Multiresponse kinetics with estimation of the experimental variance-covariance matrix. A full Bayesian analysis using Stan

Abstract

Multiresponse analysis is a powerful technique to unravel complex kinetic chemical reaction schemes. Box and Draper (1965) showed that least-squares analysis of multiresponse measurements can be inappropriate if measurements are correlated, which will be frequently the case. Insight into the experimental variance-covariance matrix is therefore important, but this matrix is usually unknown. Using a Bayesian analysis, Box and Draper (1965) integrated out the experimental variance-covariance matrix to find an approximate solution for parameter estimates, called the determinant criterion. Nowadays, a full Bayesian analysis is possible via Markov Chain Monte Carlo (MCMC) sampling that does allow to estimate the full experimental variance-covariance matrix, next to the common kinetic parameter estimates. Two multiresponse examples are discussed to investigate this, using the probabilistic language Stan coupled to R. One example is the simulated data set used by Box and Draper (twelve runs with three components), the other a much-used real data set on heat-induced isomerization of $\alpha$-pinene (nine runs with six components). One main result is that with both data sets the experimental variance-covariance matrix could be estimated, which gives much insight in the properties of the experimental data and further statistical analysis. Another main result is that, at least for the pinene data set, least-squares analysis leads to unrealistic precise estimates compared to the Bayesian solution, showing that it is worthwhile to use the Bayesian solution. An additional result is that the generally accepted mechanism for the pinene data set is questionable. Overall, the results show that Stan is a great addition to the toolkit of the kineticist for multiresponse kinetics analysis.