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Correlating two continuous variables subject to detection limits in the context of mixture distributions

Author

Listed:
  • Haitao Chu
  • Lawrence H. Moulton
  • Wendy J. Mack
  • Douglas J. Passaro
  • Paulo F. Barroso
  • Alvaro Muñoz

Abstract

Summary. In individuals who are infected with human immunodeficiency virus (HIV), distributions of quantitative HIV ribonucleic acid measurements may be highly left censored with an extra spike below the limit of detection LD of the assay. A two‐component mixture model with the lower component entirely supported on [0, LD] is recommended to model the extra spike in univariate analysis better. Let LD1 and LD2 be the limits of detection for the two HIV viral load measurements. When estimating the correlation coefficient between two different measures of viral load obtained from each of a sample of patients, a bivariate Gaussian mixture model is recommended to model the extra spike on [0, LD1] and [0, LD2] better when the proportion below LD is incompatible with the left‐hand tail of a bivariate Gaussian distribution. When the proportion of both variables falling below LD is very large, the parameters of the lower component may not be estimable since almost all observations from the lower component are falling below LD. A partial solution is to assume that the lower component's entire support is on [0, LD1]×[0, LD2]. Maximum likelihood is used to estimate the parameters of the lower and higher components. To evaluate whether there is a lower component, we apply a Monte Carlo approach to assess the p‐value of the likelihood ratio test and two information criteria: a bootstrap‐based information criterion and a cross‐validation‐based information criterion. We provide simulation results to evaluate the performance and compare it with two ad hoc estimators and a single‐component bivariate Gaussian likelihood estimator. These methods are applied to the data from a cohort study of HIV‐infected men in Rio de Janeiro, Brazil, and the data from the Women's Interagency HIV oral study. These results emphasize the need for caution when estimating correlation coefficients from data with a large proportion of non‐detectable values when the proportion below LD is incompatible with the left‐hand tail of a bivariate Gaussian distribution.

Suggested Citation

  • Haitao Chu & Lawrence H. Moulton & Wendy J. Mack & Douglas J. Passaro & Paulo F. Barroso & Alvaro Muñoz, 2005. "Correlating two continuous variables subject to detection limits in the context of mixture distributions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(5), pages 831-845, November.
  • Handle: RePEc:bla:jorssc:v:54:y:2005:i:5:p:831-845
    DOI: 10.1111/j.1467-9876.2005.00512.x
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    Cited by:

    1. Zonghui Hu & Jing Qin & Dean Follmann, 2008. "Semiparametric two‐sample changepoint model with application to human immunodeficiency virus studies," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 57(5), pages 589-607, December.
    2. Huiyun Wu & Qingxia Chen & Lorraine B. Ware & Tatsuki Koyama, 2012. "A Bayesian approach for generalized linear models with explanatory biomarker measurement variables subject to detection limit: an application to acute lung injury," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(8), pages 1733-1747, March.
    3. R. Bajorunaite & V. Brazauskas, 2008. "Method of trimmed moments for robust fitting of parametric failure time models," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 341-360.

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