A Thurstonian Model and Statistical Inference for the 2-Alternative Choice Test with Both Test Pairs and Placebo Pairs

Abstract

For the paired difference or preference test applied to consumer research, the 2-Alternative Choice (2-AC) method that allows a “no difference” or “no preference” option is more suitable. For this method, using both test and placebo pairs is useful for estimating the degree of response bias among the consumers using the test and accurately estimating the degree of difference or preference between the test pairs. This paper proposes a Thurstonian model for the 2-AC for both test pairs and placebo pairs in a test method. Maximum likelihood estimations are used for the parameters: d′, a perceived distance of difference or preference, and τ, a criterion and a decision parameter. Three methods are used for estimations of the covariance matrix for the parameter estimators: (1) using the built-in functions in the S-Plus and R packages, which are based on maximum likelihood and observed Fisher information; (2) the delta method, which is based on the Taylor series approximation; and (3) the bootstrap method, which is computer-intensive with resampling. Statistical tests are discussed for d′ and vectors of responses for test pairs and placebo pairs in both a monadic design and a paired design. R/S-Plus codes are developed and provided for the calculations. Practical Applications This paper proposes a Thurstonian model for the 2-Alternative Choice (2-AC) difference test or preference test method with both test pairs and placebo pairs. This test method provides more information than the 2-AC method without an embedded placebo pair control. Applying this model, especially for the paired preference test method with a “no-preference” option can be accurately and readily used for consumer research to achieve various business objectives such as reformulation and claim substantiation. Using R/S-Plus and built-in functions and codes developed and provided in the paper, maximum likelihood estimation of parameters in the model, covariance matrix of the parameter estimators and statistical tests can be obtained and conducted easily and quickly.