Happy π-Day!

To celebrate the International Day of Mathematics, we're serving a special collection of mathematical cakes in the Academy. Each one is inspired by a formula used in PLA 3.0 and turns a piece of bioassay analysis into something worth savoring.

These mathematical recipes (the formulas) tell PLA 3.0 how to mix the ingredients (your experimental data) so that everything comes together just right. From curve fitting to evaluating model quality, they guide the process from raw data to well-baked results.

Choose your favorite cake and enjoy the mathematics behind your data analysis.

Restricted 4-parameter logistic model

A carefully balanced cake for smooth curves

Ingredients
Dose–response data from the reference standard
Dose–response data from the test sample

Preparation

Use the logarithm base B to obtain the transfomend dose (z). Then fit the data of the reference standard and the test sample simultaneously using a restricted regression model.
The restriction ensures that both curves share the same asymptotes (a and d) and slope (b), resulting in identical dynamic ranges.

To achieve this, the indicator function I₂(zᵢ) is used in the regression formula. It is equal to 1 if the observation belongs to the test sample and 0 otherwise.

Bake

Using this formulation, the parameters a, b, and d are estimated from the combined data of both samples.
The location parameter becomes c for the reference standard and c + r for the test sample. The logarithmic relative potency r can then be determined directly from the fitted model.

Chef's tip

Depending on the assay characteristics, an asymmetric 5-parameter logistic model or a parallel line model may also be used as alternative regression approaches.

Coefficient of Determination (R²)

A cake that shows how much of your data your model explains

Ingredients
Residuals between observed and predicted values

Preparation

Fit the model to the data and calculate the residuals between the observed values and the model predictions.
Using these quantities, the coefficient of determination R² is computed using the ratio between explained and total variability in the data.

Bake

When the residuals of the fitted model become small, the numerator of the formula approaches zero and the R² value approaches 1, indicating that the model explains a large portion of the variability in the observations.

Chef's caution

A large slice does not always mean the recipe is correct.
Because the number of model parameters is not taken into account, R² can increase when additional parameters are added, potentially leading to overfitting. For this reason, R² should be interpreted with care.

Serving suggestion

Despite these limitations, verifying that the R² value of the fitted model exceeds a required threshold remains a widely used criterion for assessing the validity of an assay system.

Fieller Formula

where

A cake for those who love their ratios well served

Ingredients
Standard error of the regression

Preparation

When estimating ratios of parameters, standard approaches for confidence intervals may no longer be sufficient. The Fieller formula provides a method for calculating confidence intervals for ratios by incorporating the variance and covariance of the parameter estimates.

In PLA 3.0, this approach is used when determining confidence intervals for the relative potency in a parallel-line potency assay.

Bake

Applying the Fieller formula yields the lower and upper confidence limits for the ratio estimate. Unlike simpler approaches such as the delta method, the resulting confidence intervals are exact rather than approximations.

Chef's note

Because the interval is derived from a ratio of parameters, the resulting confidence region may sometimes have non-intuitive shapes. Nevertheless, the Fieller method remains a rigorous approach for estimating uncertainty in ratio estimates.

Confidence intervals for fit parameters

A cake that reminds us every estimate comes with a margin

Ingredients
Standard error of the estimate
Degrees of freedom of the regression model

Preparation

Estimate the parameter values by fitting the regression model to the data and determine the corresponding standard errors.

Using the degrees of freedom of the model, obtain the appropriate critical value from the Student’s t-distribution. These quantities define the lower and upper bounds of the confidence interval for each parameter.

Bake

The confidence interval specifies the range in which the true parameter value is likely to lie. It therefore quantifies the uncertainty of the estimate and provides an indication of how reliable the fitted parameter really is.

Chef's test

After the confidence interval has been calculated, equivalence testing can be used to assess the sample. If the confidence interval does not lie fully within the predefined acceptance region, the sample may be considered invalid.