What is the 5PL Formula?

In bioassay analyses, there are three statistical models that are commonly fit to dose-response data: the linear model, the 4PL model, and the 5PL model. We’ve examined the properties of the 4PL model in great detail previously, and the time has come to give the 5PL model a similar treatment. The 5PL model is the most complex and the least used of the three main model classes. We hope to demonstrate that it should be nothing to be afraid of and can indeed be a very useful tool in many situations.

The 5-Parameter Logistic model, or 5PL, is a statistical model which is often used to fit bioassay dose-response data. Similar to the 4PL model, it takes the form of an “S-shaped” curve, with a flat zero-dose asymptote transitioning into a steep linear section, before flattening back out into an infinite dose asymptote.

The major difference between the 4PL and 5PL is that the latter can exhibit asymmetry. We can think about this in terms of the curvature of the models at the transitions between the linear section and the two asymptotes. A good way to visualise this qualitatively is by imagining the curved section as part of a circle. The smaller the diameter of the circle, the greater the curvature of the transition.

A 4PL is symmetric, meaning the curvature at the zero-dose transition is identical to that at the infinite-dose transition. We can see this in the figure below: the dashed circles have the same diameter.

By contrast, a 5PL curve is asymmetric: the curvature of the two transitions are different to one another. In the first of the plots below, the curvature for the infinite-dose transition is less than for the zero-dose transition, indicated by the increased diameter of the associated circle. By contrast, the curvature of the infinite-dose transition is greater than that of the zero-dose transition in the second plot. This is shown by the reduced diameter of the circle at that transition.

The “S-shaped” curves of the 4 and 5PL models are appropriate for statistically modelling bioassays as they provide a good representation of how many biological reactions behave when the concentration of one of a sample is varied.

Let’s assume we have an assay where the observed response increases with the concentration of a certain sample. Typically, little to no response is observed until the concentration of the sample reaches a certain threshold. Then, the response increases quickly as the concentration increases: in this phase, the reaction depends strongly on the concentration of the sample. Eventually, we reach a concentration where the observed response is maximised. After this, we see very little change in the response as the sample concentration increases.

For the dose-response curve to be symmetric in this case, the rate at which the response changes with increasing sample concentration must be the same both entering the linear phase from the minimal response and exiting the linear phase into the maximal response. In some cases, these rates will not be equal, meaning there is asymmetry in the dose-response curve that should be accounted for by the statistical model.

As the name suggests, where the 4PL has four free parameters, whose values are estimated during model fitting, the 5PL has five. The formula for the 5PL for response y associated with a dose x is:

Where:

is the zero-dose asymptote;

is the slope parameter;

is related to the EC₅₀ (more on this later);

is the infinite-dose asymptote;

is the asymmetry parameter.

Note that the above designations are for . If , then the parameter controls the infinite-dose asymptote and the parameter controls the zero-dose asymptote.

Regular readers will note that this formula is very similar to the formula for the 4PL but with one key difference: the parameter. This is the dial which is tuned to change the amount of asymmetry in the model. Broadly speaking, the further is from one, then the more asymmetric the model. When , the 5PL is identical to a 4PL and exhibits no asymmetry. In this sense, the 4PL can be thought of as a special case of a 5PL where .

As with the 4PL, the slope parameter is proportional to the slope of the curve at the dose given by the parameter . In this case, however, the relationship not only depends on the asymptotes and , but also the asymmetry parameter . If the slope at dose is given by :

Unlike the 4PL, however, the parameter does not simply correspond with the . Instead, there is a shift dependent on the and parameters given by:

For both these parameters, we see that as , they tend towards the relationships we expect to see for the 4PL, namely:

and

All this means that it’s often quite hard to visualise the effects of varying  on the shape of the curve. In the animation below, the and the slope of the curve at the are constant across the curves. This isolates the effect of changing on the asymmetry of the model.

The coloured curves here are 5PL curves with the parameter set to varying values between 0.3 and 2.0 with consistent asymptotes, slope, and . The black curve is taken from the same class of curves, but with . This means it is equivalent to a 4PL curve with the same asymptotes, slope, and .

As we mentioned, the main effect of is to control the steepness of the transitions between the linear section and the two asymptotes. As increases, both steepen, with the infinite-dose transition steepening faster than the zero-dose transition. Once , the infinite-dose transition becomes steeper than the zero-dose transition.

Note that the properties of the curve with varying do not necessarily generalise easily to other examples. In particular, a negative parameter or a curve where would have different properties to the curves we’ve examined above.

So, if some assay response data may exhibit asymmetry, a conservative or “safe” approach might appear to be to fit a 5PL model for every assay, especially since the 4PL model is a special case of the 5PL.

Unfortunately, it’s not quite that simple: 5PL models can be significantly harder to fit than 4PL models. The additional parameter of the 5PL model means it has a lot more flexibility than a 4PL. While this may sound solely a good thing, it has the consequence that an optimised parameter combination can be more difficult to find. Even large changes in parameter estimates may result in models which fit the data similarly. This  means the true optimal model fit may be difficult to find.

Another challenge of using a 5PL is that more data is required to fully characterise the model than when using a 4PL. Whereas a 4PL only needs 5 data points, the extra parameter means that a 6^th data point is necessary to fit a 5PL. And, since the 5PL is asymmetric, it is critical to have data points that fall on both flat portions of the curve. For a 4PL, while not ideal, we can use its symmetric properties to infer one of the asymptotes if we do not have a sufficient range of data to see both. For a 5PL, however, this is not possible.

So, how can you test which model fit is best for your assay data? One way is to look at the confidence interval on the fitted E parameter. Here, we have a choice of approaches:

• A significance approach: does the confidence interval contain 1? If not, then we can say that the parameter is significantly different from 1, and the data would fit a 5PL model better than a 4PL.
• An equivalence approach: does the confidence interval fall entirely within pre-determined limits about 1? If it does not, then the parameter is not statistically equivalent to 1, and the data fits a 5PL better than a 4PL.

The 5PL is certainly the most complex of the models commonly used for analysing bioassay data and, for the reasons we’ve outlined here, can be difficult to fit. Thankfully, help is on hand! QuBAS, the bioassay analysis software package developed by Quantics, can easily fit all common bioassay models with no coding required. Follow the link to our QuBAS page, or get in touch to find out more.