Analysis of Multinomial Data With Overdispersion Diagnostics and Application

Abstract

In agronomic experiments, the presence of polytomous variables is common, and the generalized logit model can be used to analyze these data. One of the characteristics of the generalized logit model is the assumption that the variance is a known function of the mean, and the observed variance is expected to be close to that assumed by the model. However, it is not uncommon for extra-multinomial variation to occur, due to the systematic observation of data that are more heterogeneous than the variance specified by the model, a phenomenon known as overdispersion. In this context, the present work discusses a diagnostic of overdispersion in multinomial data, with the proposal of a descriptive measure for this problem, as well as presenting a methodological alternative through the Dirichlet- multinomial model. The descriptivemeasure is evaluated through simulation, based on two particular scenarios. As a motivational study, we report an experiment applied to fruit growing, whose objective was to compare the flowering of adult plants of an orange tree, grafted on “Rangpur”lime or “Swingle” citrumelo, with as response variable the classification of branches into three categories: lateral flower, no flower or aborted flower, terminal flower. Through the proposed descriptive measure, evidence of overdispersion was verified, indicating that the generalized logit model may not be the most appropriate. Thus, as a methodological alternative, the Dirichlet-multinomial model was used. Compared to the generalized logit model, the Dirichlet-multinomial proved to be more suitable to fit the data with overdispersion, by allowing the inclusion of an additional parameter to accommodate the excessive extra-multinomial dispersion.

Overdispersion Models for Clustered Toxicological Data in a Bioassay of Entomopathogenic Fungus

“…Here, we took a simpler approach basing overdispersion parameter estimation on the overall residual lack of fit, as is common in applications of overdispesion models. Diagnostic assessment of model fit and the adequacy of specific overdispersion models could also, in principle, be explored using half-normal plots as in Salvador et al (2022) for standard multinomial and Dirichlet-multinomial models. To apply such methods here would require extending the approach to the cumulative type of response considered here through, for example, providing these new model types for the hnp package (Moral et al, 2017) in R. can be written in a fairly simple overall form.…”

“…This model compounds the multinomial distribution for the observed counts with a Dirichlet distribution for the vector of underlying probabilities, leading to a Dirichlet-multinomial distribution, which has a different form for the variance function generalizing that for the standard multinomial distribution. Note that the Dirichlet-multinomial generalization has been applied in many different settings, including to nominal polytomous data (Salvador et al, 2022), as an extension to deep learning for RNA sequencing data (Corsini & Viroli, 2022). The third model is a random intercept model, where we incorporate an additive random effect in the linear predictor to give a random location shift for each distinct multinomial sample.…”