## 5 Partial Parallelism Plots  
In a partial parallelism plot, the laboratory test results are plotted against each other on the \(x\)- and \(y\)-axes. A line of unity is then added to the plot to represent perfect parallel agreement between the measurements. The slope of the line, normalised to the first dilution step of the standard curve, is zero with an intercept of one. Therefore, a horizontal reference line at once permits the comparison of the slope of the samples to visually analyse the degree of parallelism for the biomarker in question. If the line of unity (horizontal reference line) is significantly different from the slope of the line of best fit for the sample, it suggests that partial parallelism is absent. One advantage of partial parallelism plots is that they can be used to assess parallelism between samples over a limited, thought to represent the clinically useful, range of concentrations. This can provide a more practical evaluation of parallelism as relevant for routine healthcare practice. As a first step towards this goal, Figures 3 and 4 introduce the graphical representation of the line of unity. The result of normalisation for subsequent dilution steps (Formula (7)) is shown graphically for the data from Table 1. In this presentation, there is a similar graphical pattern for the plots in Figure 3A,B. The difference between the two plots can be seen on the \(x\)-axis. Note that the \(x\)-axis in Figure 3A is log based. Whilst mathematically correct, this presentation does not make for an easy laboratory, clinical, or health authority-tuned assessment of the biomarker concentration. A much more common notation is the dilution step as used on the categorical scaled \(x\)-axis in Figure 3B. A limitation of both graphical presentations is that it cannot readily be seen that parallelism between Sample-B and the line of unity is only achieved after a dilution step of 1:32. The graphical presentation can be improved to better visualise when parallelism is achieved. Figure 4A gives a graphical representation of the same two plots as in Figure 3, adjusted for the dilution steps. For generalisation, the intercept is normalised to one at the baseline in Figure 4B. This graphical presentation is the basis for the development of partial parallelism plots. The term partial parallelism plot shall be defined as a defined range of biomarker concentrations for which parallelism between sample and standard can be demonstrated. In laboratory practice, parallelism may only be achieved after a certain dilution step because of, for example, a matrix effect (Table 3). Figure 5A illustrates a theoretical situation with a small matrix effect which persists up to a dilution of 1:4 (see vertical reference line). The graphical presentation for a mildly stronger matrix effect persisting up to a dilution of 1:8 is shown in Figure 5B. Importantly, lack of parallelism can also be present at later stages of the dilution curve (Table 4), for example, because the concentration of the biomarker is below the detection limit of the assay (i.e., \(d\) in Formula (3)). Figure 6A shows the graphical presentation for lack of parallelism after a dilution step of 1:128. It would be physically impossible to see the developing lack of parallelism with the curves presented in Figure 3. Finally, Figure 6B illustrates the presence of partial parallelism between a dilution of 1:8 to 1:128. At lower or higher dilution steps there is non-parallelism. Again, this pattern cannot be visually extracted from the graphical presentation in Figure 3.  
Figure 5: These two examples show that parallelism is achieved after (**A**) a dilution of 1:4 and (**B**) a dilution of 1:8. The data are from Table 3.  
Figure 6: These two examples shows that parallelism is lost after (**A**) a dilution of 1:128. Finally, (**B**) illustrates that parallelism was only achieved between a dilution of 1:8 to 1:128. Data are from Table 4.