# Thurstonian Scaling for Sensory Discrimination Methods
## 5 Conclusions  
Thurstonian scaling, i.e., Thurstonian discriminant distance (\(\delta\) or \(d^{\prime}\)) is an advanced technique to measure sensory difference/similarity between two samples. The index \(d^{\prime}\) can be produced from a Thurstonian model, i.e., a psychometric function of any one of the sensory discrimination methods. There are some published papers on Thurstonian scaling in the sensory literature; however, there are no papers on general statistical inference for \(d^{\prime}\) data. This paper briefly and comprehensively explores how to estimate and test Thurstonian discrimination distances (\(d^{\prime}\) values) and generally discusses the application and significance of Thurstonian scaling for sensory discrimination methods. R codes are developed and provided to calculate \(d^{\prime}\) values and their variances from 10 basic sensory discrimination methods. The 10 basic sensory discrimination methods include 6 force-choice methods (2-AFC, 3-AFC, triangle, duo-trio, specified tetrad, and unspecified tetrad) and 4 methods with response bias (A-Not A, Same-Different, ratings of A-Not A, and ratings of Same-Different). With the estimated \(d^{\prime}\) values and their variances, differencetesting and equivalence/similarity testing for individual \(d^{\prime}\) value, or two or multiple \(d^{\prime}\) values, as well as multiple comparisons for multiple \(d^{\prime}\) values, can be conducted by using R codes, the S-Plus program, and R packages as shown in the paper. We believe that the technique of Thurstonian scaling provided and discussed in the paper can be used as a useful tool in several scientific research fields including the area of detecting and monitoring food contamination. As mentioned earlier, advantages include (1) improved sensitivity, (2) quantitative interpretation, (3) statistical robustness and reliability, (4) clearer decision-making across methods based on a probabilistic framework, and (5) compatibility across methods using a common parameter \(d^{\prime}\). Understanding Thurstonian scaling assumptions and requirements is necessary to apply it successfully. Disadvantages in adopting Thurstonian research might be complexity and accessibility, normality assumptions, and requirement for large, high-quality datasets. Future efforts, improvements and advancements in using Thurstonian scaling for monitoring foods might include the following: (1) integration with digital technologies, nanotechnologies, IOT, AI, ML, multimodal data, and other types of data (sensor data, analytical, microbial, etc.) to provide deeper insights, (2) incorporation of non-linear relationships to handle diverse data, (3) Bayesian extensions to allow for better incorporation of prior information, (4) relaxing distributional assumptions to make Thurstonian models more robust to real-world data, (5) Thurstonian models to analyze data over time or in dynamic contexts (i.e., shelf-life prediction models), and (6) creation of software tools to lower barriers for researchers to apply the models, thus making Thurstonian modeling more versatile, accessible, and capable of addressing modern challenges for food safety. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15020991/s1](https://www.mdpi.com/article/10.3390/app15020991/s1)]([https://www.mdpi.com/article/10.3390/app15020991/s1](https://www.mdpi.com/article/10.3390/app15020991/s1)). The R codes used in this paper and some statistical output can be found in the Supplementary Materials in the online version of this paper. Conceptualization, J.B. and C.K. ; software, J.B.; formal analysis, J.B.; writing--original draft preparation, J.B. and C.K. ; writing--review and editing, J.B. and C.K. ; visualization, J.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article/Supplementary Materials, further inquiries can be directed to the corresponding author.