Journal of Statistical Theory and Practice

, Volume 11, Issue 2, pp 269–295 | Cite as

Approximations of the information matrix for a panel mixed logit model

  • Wei ZhangEmail author
  • Abhyuday Mandal
  • John Stufken


Information matrices play a key role in identifying optimal designs. Panel mixed logit models are more flexible than multinomial logit models for discrete choice experiments. For panel mixed logit models, the information matrix does not have a closed-form expression and is difficult to evaluate. We propose three methods to approximate the information matrix, namely, importance sampling, Laplace approximation, and joint sampling. The three methods are compared through simulations. Since our ultimate goal is to find optimal designs, the three methods are compared on whether they rank designs similarly, not on how accurate the approximations are. Although the Laplace approximation is not as accurate as the other two methods, it can still be used to rank designs accurately and it is much faster than the other two methods. When an optimal design search using an exchange algorithm takes days to run, the Laplace approximation may be the only viable choice to use in practice.


Discrete choice experiments optimal designs Laplace’s method importance sampling joint sampling A-optimality D-optimality 

AMS Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arora, N., and J. Huber. 2001. Improving parameter estimates and model prediction by aggregate customization in choice experiments. Journal of Consumer Research 28 (2):273–83.CrossRefGoogle Scholar
  2. Atkinson, A. C., A. N. Donev, and R. D. Tobias. 2007. Optimum experimental designs, with SAS. New York, NY: Oxford University Press.zbMATHGoogle Scholar
  3. Bhat, C. 1998. Accommodating variations in responsiveness to level-of-service variables in travel mode choice models. Transportation Research A 32:455–507.CrossRefGoogle Scholar
  4. Bhat, C. 2000. Incorporating observed and unobserved heterogeneity in urban work mode choice modeling. Transportation Science 34:228–38.CrossRefGoogle Scholar
  5. Bliemer, M. C., and J. M. Rose. 2010. Construction of experimental designs for mixed logit models allowing for correlation across choice observations. Transportation Research Part B: Methodological 44 (6):720–34.CrossRefGoogle Scholar
  6. Booth, J. G., and J. P. Hobert. 1999. Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society, Series B, Statistical Methodology 61 (1):265–85.CrossRefGoogle Scholar
  7. Breslow, N. E., and D. G. Clayton. 1993. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88 (421):9–25.zbMATHGoogle Scholar
  8. Brownstone, D., and K. Train. 1999. Forecasting new product penetration with flexible substitution patterns. Journal of Econometrics 89:109–29.CrossRefGoogle Scholar
  9. Chernoff, H. 1953. Locally optimal designs for estimating parameters. Annals of Mathematical Statistics 24 (4):586–602.MathSciNetCrossRefGoogle Scholar
  10. Erdem, T. 1996. A dynamic analysis of market structure based on panel data. Marketing Science 15:359–78.CrossRefGoogle Scholar
  11. Hensher, D. A., J. M. Rose, and W. H. Greene. 2005. Applied choice analysis: A primer. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  12. McCulloch, C. E. 1997. Maximum likelihood algorithms for generalized linear mixed models. Journal of the American Statistical Association 92 (437):162–70.MathSciNetCrossRefGoogle Scholar
  13. McFadden, D. 1974. Conditional logit analysis of qualitative choice behavior. In Frontiers in econometrics, ed. P. Zarembka, 105–42. New York, NY: Academic Press.Google Scholar
  14. McFadden, D., and K. Train. 2000. Mixed MNL models of discrete response. Journal of Applied Econometrics 15:447–70.CrossRefGoogle Scholar
  15. Moerbeek, M., and C. J. Maas. 2005. Optimal experimental designs for multilevel logistic models with two binary predictors. Communications in Statistics—Theory and Methods 34 (5):1151–67.MathSciNetCrossRefGoogle Scholar
  16. Revelt, D., and K. Train. 1998. Mixed logit with repeated choices: households’ choices of appliance efficiency level. Review of Economics and Statistics 80 (4):647–57.CrossRefGoogle Scholar
  17. Rossi, P. E., G. M. Allenby, and R. McCulloch. 2006. Bayesian statistics and marketing. London, UK: John Wiley and Sons, Ltd.zbMATHGoogle Scholar
  18. Sándor, Z., and M. Wedel. 2002. Profile construction in experimental choice designs for mixed logit models. Marketing Science 21 (4):455–75.CrossRefGoogle Scholar
  19. Tierney, L., and J. B. Kadane. 1986. Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association 81 (393):82–86.MathSciNetCrossRefGoogle Scholar
  20. Tierney, L., R. E. Kass, and J. B. Kadane. 1989. Fully exponential Laplace approximations to expectations and variances of nonpositive functions. Journal of the American Statistical Association 84 (407):710–16.MathSciNetCrossRefGoogle Scholar
  21. Toubia, O., J. R. Hauser, and D. I. Simester. 2004. Polyhedral methods for adaptive choice-based conjoint analysis. Journal of Marketing Research 41 (1):116–31.CrossRefGoogle Scholar
  22. Train, K. E. 2009. Discrete choice methods with simulation. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  23. Waite, T. W., and D. C. Woods. 2014. Designs for generalized linear models with random block effects via information matrix approximations. Southampton, UK: Southampton Statistical Sciences Research Institute. Southampton Statistical Sciences Research Institute Methodology Working Papers, M12/01.zbMATHGoogle Scholar
  24. Wand, M. P. 2007. Fisher information for generalized linear mixed models. Journal of Multivariate Analysis 98 (7):1412–16.MathSciNetCrossRefGoogle Scholar
  25. Yu, J., P. Goos, and M. Vandebroek. 2011. Individually adapted sequential Bayesian conjoint-choice designs in the presence of consumer heterogeneity. International Journal of Research in Marketing 28 (4):378–88.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of Georgia AthensGeorgiaUSA
  2. 2.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

Personalised recommendations