Non-linear Transformation of Enzyme-linked Immunosorbent Assay (Elisa)
In research questions such as resistance selection against beet necrotic yellow vein virus, it is of interest to compare the virus concentrations of samples from different groups. The enzyme immunoassay (ELISA) is the standard tool for measuring virus concentrations. Simple data analysis methods such as analysis of variance (ANOVA), however, are impaired due to non-normality of the resulting optical density (OD) values as well as unequal variances in different groups.
To understand the relationship between OD values from an ELISA assay and virus concentration per sample, we used a large serial dilution and modeled its nonlinear shape using a five-parameter logistic regression model. . Additionally, we investigated whether the quality of the model can be increased if one or more of the model parameters are defined beforehand. Subsequently, we used the inverse of the best model to estimate the virus concentration for each measured OD value.
We show that the transformed data are essentially normally distributed but provide unequal variances by group. Thus, we propose a generalized least squares model that allows unequal variances of the groups to analyze the transformed data.
ANOVA requires normally distributed data as well as equal variances. Both requirements are not met with raw OD values from an ELISA test. A transformation with an inverse logistic function, however, gives the possibility of using linear models for the analysis of virus concentration data. We conclude that this method can be applied in every assay where the virus concentrations of samples from different groups need to be compared via the OD values of an ELISA test. To encourage researchers to use this method in their studies, we provide an R script for data transformation along with our trial data.
Since its invention in 1972, the enzyme immunoassay (ELISA)  has been used until today as a reliable tool to detect and quantify virus concentrations in humans [2,3,4,5 ,6,7,8] , animals [9,10,11] and plants [12,13,14,15,16,17,18]. To quantify virus concentrations in a sample, the double antibody sandwich ELISA (DAS-ELISA) is a common tool in which an enzyme-linked antibody specifically binds to the coat protein of the virus, which is again bound to the surface of a microtiter plate by another specific antibody. Subsequently, a colorless substrate is given to each sample which breaks down over time, resulting in lower light transmission.
This reaction is catalyzed by the enzyme [13, 19]. To measure the transmission level of the sample at a certain time, light with a certain wavelength can be sent through the sample. It can then be measured via a sensor the amount of light absorbed by the sample, which results in an optical density (OD) value for each sample . Due to the measurement of transmission levels, the resulting DOs can only take on values within a certain range. In the case of the ELISA machine in this trial, this range was zero to four.
The numerical form of the ODs allows the ODs of different groups of samples to be compared to find significant differences in the protein concentrations of these groups. Such studies have been carried out in the fields of medicine [21,22,23,24,25,26], veterinary medicine , neurosciences [28,29], pharmacology  and agriculture [31,32,33,34]. To find significant differences in the ODs of different groups, analysis of variance (ANOVA) is the established standard. ANOVA can only be performed when the normal distribution of response variables in each group can be assumed . This requirement is rarely tested in OD data analysis, but if tested, it has been rejected for at least one group [27, 29, 34].
Here we present data from a trial where lined haploid lines of sugar beet (Beta vulgaris L.) were grown in soil infested with beet necrotic yellow vein virus (BNYVV) in different environments and harvested at different times. Due to the relatively large sample sizes per group and the use of genetically identical individuals, we show that the OD data per group are not normally distributed. This coincides with the observations of a similar trial  and explains the result of the distribution tests.