assumptions. Such assumptions are required by computationally convenient alternative methods like regression calibration (RC) and ML based on probit approximations. We summarize simulations demonstrating considerable potential for bias in the latter two approaches, while supporting the use of PL across a variety of scenarios. We also provide accessible strategies for obtaining adjusted standard errors to accompany RC and PL estimates.
Similar content being viewed by others
References
Carroll, R. J. (1989), “Covariance Analysis in Generalized Linear Measurement Error Models,” Statistics in Medicine, 8, 1075–1093.
Carroll, R. J., and Stefanski, L. A. (1990), “Approximate Quasilikelihood Estimation in Models with Surrogate Predictors,” Journal of the American Statistical Association, 85, 652–663.
Carroll, R. J., Spiegelman, C. H., Lan, K. K. G., Bailey, K. T., and Abbott, R. D. (1984), “On Errors-in-Variables for Binary Regression Models,” Biometrika, 71, 19–25.
Carroll, R. J., Ruppert, D., Stefanski, L. A., and Crainiceanu, C. M. (2006), Measurement Error in Nonlinear Models: A Modern Perspective (2nd ed.), London: Chapman and Hall.
Clayton, D. (1992), “Models for the Longitudinal Analysis of Cohort and Case-Control Studies with Inaccurately Measured Exposures,” in Statistical Models for Longitudinal Studies of Health, ed. J. H. Dwyer, Oxford: Oxford University Press, pp. 301–325.
Crouch, E. A., and Spiegelman, D. (1990), “The Evaluation of Integrals of the Form \(\int_{ - \infty}^{\infty} f(t)\exp( - t^{2})\,dt\): Applications to Logistic-Normal Models,” Journal of the American Statistical Association, 85, 464–467.
Demidenko, E. (2004), Mixed Models: Theory and Applications, New York: Wiley.
Eisen, E. A., Holcroft, C. R., Greaves, I. A., Wegman, D. H., Woskie, S. R., and Monson, R. R. (1997), “A Strategy to Reduce Healthy Worker Effect in a Cross-Sectional Study of Asthma and Metalworking Fluids,” American Journal of Industrial Medicine, 31, 671–677.
Eisen, E. A., Smith, T. J., Kreibel, D., Woskie, S. R., Myers, D. J., and Kennedy, S. M. et al. (2001), “Respiratory Effects of Machining Fluids: Pulmonary Function,” American Journal of Industrial Medicine, 39, 443–453.
Fuller, W. A. (1987), Measurement Error Models, New York: Wiley.
Gong, G., and Samaniego, F. J. (1981), “Pseudo Maximum Likelihood Estimation: Theory and Applications,” Annals of Statistics, 9, 861–869.
Greaves, I. A., Eisen, E. A., Smith, T. J., Pothier, L. J., Kreibel, D., and Woskie, S. R. et al. (1997), “Respiratory Health of Automobile Workers Exposed to Metal-Working Fluid Aerosols: Respiratory Symptoms,” American Journal of Industrial Medicine, 26, 621–634.
Hwang, J. T. (1986), “Multiplicative Errors-in-Variables Models with Applications to Recent Data Released by the U.S. Department of Energy,” Journal of the American Statistical Association, 81, 680–688.
Iturria, S., Carroll, R. J., and Firth, D. (1999), “Multiplicative Measurement Error Estimation: Estimating Equations,” Journal of the Royal Statistical Society, Series B, 61, 547–562.
Kuha, J. (1994), “Corrections for Exposure Measurement Error in Logistic Regression Models with an Application to Nutritional Data,” Statistics in Medicine, 13, 1135–1148.
Liang, K.-Y., and Liu, X.-H. (1991), “Estimating Equations in Generalized Linear Models with Measurement Error,” in Estimating Functions, ed. V. P. Godambe, Oxford: Oxford University Press.
Lyles, R. H., and Kupper, L. L. (1997), “A Detailed Evaluation of Adjustment Methods for Multiplicative Measurement Error in Linear Regression with Applications in Occupational Epidemiology,” Biometrics, 53, 1008–1025.
Messer, K., and Natarajan, L. (2008), “Maximum Likelihood, Multiple Imputation and Regression Calibration for Measurement Error Adjustment,” Statistics in Medicine, 27, 6332–6350.
Monahan, J., and Stefanski, L. A. (1991), “Normal Scale Mixture Approximations to F ∗(z) and Computation of the Logistic-Normal Integral,” in Handbook of the Logistic Distribution, ed. N. Balakrishnan, New York: Dekker, pp. 529–540.
Pierce, D. A., Stram, D. O., Vaeth, M., and Schafer, D. W. (1992), “The Errors-in-Variables Problem: Considerations Provided by Radiation Dose-Response Analysis of the A-Bomb Survivor Data,” Journal of the American Statistical Association, 87, 351–359.
Rappaport, S. M. (1991), “Assessment of Long-Term Exposures to Toxic Substances in Air,” Annals of Occupational Hygiene, 35, 61–121.
Rappaport, S. M., and Kupper, L. L. (2008), Quantitative Exposure Assessment, Raleigh, NC: Lulu Press.
Rosner, B., Willett, W. C., and Spiegelman, D. (1989), “Correction of Logistic Regression Relative Risk Estimates and Confidence Intervals for Systematic Within-Person Measurement Error,” Statistics in Medicine, 8, 1051–1069.
Rosner, B., Spiegelman, D., and Willett, W. C. (1990), “Correction of Logistic Regression Relative Risk Estimates and Confidence Intervals for Measurement Error: The Case of Multiple Covariates Measured with Error,” American Journal of Epidemiology, 132, 734–745.
Rubin, D. B. (1987), Multiple Imputation for Nonresponse in Surveys, New York: Wiley.
SAS Institute, Inc. (2004). SAS IML 9.1 User’s Guide, Cary, NC: SAS Institute.
Savalei, V. (2006), “Logistic Approximation to the Normal: The KL Rationale,” Psychometrika, 71, 763–767.
Schafer, D. W. (1987), “Covariate Measurement Error in Generalized Linear Models,” Biometrika, 80, 889–904.
Schafer, J. L. (1999), “Multiple Imputation: A Primer,” Statistical Methods in Medical Research, 8, 3–15.
Spall, J. C. (1989), “Effects of Imprecisely Known Nuisance Parameters on Estimates of Primary Parameters,” Communications in Statistics: Theory and Methods, 18, 219–237.
Spiegelman, D., Carroll, R., and Kipnis, V. (2001), “Efficient Regression Calibration for Logistic Regression in Main Study/Internal Validation Study Designs with an Imperfect Reference Instrument,” Statistics in Medicine, 10, 139–160.
Stefanski, L. A., and Carroll, R. J. (1990), “Structural Logistic Regression Measurement Error Models,” in Proceedings of the Conference of Measurement Error Models, eds. P. J. Brown and W. A. Fuller, Providence, RI: American Mathematical Society.
Thomas, D., Stram, D., and Dwyer, J. (1993), “Exposure Measurement Error: Influence on Exposure–Disease Relationships and Methods of Correction,” Annual Review of Public Health, 14, 69–93.
Thoresen, M., and Laake, P. (2000), “A Simulation Study of Measurement Error Correction Methods in Logistic Regression,” Biometrics, 56, 868–872.
Thurston, S. W., Spiegelman, D., and Ruppert, D. (2003), “Equivalence of Regression Calibration Methods in Main Study/external Validation Study Designs,” Journal of Statistical Planning and Inference, 113, 527–539.
Weller, E. A., Milton, D. K., Eisen, E. A., and Spiegelman, D. (2007), “Regression Calibration for Logistic Regression with Multiple Surrogates for One Exposure,” Journal of Statistical Planning and Inference, 137, 449–461.
Woskie, S. R., Smith, T. J., Hallock, M. F., Hammond, S. K., Rosenthal, F., and Eisen, E. A. et al. (1994), “Size-Selective Pulmonary Dose Indices for Metal-Working Fluid Aerosols in Machining and Grinding Operations in the Automobile Manufacturing Industry,” American Industrial Hygiene Association Journal, 55, 20–29.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lyles, R.H., Kupper, L.L. Approximate and Pseudo-Likelihood Analysis for Logistic Regression Using External Validation Data to Model Log Exposure. JABES 18, 22–38 (2013). https://doi.org/10.1007/s13253-012-0115-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13253-012-0115-9