Journal of Pharmacokinetics and Pharmacodynamics

, Volume 38, Issue 6, pp 833–859 | Cite as

Extending the latent variable model for extra correlated longitudinal dichotomous responses



Since generalized nonlinear mixed-effects modeling methodology of ordered categorical data became available in the pharmacokinetic/pharmacodynamic (PK/PD) literature over a decade ago, pharmacometricians have been increasingly performing exposure–response analyses of such data to inform drug development. Also, as experiences with and scrutiny of these data have increased, pharmacometricians have noted fewer transitions (or greater correlations) between response values than predicted by the model. In this paper, we build on the latent variable (LV) approach, which is convenient for incorporating pharmacological concepts such as pharmacodynamic onset of drug effect, and present a PK/PD methodology which we term the multivariate latent variable (MLV) approach. This approach uses correlations between the latent residuals (LR) to address extra correlation or a fewer number of transitions, relative to if the LR were independent. Four approximation methods for handling dichotomous MLV data are formulated and then evaluated for accuracy and computation time using simulation studies. Some analytical results for models linear in the subject-specific random effects are also presented, and these provide insight into modeling such repeated measures data. Also, a case study previously analyzed using the LV approach is revisited using one of the MLV approximation methods and the results are discussed. Overall, consideration of the simulation and analytical results lead us to some conclusions we feel are applicable to many of the models and situations frequently encountered in analysis of such data: the MLV approach is a flexible method that can handle many different extra correlated data structures and therefore can more accurately predict the number of transitions between response values; incorrect modeling of the population covariances by implementing an LV model when extra correlation exists is not likely to (and in many cases does not) influence accuracy of the population (marginal) mean predictions; adequate prediction of the population mean probabilities achieves adequate predictions of the population variances, regardless of the correct specification of the population covariances—that is, if the LV model accurately predicts the means in the presence of extra correlation, it will accurately predict the variances; the between subject random effects component to the model describe the marginal covariances in responses—not the marginal variances as with continuous-type data. From these conclusions we make a general statement that it may not be necessary to model the extra correlation in every case using the MLV model, which requires technical implementation with currently available commercially or publically available software. The LV model may be sufficient for answering many of the typical questions arising during drug development. The MLV approach should be considered however if prediction or simulation of individual level data is an objective of the analysis.


Latent variable Probit Multivariate probit Generalized nonlinear mixed-effects models 


  1. 1.
    Sheiner LB (1994) A new approach to the analysis of analgesic drug trials, illustrated with bromfenac data. Clin Pharmacol Ther 56:309–322PubMedCrossRefGoogle Scholar
  2. 2.
    Sheiner LB, Beal SL, Dunne A (1997) Analysis of nonrandomly censored ordered categorical longitudinal data from analgesic trials. J Am Stat Assoc 92:1235–1244CrossRefGoogle Scholar
  3. 3.
    Hutmacher MM, Krishnaswami S, Kowalski KG (2008) Exposure-response modeling using latent variables for the efficacy of a JAK3 inhibitor administered to rheumatoid arthritis patients. J Pharmacokinet Pharmacodyn 35:139–157PubMedCrossRefGoogle Scholar
  4. 4.
    Hu C, Xu Z, Rahman MU, Davis HM, Zhou H (2010) A latent variable approach for modeling categorical endpoints among patients with rheumatoid arthritis treated with golimumab plus methotrexate. J Pharmacokin Pharmacodyn 37:309–321CrossRefGoogle Scholar
  5. 5.
    Lacroix BD, Lovern MR, Stockis A, Sargentini-Maier ML, Karlsson MO, Friberg LE (2009) A pharmacodynamic Markov mixed-effects model for determining the effect of exposure to certolizumab pegol on the ACR20. Clin Pharmacol Ther 86:387–395PubMedCrossRefGoogle Scholar
  6. 6.
    Felson DT, Anderson JJ, Boers M, Bombardier C, Furst D, Goldsmith C, Katz LM, Lightfoot R Jr, Paulus H, Strand V, Tugwell P, Weinblatt M, Williams HJ, Wolfe F, Kieszak S (1995) ACR preliminary definition of improvement in rheumatoid arthritis. Arthritis Rheum 38:727–735PubMedCrossRefGoogle Scholar
  7. 7.
    Karlsson MO, Schoemaker RC, Kemp B, Cohen AF, van Gerven JM, Tuk B, Peck CC, Danhof M (2000) A pharmacodynamic Markov mixed-effects model for the effect of temazepam on sleep. Clin Pharmacol Ther 68:175–188PubMedCrossRefGoogle Scholar
  8. 8.
    Xu H, Craig BA (2009) A probit latent class model with general correlation structures for evaluating accuracy of diagnostics tests. Biometrics 65:1145–1155PubMedCrossRefGoogle Scholar
  9. 9.
    Beal SL (2001) Ways to fit a PK model with some data below the quantification limit. J Pharmacokin Pharmacodyn 28:481–504CrossRefGoogle Scholar
  10. 10.
    Ahn JE, Karlsson MO, Dunne A, Ludden TM (2008) Likelihood based approaches to handling data below the quantification limit using NONMEM VI. J Pharmacokinet Pharmacodyn 35:401–421PubMedCrossRefGoogle Scholar
  11. 11.
    Mendell NR, Elston RC (1974) Multifactorial qualitative traits: genetic analysis and prediction of recurrence risks. Biometrics 30:41–57PubMedCrossRefGoogle Scholar
  12. 12.
    Rice J, Reich T, Cloninger CR, Wette R (1979) An approximation to the multivariate normal integral: its application to multifactorial qualitative traits. Biometrics 35:451–459CrossRefGoogle Scholar
  13. 13.
    Smith C, Mendell NR (1974) Recurrence risks from family history and metric traits. Ann Hum Genet 37:275–286PubMedCrossRefGoogle Scholar
  14. 14.
    Cappellari L, Jenkins SP (2003) Multivariate probit regression using simulated maximum likelihood. Stata J 3:278–294Google Scholar
  15. 15.
    Joe H (1995) Approximations to multivariate normal rectangle probabilities based on conditional expectations. J Am Stat Assoc 90:957–964CrossRefGoogle Scholar
  16. 16.
    SAS Institute Inc (2002) SAS software: usage and reference, version 9. SAS Institute Inc, CaryGoogle Scholar
  17. 17.
    MathSoft Inc (2005) S-Plus for Windows. Data Analysis Division, MathSoft Inc, SeattleGoogle Scholar
  18. 18.
    Beal SL, Sheiner LB, Boeckmann AJ (eds) (1989–2006) NONMEM users guides. Icon Development Solutions, Ellicott CityGoogle Scholar
  19. 19.
    Drezner Z, Wesolowsky GO (1990) On the computation of the bivariate normal integral. J Stat Comput Simul 35:101–107CrossRefGoogle Scholar
  20. 20.
    Kuhn E, Lavielle M (2005) Maximum likelihood estimation in nonlinear mixed effects models. Comput Stat Data Anal 49:1020–1038CrossRefGoogle Scholar
  21. 21.
    Bock RD, Gibbons RD (1996) High-dimensional multivariate probit analysis. Biometrics 52:1183–1194PubMedCrossRefGoogle Scholar
  22. 22.
    Kremer M, Bloom BJ, Breedveld FC, Coombs J, Fletcher MP, Gruben D, Krishnaswami S, Burgos-Vargas R, Wilkinson B, Zerbini CAF, Zwillich SH (2006) The safety and efficacy of a JAK inhibitor in patients with active rheumatoid arthritis: results of a double-blind, placebo-controlled phase IIa trial of three dosage levels of CP-690, 550 versus placebo. Arthritis Rheum 60:1895–1905CrossRefGoogle Scholar
  23. 23.
    Ochi Y, Prentice RL (1984) Likelihood inference in a correlated probit regression model. Biometrika 71:531–543CrossRefGoogle Scholar
  24. 24.
    Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22CrossRefGoogle Scholar
  25. 25.
    Jacqmin-Gadda H, Proust-Lima C, Amiéva H (2010) Semi-parametric latent process model for longitudinal ordinal data: application to cognitive decline. Stat Med 29:2723–2731PubMedCrossRefGoogle Scholar
  26. 26.
    Genz A (1992) Numerical computation of multivariate normal probabilities. J Comput Graph Stat 1:141–149CrossRefGoogle Scholar
  27. 27.
    Yano I, Beal SL, Sheiner LB (2001) The need for mixed-effects modeling with population dichotomous data. J Pharmacokinet Pharmacodyn 28:389–412PubMedGoogle Scholar
  28. 28.
    Philipson PM, Ho WK, Henderson R (2008) Comparative review of methods for handling drop-out in longitudinal studies. Stat Med 27:6276–6298PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Ann Arbor Pharmacometrics Group (A2PG)Ann ArborUSA
  2. 2.Pharamcometrics, Pfizer, Inc.GrotonUSA

Personalised recommendations