## 3 The Definition of Parallelism and Partial Parallelism  
The term parallelism, in its simplest definition, describes the relationship between the concentration of an analyte (such as a biomarker) in a sample and the signal produced by the reference standard of the laboratory test used to measure that analyte as earlier introduced [1; 2]. When the relationship between concentration and signal is linear, parallelism is said to be present. Hence, the other term used in the literature for parallelism is linearity. This is important because it means that the laboratory test accurately reflects the concentration of the analyte in the sample and, therefore, provides a reliable measurement of the biomarker. However, if there is non-parallelism (i.e., a non-linear relationship between concentration and signal), the accuracy of the laboratory test may be compromised. This can occur if there is interference from other substances in the sample, or if the laboratory test is not able to accurately detect the analyte, for example, at higher concentrations. This is a frequent problem with biomarker assays requiring use of a non-linear standard curve, as reviewed theoretically in [2] and demonstrated experimentally in [5]. In order to test for linearity, a regression analysis is performed to determine the slope of the line of best fit. The formula for the slope of the line is:  
\[slope=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})\times(y_{i}-\bar{y})}{\sum \limits_{i=1}^{n}(x_{i}-\bar{x})^{2}} \tag{1}\]  
where  
\(x_{i}\) is the concentration of the biomarker in the sample;  
\(y_{i}\) is the signal produced by the test used to measure the biomarker;  
\(n\) is the number of data points;  
\(\bar{x}\) is the mean concentration of the biomarker;  
\(\bar{y}\) is the mean signal produced by the laboratory test. If the slope is not significantly different from 1 (i.e., if \(|slope-1|\leq SE\) where \(SE\) is the standard error), then parallelism is present. The values of \(x_{i}\) are the given concentrations (i.e., ng/mL, pg/mL, g/L), and formula (1) uses those values to calculate the slope of the line of best fit, which represents the relationship between the concentration of the analyteand the signal produced by the laboratory test, as detailed in an entire book chapter [9]. It follows that for biomarker assays with proof of linearity, a parallelism coefficient close to 1 indicates that the patient sample and standard curve have similar slopes:  
\[\text{Parallelism coefficient}=\frac{\text{Slope of Patient Dilution}}{\text{Slope of Standard Curve}} \tag{2}\]  
In absence of linearity, determination of parallelism was defined for bioassay dilution curves in absence of a standard curve by a logistic-log model in which the signal for the test are optic densities (OD) as:  
\[OD=d+\frac{a-d}{1+(\frac{dilution}{c})^{b}} \tag{3}\]  
where  
\(a\) is the upper asymptote of the curve of the OD at a theoretical infinite concentrations;  
\(d\) is the lower asymptote of the curve of the OD at a theoretical zero concentrations;  
\(b\) is a curvature parameter;  
\(c\) is the symmetry point of the sigmoid. Linear transformation of the curve is achieved through a logistic function where \(OD_{min}\) and \(OD_{max}\) correspond to bespoke upper and lower asymptotes as  
\[Logit(OD)_{fs}=\log\left(\frac{OD-OD_{min}}{OD_{max}-OD}\right) \tag{4}\]  
The formula can be reduced to express a partially specified logit model introduced in [1]:  
\[Logit(OD)_{ps}=\log\left(\frac{OD}{OD_{max}-OD}\right) \tag{5}\]  
The visualisation of this approach is illustrated in Figure 1. Clearly, none of these curves in Figure 1A meet the criteria for linearity as defined at the onset of this section. Only bespoke logistic-log transformation permits to demonstrate parallelism (Curves 1-4 in Figure 1B) and lack of parallelism (Curve 5 in Figure 1B). The logistic-log transformation is the basis for the statistical analysis of a dilution series intended to facilitate visualisation as intended [1].