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MSmix is a recently developed R package implementing maximum likelihood estimation of finite mixtures of Mallows models with Spearman distance for full and partial rankings. The package is designed to implement computationally tractable estimation routines of the model parameters, with the ability to handle arbitrary forms of partial rankings and sequences of a large number of items. The frequentist estimation task is accomplished via EM algorithms, integrating data augmentation strategies to recover the unobserved heterogeneity and the missing ranks. The package also provides functionalities for uncertainty quantification of the estimated parameters, via diverse bootstrap methods and asymptotic confidence intervals. Generic methods for S3 class objects are constructed for more effectively managing the output of the main routines. The usefulness of the package and its computational performance compared with competing software is illustrated via applications to both simulated and original real ranking datasets., Comment: 29 pages, 7 figures, 3 tables

The Mallows model occupies a central role in parametric modelling of ranking data to learn preferences of a population of judges. Despite the wide range of metrics for rankings that can be considered in the model specification, the choice is typically limited to the Kendall, Cayley or Hamming distances, due to the closed-form expression of the related model normalizing constant. This work instead focuses on the Mallows model with Spearman distance. An efficient and accurate EM algorithm for estimating finite mixtures of Mallows models with Spearman distance is developed, by relying on a twofold data augmentation strategy aimed at i) enlarging the applicability of Mallows models to samples drawn from heterogeneous populations; ii) dealing with partial rankings affected by diverse forms of censoring. Additionally, a novel approximation of the model normalizing constant is introduced to support the challenging model-based clustering of rankings with a large number of items. The inferential ability of the EM scheme and the effectiveness of the approximation are assessed by extensive simulation studies. Finally, we show that the application to three real-world datasets endorses our proposals also in the comparison with competing mixtures of ranking models., Comment: 20 pages, 6 Figures, 11 Tables

The forward order assumption postulates that the ranking process of the items is carried out by sequentially assigning the positions from the top (most-liked) to the bottom (least-liked) alternative. This assumption has been recently relaxed with the Extended Plackett-Luce model (EPL) through the introduction of the discrete reference order parameter, describing the rank attribution path. By starting from two formal properties of the EPL, the former related to the inverse ordering of the item probabilities at the first and last stage of the ranking process and the latter well-known as independence of irrelevant alternatives (or Luce's choice axiom), we derive novel diagnostic tools for testing the appropriateness of the EPL assumption as the actual sampling distribution of the observed rankings. Besides contributing to fill the gap of goodness-of-fit methods for the family of multistage models, we also show how one of the two statistics can be conveniently exploited to construct a heuristic method, that surrogates the maximum likelihood approach for inferring the underlying reference order parameter. The relative performance of the proposals compared with more conventional approaches is illustrated by means of extensive simulation studies., Comment: 19 pages, 3 tables, 1 figure

Multistage ranking models, including the popular Plackett-Luce distribution (PL), rely on the assumption that the ranking process is performed sequentially, by assigning the positions from the top to the bottom one (forward order). A recent contribution to the ranking literature relaxed this assumption with the addition of the discrete-valued reference order parameter, yielding the novel Extended Plackett-Luce model (EPL). Inference on the EPL and its generalization into a finite mixture framework was originally addressed from the frequentist perspective. In this work, we propose the Bayesian estimation of the EPL with order constraints on the reference order parameter. The proposed restrictions reflect a meaningful rank assignment process. By combining the restrictions with the data augmentation strategy and the conjugacy of the Gamma prior distribution with the EPL, we facilitate the construction of a tuned joint Metropolis-Hastings algorithm within Gibbs sampling to simulate from the posterior distribution. The Bayesian approach allows to address more efficiently the inference on the additional discrete-valued parameter and the assessment of its estimation uncertainty. The usefulness of the proposal is illustrated with applications to simulated and real datasets., Comment: 20 pages, 4 figures, 4 tables. arXiv admin note: substantial text overlap with arXiv:1803.02881

Choice behavior and preferences typically involve numerous and subjective aspects that are difficult to be identified and quantified. For this reason, their exploration is frequently conducted through the collection of ordinal evidence in the form of ranking data. A ranking is an ordered sequence resulting from the comparative evaluation of a given set of items according to a specific criterion. Multistage ranking models, including the popular Plackett-Luce distribution (PL), rely on the assumption that the ranking process is performed sequentially, by assigning the positions from the top to the bottom one (forward order). A recent contribution to the ranking literature relaxed this assumption with the addition of the discrete reference order parameter, yielding the novel Extended Plackett-Luce model (EPL). Inference on the EPL and its generalization into a finite mixture framework was originally addressed from the frequentist perspective. In this work, we propose the Bayesian estimation of the EPL with order constraints on the reference order parameter. The restrictions for the discrete parameter reflect a meaningful rank assignment process and, in combination with the data augmentation strategy and the conjugacy of the Gamma prior distribution with the EPL, facilitate the construction of a tuned joint Metropolis-Hastings algorithm within Gibbs sampling to simulate from the posterior distribution. We additionally propose a novel model diagnostic to assess the adequacy of the EPL parametric specification. The usefulness of the proposal is illustrated with applications to simulated and real datasets., Comment: 14 pages, 2 tables, 3 figures

Ranking data represent a peculiar form of multivariate ordinal data taking values in the set of permutations. Despite the numerous methodological contributions to increase the flexibility of ranked data modeling, the application of more sophisticated models is limited by the related computational issues. The PLMIX package offers a comprehensive framework aimed at endowing the R statistical environment with some recent methodological advances in modeling and clustering partially ranked data. The usefulness of the novel PLMIX package can be motivated from several perspectives: (i) it contributes to fill the gap concerning Bayesian estimation of ranking models in R, by focusing on the Plackett-Luce model and its extension within the finite mixture approach as the generative sampling distribution; (ii) it addresses computational complexity by combining the flexibility of R routines and the speed of compiled C++ code, with possible parallel execution; (iii) it covers the fundamental phases of ranking data analysis allowing for a more careful and critical application of ranking models in real experiments; (iv) it provides effective tools for clustering heterogeneous partially ranked data. The functionality of the novel package is illustrated with several applications to simulated and real datasets., Comment: 33 pages, 1 figures, 3 tables

The multi-cycle organization of modern university systems stimulates the interest in studying the progression to higher level degree courses during the academic career. In particular, after the achievement of the first level qualification (Bachelor degree), students have to decide whether to continue their university studies, by enrolling in a second level (Master) programme, or to conclude their training experience. In this work we propose a binary quantile regression approach to analyze the Bachelor-Master transition phenomenon with the adoption of the Bayesian inferential perspective. In addition to the traditional predictors of academic outcomes, such as the personal characteristics and the field of study, different aspects of the student's performance are considered. Moreover, a new contextual variable, indicating the type of university regulations, is taken into account in the model specification. The utility of the Bayesian binary quantile regression to characterize the non-continuation decision after the first cycle studies is illustrated with an application to administrative data of Bachelor graduates at the School of Economics of Sapienza University of Rome and compared with a more conventional logistic regression approach., Comment: 24 pages, 7 figures and 3 tables

The elicitation of an ordinal judgment on multiple alternatives is often required in many psychological and behavioral experiments to investigate preference/choice orientation of a specific population. The Plackett-Luce model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. The present work introduces a Bayesian finite mixture of Plackett-Luce models to account for unobserved sample heterogeneity of partially ranked data. We describe an efficient way to incorporate the latent group structure in the data augmentation approach and the derivation of existing maximum likelihood procedures as special instances of the proposed Bayesian method. Inference can be conducted with the combination of the Expectation-Maximization algorithm for maximum \textit{a posteriori} estimation and the Gibbs sampling iterative procedure. We additionally investigate several Bayesian criteria for selecting the optimal mixture configuration and describe diagnostic tools for assessing the fitness of ranking distributions conditionally and unconditionally on the number of ranked items. The utility of the novel Bayesian parametric Plackett-Luce mixture for characterizing sample heterogeneity is illustrated with several applications to simulated and real preference ranked data. We compare our method with the frequentist approach and a Bayesian nonparametric mixture model both assuming the Plackett-Luce model as a mixture component. Our analysis on real datasets reveals the importance of an accurate diagnostic check for an appropriate in-depth understanding of the heterogenous nature of the partial ranking data., Comment: 36 pages, 7 figures, 11 tables

We propose the use of probability models for ranked data as a useful alternative to a quantitative data analysis to investigate the outcome of bioassay experiments, when the preliminary choice of an appropriate normalization method for the raw numerical responses is difficult or subject to criticism. We review standard distance-based and multistage ranking models and in this last context we propose an original generalization of the Plackett-Luce model to account for the order of the ranking elicitation process. The usefulness of the novel model is illustrated with its maximum likelihood estimation for a real data set. Specifically, we address the heterogeneous nature of experimental units via model-based clustering and detail the necessary steps for a successful likelihood maximization through a hybrid version of the Expectation-Maximization algorithm. The performance of the mixture model using the new distribution as mixture components is compared with those relative to alternative mixture models for random rankings. A discussion on the interpretation of the identified clusters and a comparison with more standard quantitative approaches are finally provided., Comment: (revised to properly include references)