# Thurstonian Scaling for Sensory Discrimination Methods
## 3 Results
#### 3.3.1 Difference Test for Multiple Test Samples Based on Multiple d' Values and Their Variances  
The R code 'dstest(d,v)', based on Equation (3), can be used to conduct a difference test for multiple \(\delta\) based on multiple \(d^{\prime}\) and their variances.  
For example, for a difference test for four test samples based on the four \(d^{\prime}\) values and their variances in Table 2 ('dpdat'), the test results are as below, using the R code 'dstest(d,v)'. A significant difference among the four test samples was found at an alpha level of 0.05 (\(p\)-value = 0.0425).  
\(>\)dstest(d = dpdat[,1],v = dpdat[,2])  
\(p\)-value: 0.0425  
Weighted mean: 2.0588  
Variance of Wm: 0.0261  
[1] 0.0425 2.0588 0.0261  
3.2 Multiple Comparisons for Multiple Test Samples Based on the \(d^{\prime}\) Vector and Co-Variance Matrix  
The \(d^{\prime}\) vector 'dp' and co-variance matrix 'dv' can be produced using the data 'dpdat' in Table 2 as below.  
\begin{table}
\begin{tabular}{c c c} \hline
**No.** & \(d^{\prime}\) & **Variance of \(d^{\prime}\)** \\ \hline
1 & 2.4868 & 0.0687 \\
2 & 2.0849 & 0.0662 \\
3 & 1.4422 & 0.1628 \\
4 & 0.9442 & 0.3927 \\ \hline \end{tabular}
\end{table}
Table 2: Data of estimated \(d^{\prime}\) and its variance.  
3.3.3. TOST Equivalence/Similarity Test Based on Two \(d^{\prime}\) Values and Their Variances  
The R code's2 dptest(d,v,d0)' based on Equations (4a) and (4b) can be used for the TOST equivalence/similarity test for two test samples based on two estimators, e.g., \(d^{\prime}{}_{1}\) and \(d^{\prime}{}_{2}\) for test samples T1 and T2 and their variances. The input of the code is the two \(d^{\prime}\) estimators and their variances, as well as an equivalence/similarity limit \(\Delta_{0}\). The output of the code are the test statistics Z1 and Z2 and the \(p\)-values. For example, for the data d = c(2.4868,2.0849), v = c(0.0687,0.0662), and an equivalence/similarity limit d0 = 1, the output is as below. Significant equivalence/similarity of T1 and T2 can be concluded with an equivalence/similarity limit of 1 at a significance level of 0.1 with \(p\)-values of 0.0001 and 0.0517.  
\(>\)s2 dptest(d = c(2.4868,2.0849),v = c(0.0687,0.0662),d0 = 1)  
Z1,Z2,pv1 and pv2:  
[1] 3.8169 -1.6284 0.0001 0.0517  
The R program 'TOSTmeta' in the R package 'TOSTER' (Lakens and Caldwell [64]) can also be used for the equivalence/similarity test for two test samples using the two estimators and their variances, as well as an equivalence/similarity limit \(\Delta_{0}\). For example, for the data above and an equivalence/similarity limit d0 = 1, the same results can be obtained as below. Significant equivalence/similarity of T1 and T2 can be concluded with an equivalence/similarity limit of 1 at a significance level of 0.1 with \(p\)-values of 0.0001 and 0.052.  
\(>\)library(TOSTER)  
\(>\) TOSTmeta(ES = 2.4868-2.0849, var = 0.0687 + 0.0662, low_eqbound_d = -1, high_eqbound_d = 1, alpha = 0.1)  
TOST results:  
Z-value lower bound: 3.82  
\(p\)-value lower bound: 0.00007  
Z-value upper bound: -1.63  
\(p\)-value upper bound: 0.052  
....  
Equivalence Test Result:  
The equivalence test was significant, Z = -1.628, \(p\) = 0.0517, given equivalence bounds of -1.000 and 1.000 and an alpha of 0.1.  
Figure 1: Multiple comparisons based on \(d^{\prime}\) vector and co-variance matrix.