## 4 Graphical Presentations of Parallelism  
Graphical presentations of parallelism can provide a visual representation of the accuracy and reliability of laboratory tests for biomarkers. Graphical presentations can help healthcare providers and researchers to rapidly (i.e., at a glance) identify potential issues with biomarker tests, such as interference from other substances in the sample or limitations with the analytical sensitivity of the test. They can also be useful for comparing the accuracy of different laboratory tests for the same biomarker. One common graphical presentation of parallelism is the parallelism plot, which involves plotting the signal produced by the laboratory test on the \(y\)-axis and the concentration of the biomarker on the \(x\)-axis. A seminal example from the literature was presented in Figure 1. If the lines are parallel, it suggests that the laboratory test accurately reflects the concentration of the biomarker in the sample and that parallelism is present. One development in the biomarker field, since the introduction of the logistic-log transformation [1], has been the use of calibrated and quality controlled protein standard curves. Consequently, reported biomarker concentrations are derived from the curve between the symmetry point (c as defined for Formula (3)), but never from the asymptotes. The lower asymptotes (d) indicate non-measurable data. This is either because the detection limit of the assay is insufficient or because there is nothing there to be measured. For the upper asymptote (a), the concentration of the biomarker is too high to be estimated reliably. Extrapolation is not permitted. Sample dilution is required. Taken together parallelism of a biomarker is therefore only determined for  
\[(x_{i}^{std},y_{i}^{std}),\in\{1,\ldots,n^{std}\},\text{where }y\
eq a\lor d \tag{7}\]  
Table 1 shows the data used for calculation of the graphical presentation of the curves in Figures 2 and 3. The first step of the data transformation used for the graphical presentation of the partial parallelism plots is to adjust the calculated concentration at each dilution step (\(i\)) as follows:  
\[z_{i}=x_{i}^{std}\times i \tag{8}\]  
This is followed by normalisation of each value of the transformed series to the value for the lowest dilution step (i.e., dilution 1:1, Table 2) as  
\[\bar{z}_{i}=\frac{z_{i}}{z_{1}} \tag{9}\]  
Figure 1: Comparison of logistic-log curves and their fully specified logit-log transformed counterparts. (**A**) Lines 1 and 2, logistic-log curves with identical slopes and asymptotes; Lines 3 and 4, logistic-log curves with identical slopes and different asymptotes; Line 5, logistic-log curve with different slope and asymptotes. (**B**) Corresponding straight lines formed by using the fully specified logit-log transformation. [Reproduced with permission from [1]]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dilution** & **Standard** & **Sample-A** & **Sample-B** \\ \hline
1:1 & 10 & 8 & 4 \\
1:2 & 5 & 4 & 6 \\
1:4 & 2.5 & 2 & 5 \\
1:8 & 1.25 & 1 & 4 \\
1:16 & 0.625 & 0.5 & 3 \\
1:32 & 0.3125 & 0.25 & 1.5 \\
1:64 & 0.15625 & 0.125 & 0.75 \\
1:128 & 0.078125 & 0.0625 & 0.375 \\
1:256 & 0.0390625 & 0.03125 & 0.1875 \\
1:512 & 0.01953125 & 0.015625 & 0.09375 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Raw data for the doubling dilution curves used for Figures 2 and 3. Figure 2: Conventional presentations of a doubling dilution curve for demonstration of parallelism between a standard and a sample. This graph illustrates how the concentration of a compound (\(y\)-axis) decreases with subsequent dilution steps (\(x\)-axis) either presented as a continuous variable on (**A**), a linear scale as used in [12], and (**B**) on the logarithmic scale derived from Formula (5) [1]. The standard curve (cross, dotted grey line) and dilution curves (Sample-A, open square, dashed grey line; Sample-B, open circle, black line).