# Thurstonian Scaling for Sensory Discrimination Methods
## 2 Materials and Methods  
Estimating \(d^{\prime}\) Values and Their Variances from Psychometric Functions of Some Basic Sensory Discrimination Methods  
Thurstonian scaling, i.e., Thurstonian discriminant distance \(\delta\) or \(d^{\prime}\) can be derived from the Thurstonian models, i.e., psychometric functions of sensory discrimination methods. Any sensory discrimination method should have its unique psychometric function. The psychometric functions for the forced-choice methods describe the relationship between \(\delta\) and the probability of correct response, \(P_{c}\). The psychometric functions for the methods with response bias describe the relationship between \(\delta\) and probabilities of hits and false alarms. The probability of hits is the probability of response \"A\" for sample A in the A-Not A method or the probability of response \"same\" for the matched sample pair in the Same-Different method. The probability of false alarms is the probability of response \"A\" for sample Not A in the A-Not A method or the probability of response \"same\" for the unmatched sample pair in the Same-Different method. Thurstonian models, i.e., psychometric functions, can be derived theoretically based on different decision rules and/or be produced by simulations. [PERSON] [39] developed a simulation-derived psychometric function for the \"Two-out-of-Five\" method. [PERSON] et al. [36] developed the simulation-derived psychometric functions for various forced-choice sensory discrimination methods, i.e., \"M+N\" methods. Estimation of variances of \(d^{\prime}\) estimators is another important topic for Thurstonian modeling. With the estimated \(d^{\prime}\) and its variance, statistical testing for \(\delta\) or \(d^{\prime}\) is possible. [PERSON] et al. [28] was probably the first in the sensory literature to discuss the topic. The \(B\)-values in ASTM-E2262-03 [7] for the forced-choice methods are adapted from [PERSON] et al. [28]. [PERSON] [30] originally derived the variance of \(d^{\prime}\) for the Same-Different method. [PERSON] and [PERSON] [32] also originally derived the variances of \(d^{\prime}\) for the specified and unspecified tetrads. [PERSON] [29] (Section 3.4) describes three general approaches for estimation of variance of \(d^{\prime}\). They are as follows: (1). the bootstrap method; (2). the delta method; and (3). using some S-Plus (Insightful 2001 [51]) and R programs [50] and R packages, e.g., the R package \"numDeriv\" ([PERSON] and [PERSON] [52]), based on the Hessian of the log-likelihood function evaluated at the maximum-likelihood estimation. Table 1 lists ten (10) basic sensory discrimination methods, their psychometric functions, and corresponding R codes. The methods include three (3) specified forced-choice methods, i.e., the 2-AFC, the 3-AFC, and the specified tetrad; three (3) unspecified forced-choice methods, i.e., the triangle, the duo-trio, and the unspecified tetrad; four (4) methods with response bias, i.e., the A-Not A, the Same-Different, the ratings of A-Not A, and the ratings of Same-Different. The last two methods can be regarded as two versions of the degree of difference (DOD) method. ASTM-E2262-03 [7] provides the \(d^{\prime}\) values and corresponding \(B\)-values (note: variance = \(B/n\), where \(n\) is the sample size) for four forced-choice methods (i.e., the triangle, the duo-trio, the 3-AFC, and the 2-AFC) and two methods with response bias (i.e., the A-Not A and the Same-Different methods). [PERSON] et al. [53] provide the \(d^{\prime}\) values for the specified and unspecified tetrads. [PERSON] and [PERSON] [32] provide variances of \(d^{\prime}\) for the specified and unspecified tetrads. [PERSON] et al. [35] provide the \(d^{\prime}\) values and their variances for the ratings of the A-Not A, the ratings of A-Not AR, and the ratings of the Same-Different methods and regard them as three versions of the degree of difference (DOD) method.