## 4 Graphical Presentations of Parallelism
Overall, graphical presentations of parallelism are an important tool for evaluating the accuracy and reliability of laboratory tests for biomarkers, but there are important practical limitations to their interpretability. The next section will illustrate how this can be overcome in a standardised way which will simplify the interpretation of the graphical presentation. Figure 3: Graphical comparison of the partial logistic presentation on the \(x\)-axis only [1] in (**A**) switched for a categorical variable in (**B**). For any test, the standard (cross, dotted grey line) is used as the main comparator. In this example, the dilution curve for Sample-A (open square, dashed grey line) is parallel to the standard curve. There is a small offset on the \(y\)-axis between the standard and Sample-A because the starting concentration for Sample-B was less than for the standard. In contrast, the dilution curve for Sample-B (open circle, black line) is not parallel to the standard. For Sample-B, there is an increase in the concentration with the first dilution step. For dilution steps 1:4 to 1:16, the concentration in Sample-B reduces to a lesser degree than for the standard. After Dilution Step 1:32, there is parallelism between Sample-B and the Standard, but this is not clearly visible with this format of graphical presentation. Figure 4: **(A)** illustrates that the parallelism between the Standard and Sample-A is visually more intuitive compared to Figure 2. For this presentation, the value of the concentration was corrected by multiplication with the dilution. The offset on the _y_-axis between the Standard and Sample-A is explained by the difference in concentration. This can be a problem for the graphical representation if this difference is very large. Therefore, in example (**B**), all values were normalised to the baseline concentration. Now parallelism between the Standard and Sample-A is illustrated by the overlay of the horizontal lines at the _y_-axis value of one. The consequence of the absence of parallelism for Sample-B in the initial dilution steps leads to an overestimation of a factor of \(\approx\)12.5 once parallelism is achieved after a dilution of 1:32. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Di.** & **Std\({}^{1}\)** & **A\({}^{1}\)** & **B\({}^{1}\)** & **Std\({}^{2}\)** & **\(\overline{\textbf{A}^{2}}\)** & **\(\overline{\textbf{B}^{2}}\)** \\ \hline
1:1 & 10 & 8 & 4 & 1 & 1 & 1 \\
1:2 & 10 & 8 & 12 & 1 & 1 & 3 \\
1:4 & 10 & 8 & 20 & 1 & 1 & 5 \\
1:8 & 10 & 8 & 32 & 1 & 1 & 8 \\
1:16 & 10 & 8 & 48 & 1 & 1 & 12 \\
1:32 & 10 & 8 & 48 & 1 & 1 & 12 \\
1:64 & 10 & 8 & 48 & 1 & 1 & 12 \\
1:128 & 10 & 8 & 48 & 1 & 1 & 12 \\
1:256 & 10 & 8 & 48 & 1 & 1 & 12 \\
1:512 & 10 & 8 & 48 & 1 & 1 & 12 \\ \hline \hline \end{tabular}  
* Data of concentrations from Table 1 multiplied by dilution step from series (i.e., \(5\times 2=10\), \(2.5\times 4=10\), etc.) as summarised in Formula (8). \({}^{2}\) Data normalised to concentration at lowest dilution step of the series (i.e., \(\frac{10}{10}=1\), etc.) as summarised in Formula (9).  
\end{table} 
Table 2: Transformed data from Table 1 as needed to develop the partial parallelism plots shown in Figure 4. Abbreviations: Standard = Std, Sample-A = a, Sample-b = b. The horizontal bar above the abbreviation (e.g., \(\overline{\text{Std}}\)) indicates the normalised values.