Examining the Three-Parameter Logistic (3PL) curve for Bioassay Modelling

One of the most frequent topics of discussion on this blog has been the correct and optimal utilisation of different classes of statistical models with bioassay data. Most common among these are the four- and five-parameter (4 and 5PL) models, alongside the simple linear model. There is, however, stranger at the feast: the three-parameter logistic, or 3PL model. The 3PL model is available as an option on some bioassay statistics software packages, and we often see it in use by our clients. There are several reasons why this is not an ideal solution, however, and it is not something we often recommend. Here, we will examine these reasons, and discuss alternative solutions to the problems the 3PL model appears to solve.

As with most models used for analysing bioassay data, the 3PL model aims to fit the typical s-shaped dose-response curve. This is almost identical to the more commonly used 4PL model: two asymptotes, with a steep linear region connecting them.

The main difference between the 3PL and the 4PL model is that one of the asymptotes is not considered a free parameter in the model. That is, instead of being fit to collected data, we specify in advance where this asymptote will fall. We have, therefore, only three free parameters which are determined when the model is fitted to the data.

The formula for a 3PL model, with response , raw dose , and assuming the zero-dose asymptote is to be fixed, is given by:

Where:

is the slope parameter

is the

is the infinite-dose asymptote

is a pre-determined constant which fixes the zero-dose asymptote. is not estimated during model fitting.

By comparison with the formula for a 4PL model, it is clear that the constant is playing the role of the parameter, and defines the zero-dose asymptote in this model. In the case of a 4PL, however, the value of the parameter is estimated in model fitting, and its value changes based on the bioassay data. If the infinite-dose asymptote is to be fixed instead, would instead play the role of the parameter, and the formula would be given by:

The most obvious scenario when one might look to use a 3PL model is when it is believed that one of the asymptotes will take a certain value in every run of the assay. For example, imagine a hypothetical assay which measures the percentage of cells killed by a certain dose of a sample. It might be the case that, at higher doses, all of the cells are expected to be killed by the sample. It might seem sensible to fix the zero-dose asymptote at 100 in that scenario.

An alternative case could be if the chosen dose range was such that one could only see one of the asymptotes. For example, if only the lower asymptote and a section of the linear region is visible using a certain dose range, then it might be difficult or impossible to fully characterise a 4PL. It might be tempting to use a 3PL with the upper asymptote set to a sufficiently high value that the resulting linear section proves a good fit to the data.

The first is on the grounds of a fundamental principle of statistics: our observations in any experiment may well be measurements of the properties of a sample, but we can never access their “true” values. Even our most precise and accurate measurements will remain an estimate of reality. In fixing an asymptote, as is required to use a 3PL, we are making a decision about what we believe to be the “true” value of the asymptote based entirely on estimates.

Further, to use our choice, we are making the assumption that there will be no variability in the asymptote we have chosen to fix on repeated runs of our bioassay. Needless to say, thanks to the inherent variability associated with bioassays, this can often be a very strong assumption. Even in an assay for which we, say, expect the response at low dose to be zero, one can easily imagine a scenario where we see a non-zero response. For example, the temperature in the lab might be slightly different, or a different operator might be running the assay. These would not be unexpected changes to the assay system, but they could call serious issues if there was a change to the asymptote which, by fixing to a specified value, cannot move with the variable assay responses.

Specifically, if we have variability in the asymptote which we expect to be constant, then this can result in severe model fitting problems. These can cause a bias in the estimate of relative potency. Other scenarios may result in an under-estimate of the variance and could lead to spuriously narrow confidence intervals for the reportable result. Both these problems could result in making poor release decisions.

So, if you’re in a situation where you’re considering using a 3PL model, what should you do instead? In most common scenarios, the answer is very simple: use a 4PL! Even if you expect one of the asymptotes to be very consistent or even constant, this will still be the case when using a 4PL. While there will be a small extra computational cost as a result of the additional parameters, this is far outweighed by the cost of potential model fitting issues when using a 3PL in most scenarios.

In a situation where you can only see one asymptote, however, a 4PL may not be appropriate. For a 4PL to be fit accurately, we would ideally like to observe at least one data point on each asymptote. Therefore, a 4PL could be problematic in such cases. Here, it might be beneficial to consider the dose range which has been chosen in the assay. For instance, one could expand the dose range such that some of the missing asymptote – or even just the transition into the missing asymptote – is now visible. This will result in a 4PL curve that is better characterised.

If this is not possible, an alternative approach would be to instead restrict the dose range such that only the linear portion of the dose-response curve is visible in the assay data. A linear model can then be fit to the data. However, this is not an ideal solution if samples with a broad range of potencies are expected.

The decisions we’ve discussed here today are a clear example of the benefits of ensuring your statistician is involved early in assay development. While it is rare for an assay design to remain unchanged through development, it is likely that those changes will be less drastic if they are made to a statistically well-informed design. Crucially, any changes are also more likely to take place earlier in the development process when they are easier and less expensive to make. So, don’t be tempted to wade into the bioassay waters without a statistical guide, especially when complex situations as we have discussed here are in play!