Bayesian hierarchical linear regression for the validation of analytical methods

Analytical quantitative methods are widely used to quantify analytes of interest, for instance in pharmaceutical formulations, linking an observed response to the concentration of one compound of the formulation.
Current methodologies to validate these analytical methods are based on one-way ANOVA random effect model in order to estimate repeatability and intermediate precision variances. This model is then applied several times at different concentration levels over a range of concentrations where the method is intended to be used, assuming independency between the levels. In this way, the capacity of the method to be able to quantify accurately is assessed at various concentration levels, and the method is said to be fitted for purpose (or valid) at the concentration level(s) where it shows trueness and precision that are fully acceptable, i.e. within predefined acceptance limits. Problem of such approach is the amount of data required and the time needed to collect them, while small sample sizes (small number of series and of replicates per series) are often preferred and practiced by laboratories.
A better use of the data could then be envisaged. In this presentation, we take into account the response-concentration relationship that exists by the use of a hierarchical linear regression model. Instead of fitting a model at each concentration level that is assessed, only one model is studied. We show how the Bayesian framework is well adapted to this task. Also, as a predictive tool, we naturally derive beta-expectation and beta-gamma content tolerance intervals by means of MCMC simulations. The Bayesian modeling can also include informative prior information whenever justified, leading to reliable decisions given the domain knowledge.