Estimating and Testing Parameters of the Thurstonian Model for Torgerson's Method of Triads

Abstract

Torgerson's complete method of triads is a famous and useful method for comparison of multiple samples. However, this method has not been used widely in the sensory field. One possible reason is that the Thurstonian model of the method is complicated and there is no published tables and available computer software for applications of the method. This paper discusses a Thurstonian model for a special case of the method with one control sample and two test samples. A table for Thurstonian discriminal distances and proportions of responses is produced. A set of R/S-Plus codes was developed and provided for applications of the method including estimations of parameters and covariance matrix, difference and similarity tests. A case study is provided for illustration of applications of the methods. The main novel contributions of this paper to the method are (1) using a computer-intensive approach to produce a simulation-based psychological function for the special case of the method and (2) using a delta method to derive a covariance matrix of the estimators of the two Thurstonian discriminal distances. Practical Applications In the situation of one control sample and two test samples, investigators are often interested in measuring perceptive distances between the control sample and test samples, and in testing if the samples are significantly different or similar. The Thurstonian model of Torgerson's method of triads can be used for the measurement. The table and R/S-Plus codes developed and provided in the paper can be used for the following objectives: (1) Estimating proportions of responses that one sample is more similar to a sample than to another sample; (2) Estimating and and the covariance matrix of the estimators for given proportions of responses; (3) Difference and similarity testing for the three samples; (4) Testing powers of the statistical tests and sample sizes; (5) Simulation experiments for the Thurstonian model of the special case of Torgerson's method and establishment of an empirical psychometric function for the model.