, Volume 83, Issue 2, pp 321–332 | Cite as

On Bayesian Testing of Additive Conjoint Measurement Axioms Using Synthetic Likelihood

  • George KarabatsosEmail author


This article introduces a Bayesian method for testing the axioms of additive conjoint measurement. The method is based on an importance sampling algorithm that performs likelihood-free, approximate Bayesian inference using a synthetic likelihood to overcome the analytical intractability of this testing problem. This new method improves upon previous methods because it provides an omnibus test of the entire hierarchy of cancellation axioms, beyond double cancellation. It does so while accounting for the posterior uncertainty that is inherent in the empirical orderings that are implied by these axioms, together. The new method is illustrated through a test of the cancellation axioms on a classic survey data set, and through the analysis of simulated data.


axiom testing conjoint measurement approximate Bayesian computation 



The author is grateful for the detailed comments and suggestions by two anonymous reviewers and the Editor. They have helped improve the presentation of this article. Funding was provided by National Science Foundation (Grant Nos. SES-0242030 and SES-1156372).

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  1. Bernardo, J. (1979). Reference posterior distributions for Bayesian inference. Journal of the Royal Statistical Society, Series B, 41, 113–147.Google Scholar
  2. Domingue, B. (2014). Evaluating the equal-interval hypothesis with test score scales. Psychometrika, 79, 1–19.CrossRefPubMedGoogle Scholar
  3. Gelfand, A., Smith, A., & Lee, T.-M. (1992). Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. Journal of the American Statistical Association, 87, 523–532.CrossRefGoogle Scholar
  4. Gutmann, M., & Corander, J. (2016). Bayesian optimization for likelihood-free inference of simulator-based statistical models. Journal of Machine Learning Research, 17, 1–47.Google Scholar
  5. Hoffman, P., & Beck, J. (1974). Parole decision-making: A salient factor score. Journal of Criminal Justice, 2, 195–206.CrossRefGoogle Scholar
  6. Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2, 389–423.PubMedGoogle Scholar
  7. Karabatsos, G. (2006). Bayesian nonparametric model selection and model testing. Journal of Mathematical Psychology, 50, 123–148.CrossRefGoogle Scholar
  8. Kruskal, W. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
  9. Kyngdon, A. (2011). Plausible measurement analogies to some psychometric models of test performance. British Journal of Mathematical and Statistical Psychology, 64, 478–497.CrossRefPubMedGoogle Scholar
  10. Kyngdon, A., & Richards, B. (2007). Attitudes, order and quantity: Deterministic and direct probabilistic tests of unidimensional unfolding. Journal of Applied Measurement, 8, 1–34.PubMedGoogle Scholar
  11. Li, Z. (2008). Some problems in statistical inference under order restrictions (Unpublished doctoral dissertation). University of Michigan.Google Scholar
  12. Liu, J. (2001). Monte Carlo Strategies in Scientific Computing. New York: Springer.Google Scholar
  13. Luce, R., & Steingrimsson, R. (2011). Theory and tests of the conjoint commutativity axiom for additive conjoint measurement. Journal of Mathematical Psychology, 55, 379–385.CrossRefGoogle Scholar
  14. Luce, R., & Tukey, J. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1, 1–27.CrossRefGoogle Scholar
  15. McCulloch, R. (1989). Local model influence. Journal of the American Statistical Association, 84, 473–478.CrossRefGoogle Scholar
  16. Michell, J. (1988). Some problems in testing the double cancellation condition in conjoint measurement. Journal of Mathematical Psychology, 32, 466–473.CrossRefGoogle Scholar
  17. Michell, J. (1990). An introduction to the logic of psychological measurement. New York: Psychology Press.Google Scholar
  18. Perline, R., Wright, B., & Wainer, H. (1979). The Rasch model as additive conjoint measurement. Applied Psychological Measurement, 3, 237–255.CrossRefGoogle Scholar
  19. Price, L. F., Drovandi, C. C., Lee, A., & Nott, D. J. (2017). Bayesian synthetic likelihood. Journal of Computational and Graphical Statistics. doi: 10.1080/10618600.2017.1302882.
  20. Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research: Danish Institute for Educational Research (Expanded edition, 1980. Chicago: University of Chicago Press).Google Scholar
  21. Robert, C., & Casella, G. (2004). Monte Carlo statistical methods (2nd ed.). Springer: New York.CrossRefGoogle Scholar
  22. Robertson, T., & Warrack, G. (1985). An application of order restricted inference methodology to a problem in psychiatry. Psychometrika, 50, 421–427.CrossRefGoogle Scholar
  23. Scott, D. (1964). Measurement models and linear inequalities. Journal of Mathematical Psychology, 1, 233–247.CrossRefGoogle Scholar
  24. Silverman, B. (1986). Density estimation for statistics and data analysis. Boca Raton, FL: Chapman and Hall.CrossRefGoogle Scholar
  25. Wood, S. (2010). Statistical inference for noisy nonlinear ecological dynamic systems. Nature, 466, 1102–1104.CrossRefPubMedGoogle Scholar
  26. Zhu, W., Marin, J., & Leisen, F. (2016). A bootstrap likelihood approach to Bayesian computation. Australian and New Zealand Journal of Statistics, 58, 227–244.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2017

Authors and Affiliations

  1. 1.University of Illinois-ChicagoChicagoUSA

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