Many regression models for categorical data have been introduced in various applied fields, motivated by different paradigms. But these models are difficult to compare because their specifications are not homogeneous. The first contribution of this paper is to unify the specification of regression models for categorical response variables, whether nominal or ordinal. This unification is based on a decomposition of the link function into an inverse continuous cdf and a ratio of probabilities. This allows us to define the new family of reference models for nominal data, comparable to the adjacent, cumulative and sequential families of models for ordinal data. We introduce the notion of reversible models for ordinal data that enables to distinguish adjacent and cumulative models from sequential ones. Invariances under permutations of categories are then studied for each family. The combination of the proposed specification with the definition of reference and reversible models and the various invariance properties leads to an in-depth renewal of our view of regression models for categorical data. Finally, a family of new supervised classifiers is tested on three benchmark datasets and a biological dataset is investigated with the objective of recovering the order among categories with only partial ordering information. Keywords. invariance under permutation, link function decomposition, models equivalence, nominal variable, ordinal variable, reversibility. arXiv:1404.7331v2 [stat.ME] 12 May 2014Property 4. Let σ J be a permutation of {1, . . . , J} such that σ J (J) = J and let P σ J be the restricted permutation matrix of dimension J − 1 (P σ J ) i,j = 1 if i = σ J (j), 0 otherwise, for i, j ∈ {1, . . . , J − 1}. Then we have (reference, F, Z) σ J = (reference, F, P σ J Z), for any F ∈ F and any Z ∈ Z.Proof. For the reference ratio we havefor j ∈ {1, . . . , J − 1}. Thus we simply need to permute the linear predictors using P σ J and we obtain η = P σ J η = P σ J Zβ.Noting that P σ J is invertible with P −1 σ J = P σ −1 J , we get:Corollary 3. The family of reference models is stable under the (J − 1)! permutations that fix the reference category.Corollary 4. Let F ∈ F. The particular (reference, F , complete) and (reference, F , proportional) models are invariant under the (J − 1)! permutations that fix the reference category.

Because irregular bearing generates major agronomic issues in fruit-tree species, particularly in apple, the selection of regular cultivars is desirable. Here, we aimed to define methods and descriptors allowing a diagnostic for bearing behaviour during the first years of tree maturity, when tree production is increasing. Flowering occurrences were collected at whole-tree and (annual) shoot scales on a segregating apple population. At both scales, the number of inflorescences over the years was modelled. Two descriptors were derived from model residuals: a new biennial bearing index, based on deviation around yield trend over years and an autoregressive coefficient, which represents dependency between consecutive yields. At the shoot scale, entropy was also considered to represent the within-tree flowering synchronicity. Clusters of genotypes with similar bearing behaviours were built. Both descriptors at the whole-tree and shoot scales were consistent for most genotypes and were used to discriminate regular from biennial and irregular genotypes. Quantitative trait loci were detected for the new biennial bearing index at both scales. Combining descriptors at a local scale with entropy showed that regular bearing at the tree scale may result from different strategies of synchronization in flowering at the local scale. The proposed methods and indices open an avenue to quantify bearing behaviour during the first years of tree maturity and to capture genetic variations. Their extension to other progenies and species, possible variants of descriptors, and their use in breeding programmes considering a limited number of years or fruit yields are discussed.