Choosing a statistical model: Continuous response data
Previously, we discussed the definition of relative potency and how it relates to biological similarity, and therefore the need for parallelism between reference and test dose–response curves. In order to test for parallelism and calculate a relative potency, the assay data must be mathematically modelled.
From a modelling perspective there are two fundamentally different types of assay data. In continuous assays, the response is effectively continuous (for example optical density or chemical concentration). In quantal assays, the response is binary or “all-or-none” (for example alive/dead or reacted/not reacted). This distinction is important because different mathematical models are required.
Key Takeaways
- Continuous bioassay data are commonly modelled using the four-parameter logistic (4PL) model, which provides a flexible and biologically meaningful fit.
- Although linear models may appear attractive, relative potency shifts can cause non-parallelism and unnecessary assay failures.
- Using log-transformed doses (and often responses) helps stabilise variability and better satisfy model assumptions.
Continuous assay data
Although modern computational techniques can model data without strong assumptions about distributions, these approaches are complex. In practice, a small number of simple assumptions are often reasonable and make modelling far more straightforward.
In the example considered here, there are eight doses with three replicates at each dose. The doses span almost the full response range from zero to maximum, and the dose axis is plotted on a logarithmic scale.
A simple, symmetrical S-shaped curve appears to fit these data well. This curve is known as the four-parameter logistic (4PL) model, because four parameters are required to define it fully: the lower and upper asymptotes (A and D), the slope-related parameter B, and the horizontal position C.
The equation for the 4PL, for response y and log(dose) x, is:
\[y = D + \frac{A – D}{1 + e^{B(x – C)}}\]
Simpler models
Although the 4PL often fits well, in practice we may only observe data in the central region of the curve, which can appear approximately linear. This makes linear models tempting, particularly when statistical support is limited.
However, choosing a linear model can have serious practical consequences. The same model must be fitted to both reference and test samples. When the relative potency of the test sample decreases, its curve shifts to the right on the dose axis. Under a linear model, this causes the fitted lines to become non-parallel, leading to a failed parallelism test.
If a 4PL model were used instead, the same data would often pass the parallelism test. Linear models should therefore be used with caution if unnecessary assay failures are to be avoided.
Assumptions of the 4PL model
The standard 4PL model relies on assumptions that are often approximately satisfied in bioassay data, but which are important to understand. In particular, the model assumes constant variability across dose groups and normally distributed errors.
In real bioassays, variability frequently increases at higher response levels. Applying a logarithmic transformation to the response often stabilises this variability. For this reason, it is standard practice to use log(dose), and to consider log(response) unless there is strong justification for working on the raw scale.
Normality also assumes symmetry. As responses approach zero, this assumption breaks down on the raw scale. Using log(response) resolves this issue, since very small responses correspond to large negative values on the log scale.
Other transformations are possible, but the logarithm is by far the most common. In this context, the choice of logarithmic base (e.g. log10 or log2) is not important.
