Random Effects Ordinal Time Models for Grouped Toxicological Data from a Biological Control Assay

  • Marie-José Martinez
  • John P. HindeEmail author
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


Discrete survival times can be viewed as ordered multicategorical data. Here a continuation-ratio model is considered, which is particularly appropriate when the ordered categories represent a progression through different stages, such as survival through various times. This particular model has the advantage of being a simple decomposition of a multinomial distribution into a succession of hierarchical binomial models. In a clustered data context, random effects are incorporated into the linear predictor of the model to account for uncontrolled experimental variation. Assuming a normal distribution for the random effects, an EM algorithm with adaptive Gaussian quadrature is used to estimate the model parameters. This approach is applied to the analysis of grouped toxicological data obtained from a biological control assay. In this assay, different isolates of the fungus Beauveria bassiana were used as a microbial control for the Heterotermes tenuis termite, which causes considerable damage to sugarcane fields in Brazil. The aim is to study the pathogenicity and the virulence of the fungus in order to determine effective isolates for the control of this pest population.


Adaptive quadrature Clustered data Multicategorical data Ordinal regression Random effects 



This work was part-funded under Science Foundation Ireland’s Research Frontiers Programme (07/RFP-MATF448) and Science Foundation Ireland’s BIO-SI project, grant number 07/MI/012.


  1. Agresti, A. (2002). Categorical data analysis (2nd ed.). New York: Wiley.CrossRefzbMATHGoogle Scholar
  2. Albert, P., & Follmann, D. (2000). Modeling repeated count data subject to informative dropout. Biometrics, 56, 667–677.CrossRefzbMATHGoogle Scholar
  3. Anderson, D., & Hinde, J. (1988). Random effects in generalized linear models and the EM algorithm. Communications in Statistics—Theory and Methods, 17, 3847–3856.Google Scholar
  4. Aitkin, M. (1996). Empirical Bayes shrinkage using posterior random effect means from nonparametric maximum likelihood estimation in general random effect models. In Proceedings of the 11th International Workshop on Statistical Modelling (pp. 87–94). Orvieto: Statistical Modelling Society.Google Scholar
  5. Breslow, N., & Clayton, D. (1993). Approximate inference in generalized linear mixed model. Journal of the American Statistical Association, 88, 9–25.zbMATHGoogle Scholar
  6. Brillinger, D., & Preisler, M. (1983). Maximum likelihood estimation in a latent variable problem. In S. Karlin, T. Amemiya, & L. Goodman (Eds.), Studies in econometrics, time series and multivariate statistics (pp. 31–35). New York: Academic.Google Scholar
  7. Crouch, E., & Spiegelman, D. (1990). The evaluation of integrals of the form ∫ f(t)exp(−t 2)dt: Application to logistic-normal models. Journal of the American Statistical Association, 85, 464–469.zbMATHMathSciNetGoogle Scholar
  8. De Freitas, S. (2001). Modelos para Proporções com Superdispersão proveniente de Ensaios Toxicológicos no Tempo (Ph.D. thesis). ESALQ, Universidade de São Paulo.Google Scholar
  9. Hinde, J. (1982). Compound Poisson regression models. In R. Gilchrist (Ed.), GLIM 82. New York: Springer.Google Scholar
  10. Hinde, J. (1997). Contribution to discussion of The EM algorithm - An old folk-song to a fast new tune by Meng, X-L. and Van Dyk, D. Journal of the Royal Statistical Society, Series B, 59, 549–550.Google Scholar
  11. Jansen, J. (1990). On the statistical analysis of ordinal data when extravariation is present. Applied Statistics, 39, 74–85.CrossRefGoogle Scholar
  12. Lesaffre, E., & Spiessens, B. (2001). On the effect of the number of quadrature points in a logistic random-effects model: An example. Applied Statistics, 50, 325–335.zbMATHMathSciNetGoogle Scholar
  13. Liu, Q., & Pierce, D. (1994). A note on Gauss-Hermite quadrature. Biometrika, 81, 624–629.zbMATHMathSciNetGoogle Scholar
  14. McCullagh, P., & Nelder, J. (1989). Generalized linear models. London: Chapman & Hall.Google Scholar
  15. Molenberghs, G., & Verbeke, G. (2005). Models for discrete longitudinal data. New York: Springer.zbMATHGoogle Scholar
  16. Naylor, J., & Smith, A. (1988). Econometric illustrations of novel numerical integration strategies for Bayesian inference. Journal of Econometrics, 38, 103–125.CrossRefMathSciNetGoogle Scholar
  17. Nelder, J. (1985). Quasi-likelihood and GLIM. In R. Gilchrist, B. Francis, & J. Whittaker (Eds.), Generalized linear models: Proceedings of the GLIM 85 conference. New York: Springer.Google Scholar
  18. Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2002). Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal, 2, 1–21.Google Scholar
  19. Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128, 301–323.CrossRefMathSciNetGoogle Scholar
  20. Stram, D., Wei, L., & Ware, J. (1988). Analysis of repeated categorical outcomes with possibly missing observations and time-dependent covariates. Journal of the American Statistical Association, 83, 631–637.CrossRefGoogle Scholar
  21. Ten Have, T., & Uttal, D. (1994). Subject-specific and population-averaged continuation ratio logit models for multiple discrete time survival profiles. Applied Statistics, 43, 371–384.CrossRefzbMATHGoogle Scholar
  22. Tutz, G., & Hennevogl, W. (1996). Random effects in ordinal regression models. Computational Statistics and Data Analysis, 22, 537–557.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Team MISTIS, INRIA Rhône-Alpes & Laboratoire Jean KuntzmannSaint-Ismier CedexFrance
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of Ireland, GalwayGalwayIreland

Personalised recommendations