On Bayesian Testing of Additive Conjoint Measurement Axioms Using Synthetic Likelihood

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.

[1]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[2]  ABootstrapLikelihoodapproachtoBayesianComputation,et al.  A Bootstrap Likelihood Approach to Bayesian Computation , 2015, 1510.07287.

[3]  David T. Frazier,et al.  Bayesian Synthetic Likelihood , 2017, 2305.05120.

[4]  Attitudes, order and quantity: deterministic and direct probabilistic tests of unidimensional unfolding. , 2007, Journal of applied measurement.

[5]  D. Scott.  Measurement structures and linear inequalities , 1964 .

[6]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[7]  J. Michell.  An introduction to the logic of psychological measurement , 1995 .

[8]  Howard Wainer,et al.  The Rasch Model as Additive Conjoint Measurement , 1979 .

[9]  R. McCulloch.  Local Model Influence , 1989 .

[10]  A. Kyngdon.  Plausible measurement analogies to some psychometric models of test performance. , 2011, The British journal of mathematical and statistical psychology.

[11]  Adrian F. M. Smith,et al.  Bayesian Analysis of Constrained Parameter and Truncated Data Problems , 1991 .

[12]  Ben Domingue,et al.  Evaluating the Equal-Interval Hypothesis with Test Score Scales , 2013, Psychometrika.

[13]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[14]  S. Wood.  Statistical inference for noisy nonlinear ecological dynamic systems , 2010, Nature.

[15]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[16]  Aurélien Garivier,et al.  On the Complexity of Best-Arm Identification in Multi-Armed Bandit Models , 2014, J. Mach. Learn. Res..

[17]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[18]  Zhiguo Li.  Some Problems in Statistical Inference Under Order Restrictions. , 2008 .

[19]  Peter B. Hoffman,et al.  Parole decision-making: A salient factor score , 1974 .

[20]  Fabrizio Leisen,et al.  A Bootstrap Likelihood Approach to Bayesian Computation , 2015, 1510.07287.

[21]  G. Karabatsos,et al.  The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. , 2001, Journal of applied measurement.

[22]  Joel Mitchell,et al.  Some problems in testing the double cancellation condition in conjoint measurement , 1988 .

[23]  R. Luce,et al.  Theory And Tests Of The Conjoint Commutativity Axiom For Additive Conjoint Measurement , 2011 .

[24]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[25]  Tim Robertson,et al.  An application of order restricted inference methodology to a problem in psychiatry , 1985 .

[26]  Michael U. Gutmann,et al.  Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models , 2015, J. Mach. Learn. Res..

[27]  George Karabatsos,et al.  Bayesian nonparametric model selection and model testing , 2006 .

[28]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[29]  R. Luce,et al.  Simultaneous conjoint measurement: A new type of fundamental measurement , 1964 .