Two sample tests for Semi-Markov processes with parametric sojourn time distributions: an application in sensory analysis

Developing statistical approaches that are able to compare the probability law of qualitative trajectories can be of real interest in many fields of science such as economics and sociology, quality control or epidemiology. This work is motivated by an application in sensory analysis in which subjects indicate the succession of perceived sensations over time using a list of attributes. In Lecuelle (Food Qual Prefer 67:59–66, 2018), Semi-Markov Processes (SMPs) are introduced to model such data, allowing to take into account the dynamics via the transitions from one attribute to another as well as the duration law of each attribute. One of the major challenges of sensory analysis is to determine if two tasted products are perceived differently. For that purpose, the present paper introduces a statistical testing procedure based on the likelihood ratio between two semi-Markov processes, assuming a parametric form for the sojourn time distributions. Three approaches are evaluated to compute the p-value: a first one based on the asymptotic law of the likelihood ratio, a second one based on the parametric bootstrap and a third one based on permutations. These approaches are compared on Monte-Carlo simulated data both in terms of empirical levels under the null hypothesis and statistical powers under alternatives. We also develop partial tests to compare two processes on either their initial probabilities and transition matrices or their sojourn time distributions. Simulations show that permutation approaches perform better in nearly all situations and especially for small and moderate sample sizes. Finally, the proposed tests are illustrated on real datasets which consist in perceived sensations over time during the tasting of different chocolates and cheeses.