Enzyme-Linked Immunosorbent Assays (ELISA) are a key tool for determining concentrations, forming a key component of many fields in the life sciences. From testing food products for allergens to detecting HIV in a patient’s blood and beyond, ELISA analysis is a vital component of a scientist’s arsenal. Quantics has been at the forefront of research into ELISA statistical analysis, playing a key role in the development of the Bursa-Yellowlees (BY) Method, which was first published in 2020. We want to highlight the key differences between the BY method and traditional analysis techniques, and how these differences can often lead to improved performance when used for ELISA (or any other interpolation-type) bioanalysis.
What is the Bursa-Yellowlees Method?
The BY Method is an alternative to the traditional interpolation-based method of ELISA analysis. Whenever we perform an ELISA, our goal is to determine the unknown concentration of a test sample by comparing its biological response to that of a reference standard sample of known concentration.
Traditionally, the process looks something like this:
- Measure the response of the standard at a range of concentrations, and fit a model (e.g. linear, 4PL, 5PL) to the response versus log concentration.
- Take dilutions of the test sample, and measure their response.
- Interpolate horizontally to the standard curve, and read off the corresponding concentration.
- To find the estimated concentration of the test sample, multiply the interpolated concentration by the dilution factor. For example, if we have an interpolated concentration of 0.01 for a 1:100 dilution of the test sample, then our estimated test sample concentration is 1.
Typically, multiple dilutions of the test sample are used. In this case, the geometric mean (not the arithmetic mean, as is often incorrectly used) of the concentration estimates is used to find an overall estimate.
The BY method takes a different approach. Here, we will assume our data is best fit by a 4PL model, but the method can be used with any model:
- Measure the response of the standard at a range of concentrations, and fit a model (e.g. linear, 4PL, 5PL).
- Take dilutions of the test sample, and measure their response.
- Fit a model to the test sample response data that is parallel to the standard model. The only free parameter here is the C parameter. All other parameters are set as identical to the standard curve.
- The estimated log concentration is then the difference between the C parameters of the standard and test curves (standard – test).
You might notice that the method of determining the concentration bears a striking similarity to the calculation of a relative potency. And this is no coincidence: the BY method is indeed inspired by relative potency calculations.
Why does this work? Assuming we have similarity between a test sample and the standard (more on this later), we know that any test sample is a dilution of standard. The concentration of the standard is known, meaning we can determine the concentration of the test sample by comparing its response in the assay to that of the reference.
This is precisely what the BY Method does. If a test sample is less concentrated than the standard, then its response curve will be shifted to the right relative to the standard curve. Conversely, if the test sample is more concentrated than the standard, then its response curve will be shifted to the left. In both cases, the difference in concentration is encoded into the horizontal shift, which is then used to calculate the overall concentration of the test sample. A more mathematical outline is given in the spoiler below, but fear not: a detailed understanding is not required to appreciate the benefits of the BY Method!
Let the 1:1 dilution of the standard have concentration 
and the 1:1 dilution of the test sample have concentration 
. is known, and we want to determine .
Assuming similarity, the test sample is a dilution of the standard. This means:
![\[X_T=\Delta X_S\]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-8ee15203586dd43a433f608e85067215_l3.png "Rendered by QuickLaTeX.com")
for some dilution factor 
Each response value returend in our assay is associated with a concentration 
of either the standard or the test sample. These concentrations are dilutions of the 1:1 dilutions of the stock samples, i.e.:
![\[x_S=\delta X_S \text{ and } x_t=\delta X_T \text{ for dilution factors } \delta_i>0 \text{. Since } X_T=\Delta X_S \text{:}\]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-ca5b50995ea8ac13e6e371793fd26556_l3.png "Rendered by QuickLaTeX.com")
![\[x_T=\delta_T\Delta X_S\]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-1710942cfe1c04e8566157f2dc02aa1b_l3.png "Rendered by QuickLaTeX.com")
Assuming similarity, we know that if a dilution of the standard and a dilution of the test sample elicit the same response, then the concentrations of those dilutions must be identical. So, along a line of equal response:
![\[x_S=x_T \rightarrow \delta_S X_S = \delta_T \Delta X_S \rightarrow \Delta = \frac{\delta_S}{\delta_T}\]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-66ada26b5f7f3232860782bf66196e63_l3.png "Rendered by QuickLaTeX.com")
If we plot our responses against the dilution factors of the standard and test sample on the log scale, we can say that:
![\[\log(\Delta)=\log(\delta_S)-\log(\delta_t)\]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-e1cb4877fe0d00d20c16db55fc2dd483_l3.png "Rendered by QuickLaTeX.com")
Which is the horizontal shift between the standard curve and the test sample curve along a line of equal response. So, to find the concentration of the 1:1 dilution of the test sample:
![\[X_T=\Delta X_S = \text{antilog}\left(\delta_S - \delta_T\right) X_S \]](https://www.quantics.co.uk/wp-content/ql-cache/quicklatex.com-1fe30d3633865c7720a7bd00c1ed97fa_l3.png "Rendered by QuickLaTeX.com")
As with a relative potency measurement, a key assumption is that the biological behaviour of the test sample is the same as the standard: that we have similarity between the two samples. This ensures that the horizontal shift between the curves does not change with dilution, and gives us a well-defined measurement of the concentration of the test sample. Additionally, this means we have the freedom to choose the dilutions where we measure the shift. For consistency we utilise the C parameter of the 4PL model, which gives the dilution at which we observe half the total observed response.
This assumption means that testing for suitability forms a key suitability criterion for an assay using the BY method, just as with a relative potency assay. We’ve outlined methods for effective suitability testing for similarity in that context previously, and these broadly apply here too.
Advantages of the Bursa-Yellowlees Method
Bursa et al’s 2020 paper uses a simulation study to demonstrate the advantages of the BY method for ELISA analysis. We outline some of the key takeaways here.
The BY Method provides accuracy and precision benefits over traditional methods
With the traditional method for ELISA analysis, only response values of the test sample which fall within the range of the standard – i.e. those which fall between the asymptotes of the response curve – can be included in the interpolation. This necessitates that data outside this range is excluded. An arbitrary cut-off is often chosen for inclusion of data , typically some fraction of the difference of the asymptotes. This arbitrary cut-off can lead to a bias in the calculation of the unknown concentration.
Further, even if responses near the asymptotes are included, the calculated concentration from those points is less certain. In the traditional method, the concentrations interpolated from all included responses are given equal weighting in the calculation of the final result. This means that the precision of concentration measurement is decreased, particularly if many of the measured responses fall near one of the asymptotes of the standard curve.
By contrast, the BY method utilises all available responses – even those which fall outside of the range of the standard. It also does not place undue weight on responses close to the asymptotes. The results of the simulation study, shown in the figures below, give a clear demonstration of the advantages this provides in terms of accuracy and precision.
In both plots, simulated results using the traditional method are shown in red, while those using the BY Method are in blue. Both accuracy, quantified by a percentage bias, and precision, quantified by the width of the confidence interval on the calculated concentration, are plotted against the true concentration of the test sample.
While the methods give close, if not equal, accuracy and precision performance for much of the range of tested concentrations, it is clear that this performance of the BY method is far more consistent. Notably, there are several “spikes” at certain concentrations where the performance of the traditional method is much poorer than the BY method.
These are associated with the cut-off for data inclusion. When the data was simulated, 4 specific dilutions of the test sample (1:1, 1:4, 1:16, 1:64) were initially used. If the response for any of these dilutions fell outside of the inclusion range – 90% of the difference between the asymptotes of the standard – then that point was excluded. The “spikes” occur when the true concentration of the test sample is such that at least one of the dilutions is excluded.
The precision of any measurement will decrease if the sample size decreases, so we see wider confidence intervals when any response data is excluded. The reason for the decrease in accuracy, however, is a little more subtle. Imagine a test sample had a true concentration which meant that one of the dilutions fell, on average, just above the upper inclusion cut-off. This would mean that, for the majority of runs, this response would be excluded. There would, however, be occasions where natural variability meant that this response fell just inside the cut-off and would be included. Since this only occurs when the response is lower than usual, we observe a small downward bias in the measured concentration. It also follows that we would see a small upward bias when a response falls just outside the lower inclusion bound.
By contrast, the BY method always uses all available response data, meaning we observe no “spikes” in accuracy or precision. We do observe decreases in performance at very high and very low concentrations, which is associated with measuring shifts close to the asymptotes. The decrease in accuracy here, however, is symmetric about zero, meaning we observe no overall bias.
Bursa et al also confirm the superior performance of the BY method in the simulation study quantitatively. Measurement precision is greatly improved, with a median CI width of 0.147 produced by the BY method compared to 0.183 for the traditional method. This improvement is even greater when only samples which pass suitability (more on this later) for both methods are included (0.125 vs 0.183). Accuracy gains are more marginal, with a percentage bias of 2.2% with the BY method compared to 2.4% with the traditional method.
The BY Method leads to fewer unnecessary suitability failures
In the simulation study, a concentration was considered to have passed suitability if, for the traditional method:
- At least two responses were recorded which fell within the inclusion limits.
For the BY method:
- The test sample model fit must converge.
- The test sample must pass a parallelism F Test.
For both methods, the precision factor on the result (the ratio between the upper and lower confidence limits) must be less than 4.
The results showed a clear advantage for the BY method in this case, with 96.5% of samples passing suitability compared to just 56.6% for the traditional method. This means fewer failed samples and fewer retests, greatly increasing the efficiency of the testing process. Indeed, the simulations results suggest a massive 71% increase in throughput using the BY method over the traditional method.
These benefits were emphasised at very high and very low concentrations. The traditional method returned essentially no valid concentration estimates below 0.0625 units and above 8 units. By contrast, the BY method returned valid estimates for more than 89% of simulated samples across the full range of concentrations from 0.0156 units to 32 units.
While the accuracy and precision of very high and very low concentration estimates are reduced compared to samples which fall nearer the centre of the range even with the BY method, having a valid estimate at all is certainly advantageous over returning nothing.
The BY method can increase the efficiency of assays by reducing the need for replication
Even if your ELISA performs well using traditional interpolation analysis, the increased precision offered by the BY method could prove beneficial. Specifically, it could mean that you require fewer replicates to achieve an acceptable precision on your reportable result.
We’ve previously discussed how number of replicates can be a good way to optimise your assay, specifically by ensuring that the replication is used where the variability is highest. In this case, we can reduce the number of replicates required by using the data more efficiently. Since the BY method can provide results which are significantly more precise than those provided by traditional methods, we need fewer replicates to achieve the same precision as with more replicates using the traditional method.
This provides numerous benefits, the most obvious of which is a reduction in the amount of expensive reagents and other materials required for an effective assay. Reducing replication can also mean that more test samples can be included on each plate in some circumstances, which can lead to an improved throughput.
Should I use the BY Method?
In many scenarios, the BY method will outperform traditional methods of analysing ELISA data. Its alternative approach of utilising the shift of the test sample response curve compared to that of the standard curve removes the need for arbitrary cut-offs for inclusion of response data. This means the BY method is more precise and accurate than traditional analysis in many situations. The increase in precision in particular can be used to reduce replication requirements while maintaining acceptable precision. Combined with the BY method’s ability to provide valid results over a wider range than traditional methods, leading to fewer unnecessary suitability failures, this can mean a greatly improved throughput.
In some cases, the BY method will perform no better than the traditional ELISA analysis method. For example, the BY method performs identically to traditional methods when only one dilution is used for a test sample. As we have demonstrated, however, the BY method will provide superior results in many common scenarios. Further, the BY method will never perform worse than the traditional method, meaning you can be confident that the BY method will provide optimal results in every scenario.
The BY method is included in QuBAS, the bioassay analysis software package developed by Quantics. If you want to find out more about implementing the BY method for your ELISA analysis, check out our explainer here or feel free to get in touch.
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