# Thurstonian Scaling for Sensory Discrimination Methods
## 2 Materials and Methods
[PERSON] and [PERSON] [27] and [PERSON] and [PERSON] [40] provide a version of the DOD based on the ratings of the Same-Different method. The input of the R codes for all six forced-choice methods are \(x\) and \(n\), i.e., the number of correct responses (\(x\)) and sample size (\(n\)). The input of the R code for A-Not A are \(a\), \(an\), \(n\), and \(nn\), i.e., the number of response \"\(A\)\" for sample \(A\) (\(a\)), the sample size of sample \(A\) (\(an\)), the number of response \"\(A\)\" for sample \(Not\) \(A\) (\(n\)), and the sample size of sample \(Not\) \(A\) (\(nn\)). The input of the R code for Same-Different are \(sn\), \(n1\), \(dn\), and \(n2\), i.e., the number of response \"\(Same\)\" for matched sample pairs (AA or BB) (\(sn\)), the sample size of matched sample pairs (\(n1\)), the number of response \"\(Same\)\" for the unmatched sample pairs (AB or BA) (\(dn\)), and the sample size of the unmatched sample pairs (\(n2\)). The input of the code for the ratings A-Not A are _rfal_ and _rhit_, i.e., a vector of the frequencies of the ratings of sample \(Not\) \(A\) (_rfal_) and a vector of the frequencies of the ratings of sample \(A\) (_rhit_). The input of the R code for the ratings Same-Different are _rsam_ and _rdiff_, i.e., a vector of the frequencies of the ratings of matched sample pairs (AA or BB) and a vector of the frequencies of the rating of unmatched sample pairs (AB or BA). The output of the R codes for any one of all 10 methods are the estimated \(d^{\prime}\) and its variance. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Methods** & **Psychometric Functions** & **R-Codes** \\ \hline
1 & 2-AFC & \(P_{c}=\Phi(\frac{c}{\sqrt{2}})\) & TwoAFC(\(x\),n) \\
2 & 3-AFC & \(P_{c}=\int_{-\infty}^{\infty}\Phi^{2}(x)\Phi(x-\delta)dx\) & ThreeAFC(\(x\),n) \\
3 & Triangle & \(P_{c}=p/(3,1,\frac{1}{2}\delta^{2},\text{lower\,tail}=F)\) & TRI(\(x\),n) \\
4 & Duo-trio & \(P_{c}=\frac{1}{2}p/(3,1,\frac{1}{2}\delta^{2},\text{lower\,tail}=F)+p\{1/2 \}(\frac{1}{2},1,\frac{1}{2}\delta^{2},\text{lower\,tail}=F)\) & DUTR(\(x\),n) \\
5 & Specified Tetrad & \(P_{c}=2\int_{-\infty}^{\infty}\Phi(x)^{2}\Phi(x-\delta)dx\) & STETR(\(x\),n) \\
6 & Unspecified Tetrad & \(P_{c}=2\int_{-\infty}^{\infty}(\Phi(x)^{2}\Phi(\delta-x)\Phi(x-\delta)dx+2 \int_{-\infty}^{\infty}\Phi(x)^{2}\Phi(-\delta-x)\Phi(x+\delta)dx\) & UTETR(\(x\),n) \\
7 & A-Not A & \(H=P(``A^{\prime}A\rangle+\Phi(\delta-c);F=P(``A^{\prime}A^{\prime}/N)=\Phi(-c)\) & ANAAdv(a,an,nn,nn) \\
8 & Same-Different & \(H=P(``S\"/S)=2\Phi\left(c/\sqrt{2}\right)-1;F=P(``S\"/D)=\[PERSON]\left(c\frac{c}{ \sqrt{2}}\right)-\[PERSON](\frac{c\sqrt{2}}{c^{2}})\) & SDAdv(sn,n1,dn,n2) \\
9 & Ratings of A-Not A & \(H=P(``A^{\prime}A^{\prime}/N)=\[PERSON](-c);F=P(``A^{\prime}/N)=\Phi(-c),i=1,2, \ldots k-1\) & ANAART(rfal,nhit) \\
10 & Ratings of Same-Different & \(H=P(``S\"/S)=2\[PERSON]\left(c/\sqrt{2}\right)-1;F=P(``S\"/D)=\[PERSON]\left(\frac{c}{ \sqrt{2}}\right)-\Phi\left(\frac{c-c}{\sqrt{2}}\right),i=1,2,\ldots k-1\) & SDRAT(rsam,ndii) \\ \hline \hline \end{tabular} Note: For the psychometric functions for the methods 1–2, see [PERSON] [25]; for the methods 3–4, see [PERSON] [21]; for the methods 5–6, see [PERSON] [31]; for the methods 7–10, see [PERSON] et al. [35].  
\end{table} 
Table 1: Estimations of \(d^{\prime}\) and its variance for some basic sensory discrimination methods. Statistical Testing for Test Sample vs. Control Sample Based on the Individual \(d^{\prime}\) Estimator and Its Variance  
2.1 Difference Test for Test Sample vs. Control Sample Based on Individual \(d^{\prime}\) and Their Variance  
The difference test based on the individual \(d^{\prime}\) value and its variance is used to determine if there is a difference between one test sample and a control. The null hypothesis is \(H_{0}:\ \delta=0\). The alternative hypothesis is \(H_{a}:\ \delta>0\). The difference test statistic based on individual \(d^{\prime}\) value and its variance is Equation (1).