Multilevel modeling in food science: A case study on heat-induced ascorbic acid degradation kinetics

Characterization of variation of experimental results is achieved by repeating experiments. Frequently, results are averaged before data are analysed but that may not be the best practice because valuable information is then lost. Three other ways to analyze repetitions are: (1) each experiment is analyzed on its own (no pooling of data), (2) all experiments are analyzed together in one go (complete pooling), (3) data are analyzed together while allowing for similarities as well as differences in the result (partial pooling). Multilevel modeling uses partial pooling by partitioning variance over more than one level. Level 1 consists of the measurements themselves; higher levels consist of groups or clusters of measurements (repetitions, experiments at various temperatures, at various pH values, etc.) and parameters are analyzed both at the population and at the group/cluster level. The approach is applied to a case study in which heat-induced isothermal degradation of ascorbic acid was studied with 15 repetitions in an aqueous solution, making it a two-level study. The data were analyzed using averaging and complete pooling, complete pooling without averaging, no-pooling at all, and partial pooling. The kinetic model was established by letting the data decide about the order of the reaction, while this was compared to a model where the order was fixed at 1 (first-order model). Results show that both averaging with complete pooling, as well as complete pooling without averaging, strongly underestimate variation. The no-pooling technique overestimates variation, while partial pooling partitions variation over the levels and thus gives a better impression of the variation involved. The kinetics of ascorbic acid appear to be subject to strong variation when each experiment is considered separately because it is a compound that is very sensitive to all kinds of experimental conditions. With multilevel modeling it appeared to be possible to characterize the uncertainties involved much better than with single level modeling. A Bayesian analysis was performed, in which parameters are allowed to be variable, which is useful because multilevel modeling leads to characterization of variation of parameters. The Bayesian method allows to visualize the posterior distribution of parameters, thereby giving more insight in their behaviour. Also, a Bayesian analysis focuses more strongly on predictive accuracy of models, including multilevel models. The predictive accuracy of 4 models describing the same ascorbic acid data was compared and the multilevel model with reaction order estimated from the data performed by far the best in this regard. The pros and cons of multilevel modeling are discussed and it is concluded that multilevel modeling is to be preferred whenever the data allow to perform such an analysis.