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In this article, we present the command cub, which fits ordinal rating data using combination of uniform and binomial (CUB) models, a class of finite mixture distributions accounting for both feeling and uncertainty of the response process. CUB identifies the components that define the mixture in the baseline model specification. We apply maximum likelihood methods to estimate feeling and uncertainty parameters, which are possibly explained in terms of covariates. An extension to inflated CUB models is discussed. We also present a subcommand, scattercub, for visualization of results. We then illustrate the use of cub using a case study on students’ satisfaction for the orientation services provided by the University of Naples Federico II in Italy.

The paper proposes a method to perform diagnostics of model-based trees for preference and evaluation data on the basis of surrogate residual analysis for ordinal data models. The discussion stems from the introduction of binomial regression trees and discusses how to perform local diagnostics of misspecification against alternative model extensions within the framework of mixture models with uncertainty. Three case studies concerning customer satisfaction and perceived trust for information sources illustrate usefulness and versatile applicative extent of the proposal.

Background Drugs and their metabolites can be incorporated into hair so hair testing has become s an alternative and complementary method to assess the extent of drugs abuse. The aim of our study has been to use hair samples to rebuild the epidemiology of illicit substances use in Perugia. Methods We conducted a cross-sectional study from May to July 2012 asking hairdressers of Perugia to collect hair samples and to compile a schedule for each one. The samples were analyzed in laboratory: the extraction of basic substances was performed adding methanol; the extraction of acid substances was obtained adding NAOH. After derivatization, 1μl of each solution was analyzed through gas chromatography / mass spectrometry The data were organized in a database and processed using R version 3.2.2. Results We collected 238 samples. The most detected drugs were: THC-TMS identified in 15 samples, MDMA in 9, Beg-TMS in 8. There is a statistically significant difference in drug use between centre (23,36%) and periphery (5,34%). Age and sex have not represented influencing factors. The substance with the highest concentration in the keratin matrix is ketamine (9834,86 ng/100 mg of hair). Conclusions The use of keratin matrix offers high advantages in the toxicological analysis compared to conventional biological matrices and let us analyze the situation in Perugia where the increased market of drugs caused constant rise of drug addiction. However, because of the limits of this method, only the simultaneous use of keratin matrix and other traditional indicators, could furnish more precise information.

The paper discusses the specification of finite mixture models based on the Discretized Beta distribution for the analysis of ordered discrete responses, as ratings and count data. The ultimate goal of the paper is to parameterize clusters of opposite and intermediate response outcomes. After a thorough discussion on model interpretation, identifiability and estimation, the proposal is illustrated on the wake of a case study on the probability to vote for German Political Parties and with a comparative discussion with the state of the art.

Population aging is evolving at different rates across countries and over time, and it represents a long-term challenge for both the sustainability of pension schemes and for the realization of public intergenerational transfers. In this context, this work focuses on gender differences in survival at older ages. Specifically, we implement a comparative analysis of OECD countries to assess the adequacy of the corresponding gender-specific normal retirement age when faced with growing life expectancy. The analysis hinges on several graphical representations and is motivated by recent findings on Italian longevity to determine optimal retirement age shifts necessary to match growing life expectancy at older ages while accounting for model risk for mortality projections. Our analysis determines—at the country level—the extent to which adjustments to the normal retirement age are advisable for the sustainability of the intergenerational paradigm for pensions. The study considers males and females separately because most of these countries are characterized by aging societies where men and women have different crucial characteristics, including life expectancy. It is therefore important that policymakers have information on the future evolution of the longevity gender gap so they will be able to apply policies that preserve the principles of equality and solidarity and reduce the pension gender gap. We find groups of countries where gender gap in life expectancy follows the same dynamics.

The paper is the rejoinder to a series of Discussions on the class of cub models for rating data. The main topics advanced by Discussants are reviewed and debated, with focus on the most prominent issues. As a result, the trailhead of possible future research developments is outlined.

This paper discusses a general framework for the analysis of rating and preference data that is rooted on a class of mixtures of discrete random variables. These models have been extensively studied and applied in the last 15 years thanks to a flexible and parsimonious parametrization of data generating process and to prompt interpretation of results. The approach considers the final response as the combination of feeling and uncertainty, by allowing for finer model specifications to include refuge options, response styles and possible overdispersion, also in relation to subjects’ and objects’ covariates. The article establishes the state of art of the research inherent to this paradigm, in terms of methodology, inferential procedures and fitting measures, by emphasizing capabilities and limitations yet establishing new findings. In particular, explicative power and predictive performances of cub statistical models for ordinal data are examined and new topics that could boost and support the modelling of uncertainty in this framework are provided. Possible developments are outlined throughout the whole presentation and final comments conclude the paper.

The paper is framed within the literature around Louis’ identity for the observed information matrix in incomplete data problems, with a focus on the implied acceleration of maximum likelihood estimation for mixture models. The goal is twofold: to obtain direct expressions for standard errors of parameters from the EM algorithm and to reduce the computational burden of the estimation procedure for a class of mixture models with uncertainty for rating variables. This achievement fosters the feasibility of best-subset variable selection, which is an advisable strategy to identify response patterns from regression models for all Mixtures of Experts systems. The discussion is supported by simulation experiments and a real case study.

In dealing with veracity of data analytics, fuzzy methods are more and more relying on probabilistic and statistical techniques to underpin their applicability. Conversely, standard statistical models usually disregard to take into account the inherent fuzziness of choices and this issue is particularly worthy of note in customers' satisfaction surveys, since there are different shades of evaluations that classical statistical tools fail to catch. Given these motivations, the paper introduces a model-based fuzzy analysis of questionnaire with sound statistical foundation, driven by the design of a hybrid method that sets in between fuzzy evaluation systems and statistical modelling. The proposal is advanced on the basis of \cub mixture models to account for uncertainty in ordinal data analysis and moves within the general framework of Intuitionistic Fuzzy Set theory to allow membership, non-membership, vagueness and accuracy assessments. Particular emphasis is given to defuzzification procedures that enable uncertainty measures also at an aggregated level. An application to a survey run at the University of Naples Federico II about the evaluation of Orientation Services supports the efficacy of the proposal., Comment: 22 pages, 9 figures, 8 Tables

In the analysis of questionnaire-based evaluation of sport preferences, measurements of sport participation, opinions on social implications such as resurgence of racism, violence in stadiums and doping, the need arises to establish connections among motivations, subjects’ characteristics and responses. In this setting, the article deals with a selection of statistical models suitable to analyse sport rating data in which clusters of opposite responses are observed. Specifically, a two-component mixture of inverse hypergeometric (MIHG) distributions will be introduced and tested against competing models in order to yield a multifold interpretation of results. The ultimate comparative analysis will consider discrete models with a specific focus on those accounting for both uncertainty and feeling of self-evaluation in presence of inflation at the extreme categories. After a brief review of the methods, the proposal will be discussed both on ranking and rating data on the basis of two surveys on sport preferences and on measurements of sport activity: the identification of clusters of respondents with opposite choices will be investigated also in terms of covariates by comparing fitting performances of the selected models. The conclusions and insights offered by the study can be exploited to design plans of action for some specific policy or marketing strategy.

Mixture models for ordinal responses in the tradition of cub models use the uniform distribution to account for uncertainty of respondents. A model is proposed that uses more flexible distributions in the uncertainty component: the discretized Beta distribution allows to account for response styles, in particular the preference for middle or extreme categories. The proposal is compared with traditional cub models in simulation studies and its use is illustrated by two applications.

Social inclusion and integration are prominent topics in social sciences, and specifically in the educational field, in order to contrast the experience of inequality conditions at school. Using data from the 2015 national survey on “Integration of the second generation”, the paper proposes a synthetic indicator of educational integration of second-generation children in Italy. The chosen methodology is suitable to model subjective assessments on rating scales while accounting for both agreement and heterogeneity in response patterns. L’inclusione sociale e l’integrazione rappresentano temi cruciali nelle scienze sociali ed assumono rilievo specifico nell’ambito dell’istruzione. Utilizzando i dati dell’Indagine Istat sull’integrazione delle seconde generazioni (ISG2015), lo studio propone un indicatore sintetico di integrazione scolastica che considera sia la percezione del tratto in esame che l’eterogeneità delle risposte.

Ordinal responses in the form of ratings arise frequently in social sciences, marketing and business applications where preferences, opinions and perceptions play a major role and it is common to collect along with rater’s evaluation, a set of covariates that characterize the respondent and/or the item/service. In this framework, the ordinal nature of the response has to be properly taken into account when the interest is in the understanding of different response patterns in terms of subjects’ covariates. In this spirit, a model-based tree procedure for ordinal scores is illustrated: its structure is based on a class of mixture models for ordinal rating data that implies a twofold analysis in terms of feeling and uncertainty and effective graphi- cal visualization of results. The chosen modelling framework entails that the splitting criterion can be customized according to the purposes of the study and the available data. Thus, the selection of variables yielding to the best partitioning results is driven by fitting measures or classical likelihood and deviance measurements, for instance. In order to illustrate the performances of the different splitting criterion for the cubremot procedure, we consider data from the 5th European Working Condition Survey carried out by Eu- rofound in 2010, comparison with alternative approaches that grow trees for ordinal responses is also outlined.

Traditional statistical models with random effects account for heterogeneity in the population with respect to the location of the response in a subject-specific way. This approach ignores that also uncertainty of the responses can vary across individuals and items: for example, subject-specific indecision may play a role in the rating process relative to questionnaire items. In this setting, a generalized mixture model is advanced that accounts for subjective heterogeneity in response behaviour for multivariate ordinal responses: to this aim, random effects are specified for the individual propensity to a structured or an uncertain response attitude. Simulations and a case study illustrate the effectiveness of the proposed model and its implications.

Statistical modelling of earthquakes is a challenging and delicate topic: research activity is vivid in this respect, and tailored to an improved understanding of the seismic phenomena and of their dynamics over time and space in all its shades. By surfing on some of the available literature, a critical investigation of the probability distributions best fitting earthquake sizes and inter-arrival times is performed, by using data on the Pacific Ring of Fire as illustrative example. As a by-product of our analysis, new ideas about adequate modelling of earthquake sizes and inter-event times together with the location of the earthquakes are advanced, which in turn could pave the way to further developments in a directional perspective.

Purpose This paper aims to measure the financial impact on social security system of a recently proposed indexation mechanism for retirement age by considering the Italian longevity experience. The analysis is motivated by the progressive increase in life expectancy at advanced age, which is rapidly bringing to the fore noticeable socio-economic consequences in most industrialized countries. Among those, the impact on National Social Security systems is particularly relevant if people live longer than expected; this will lead to greater financial exposure for pension providers. Design/methodology/approach Referring to the Italian population for illustrative purposes, the authors contemplate different scenarios for mortality projection methods and for the implementation of pension age shift while accounting for gender and cohort gaps and model risk. Synthetic indicators to measure the impact of the indexation mechanism on social security system are introduced on the basis of pension cash flows. Findings An indexation policy that manages gender gap while adjusting retirement age for varying life expectancy is proposed. As a result, sustainability of public retirement expenditure is improved. Originality/value The paper is a concise scenario analysis of the reduction of costs and risks that pension providers would have if the system resorted to link retirement age to life expectancy. The ideas fostered by the paper follow a recent proposal of the Authors on a flexible retirement scheme that deals with model risk for mortality projection and accounts for gender gap in mortality rates.

The focus of the contribution is on the splitting criterion for model-based treeprocedure based on the class ofCUBmixture models for the analysis of ordinal scores. Theflexibility of the chosen modelling framework allows to select the splitting criterion to growthe tree according to the purposes of the study and the available data. In particular, theselection of variables yielding to the best partitioning results can be driven by fitting measuresor classical likelihood and deviance measures. The contribution proposes to investigate thefeatures of the available decision rules by a set of Montecarlo experiments, thus implicitlyfacing the problem of selecting the model-based tree to obtain an adequate and satisfyingoverview of response profiles.

The management of National Social Security Systems is being challenged more and more by the rapid ageing of the population, especially in the industrialized countries. In order to chase the Pension System sustainability, several countries in Europe are setting up pension reforms linking the retirement age and/or benefits to life expectancy. In this context, the accurate modelling and projection of mortality rates and life expectancy play a central role and represent issues of great interest in recent literature. Our study refers to the Italian mortality experience and considers an indexing mechanism based on the expected residual life to adjust the retirement age and keep costs at an expected budgeted level, in the spirit of sharing the longevity risk between Social Security Systems and retirees. In order to combine fitting and projections performances of selected stochastic mortality models, a model assembling technique is applied to face uncertainty in model selection, while accounting for uncertainty of estimation as well. The resulting proposal is an averaged model that is suitable to discuss about the gender gap in longevity risk and its alleged narrowing over time.

UB models are a class of mixture distributions for analyzing ordinal responses in the form of ratings or marginal ranking that designs the decision process as the combination of two main components: a personal feeling and an inherent uncertainty. This methodology is receiving an increasing interest due to successful applications to the analysis of evaluations, opinions and perceptions, in various fields of research, and thanks to advantageous graphical interpretation of results. Currently the estimation of CUB models can be run by the R library ’CUB’ ver. 1.1.3. In this paper a gretl package for CUB models is presented, all the estimation examples and graphs proposed in the R vignette are replicated.

In this paper a novel approach is proposed for ranking data analysis that is based on marginal distributions. The marginal scores for one of the items to be ranked can be considered as a proxy of preference ratings and thus they can be employed to investigate drivers of the outcome. The rationale is that a model for the univariate marginal distributions should suitably convey information about the veracity of ordinal scores given that it would disregard the ranking constraints. The method relies on cubremot ( cub re gression mo del t rees), a tool for growing trees for ordinal resposes based on the local estimation of cub models, a class of models able to explain both preferences and related uncertainty. Here we propose the Uncertainty Tree to disclose preference patterns on the basis of the model uncertainty component. Two applications on real data are discussed; comparison with other model-based trees is provided and a synthetic index for the selection of the best cubremot is also advanced.

Evaluation surveys are often repeated over time in order to check for trends in subjects’ behaviors and opinions. The paper proposes a dynamic model for the serial correlation of ratings’ intrinsic components, which is discussed on the basis of time series of price expectations in Italy collected within a survey organized by ISTAT.

The paper introduces a new technique for growing trees for ordinal responses in the model-based framework. We consider the class of cub mixtures which is particularly appropriate to model perceptions and evaluations, as it designs the response process as the combination of a personal feeling and an inherent uncertainty. In our proposal, the partitioning process is based on the local estimation of cub regression models to profile responses according to feeling and uncertainty conditional to the splitting variables. In this regard, two alternative splitting criteria are introduced featuring both inferential and fitting issues. Moreover, the chosen modelling framework allows for advantageous visualization of the classification results. The proposal is illustrated using real data from a survey conducted by the Italian National Statistical Office, with focus on perceived trust towards the European Parliament. A benchmark study is conducted to settle the proposal among the available tree methods for ordinal responses.

In social sciences the need often arises to compare and rank groups of respondents by analyzing huge amounts of data and composite indicators are amongst the most effective tools. The paper aims to design an original procedure suitable for ordinal data and able to synthesize subjective evaluations while accounting for both agreement and heterogeneity in response patterns. A composite indicator for ordinal data based on cub models is introduced: the proposal discloses and preserves the heterogeneity also at an aggregated level. Empirical evidence relies on perceived discrimination analysis stemming from the Special Eurobarometer Survey 2015.

Adoption of soil conservation practices is promoted by increased engagement between researchers and stakeholders. By reporting a case study from southern Italy where farmers have been involved in research projects dealing with soil conservation over several years, we demonstrated that the rate of adoption of conservative technologies is positively linked to the degree of stakeholder participation in the project and that farmers (and other stakeholders) have been driven toward more conscious perception of the complex link between agricultural practices, environmental impacts and socioeconomic effects. The results of this study reinforce the pivotal role of effective participatory processes in soil conservation&mdash, evidencing the importance of (i) time required to build relationships and (ii) intensity (and persistence) of collaboration among similar initiatives. Empirical evidence highlights the need to combine different approaches to deal with soil degradation. As well as the participatory time length, the synergies between the methods utilized, and the range of interacting mind sets are also very important.

CUBREMOT (CUB REgression MOdel Trees) is a model-based approach to grow trees for ordinal responses that relies on a class of mixture models for evaluations and preferences (CUB). The original proposal considers deviances in loglikelihood to partition observations. In the present paper a new splitting criterion is introduced that, among the significant splitting variables, chooses the one that maximizes a dissimilarity measure. This choice is tailored to generating child nodes as far apart as possible with respect to the estimated probability distributions. An application to real data on Italians’ trust towards the European Parliament taken from the official survey on daily life conducted by the Italian National Institute of Statistics (ISTAT) in 2015 is presented and discussed in comparison with alternative methods.

Nowadays, we are witnessing a wide and spread need to create flexible retirement schemes for facing global ageing and the prolonging working lives. Many countries have set up Social Security Systems, which link retirement age and/or pension benefits to life expectancy. In this context, we consider an indexing mechanism based on the expected residual life expectancy to adjust the retirement age and keep a constant Expected Pension Period Duration (EPPD). The analysis assesses the impact of different stochastic mortality models on the indexation by forecasting mortality paths based on extrapolative methods. Nevertheless, so far, in recent literature less attention has been given to the uncertainty issue related to model selection, although having appropriate estimates for the risk in mortality projections. With respect to the state of art, our proposal considers model-assembling techniques in order to balance fitting performances and uncertainty related to model selection, as well as uncertainty of parameters estimation.

Traditional models for ordinal data (as CUB models or cumulative models with logit/probit link, among others) present limits in explaining the surplus of zero observations, especially when the zeros may relate to two distinct situations of non-participation/inactivity and infrequent participation, for instance. We propose an extension of standard models: zero-inflated CUB models and zero inflated ordered cumulative (ZIOC) probit/logit models handling the GECUB models and using a double-hurdle combination of a split (logit/probit) model and an ordered probit/logit models, respectively. Both extensions, potentially, relate to different sets of covariates. Finally, models are applied to Sport surveys. Specifically the paper explores the determinants of sport (in)activity: the frequency and the probability of sports participation. It distinguishes between genuine ‘non-participants’ and the ones who do not participate at a time but might under different circumstances.

Statistical literature is being more and more concerned with debates about hypothesis testing and p-values supporting the significance of a given variable specification. Specifically, if on one hand statistical foundations about significance are not arguable, scholars should be able to distinguish between significance and variable importance. This is a matter of serious concern in questionnaire analysis to derive respondents’ profiles and develop targeted marketing strategies, for instance. To this aim, this contribution proposes a hypothesis system that considers the normalized dissimilarity measure to assess the importance of explanatory variables in the setting of mixture models for ordinal data to account for uncertainty of choice.

In sample surveys where people are asked to express their opinions, a high level of indecision among respondents may generate sub-optimal statistical analyses caused by large heterogeneity in the responses. We discuss a model belonging to the class of generalized cub models that is suitable for this kind of surveys. Then, we examine a real case study where the observed heterogeneity as well as respondents’ indecision can be analyzed within the theoretical framework of the proposed model leading to convincing interpretations. A comparison with current literature and some concluding remarks end the paper.

International audience; The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to $3$. The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit thresholds, as well as establish several connections with the so-called universality phenomenon of probability theory. Both in the classical and free probability frameworks, our results extend and unify previous Fourth Moment Theorems for Gaussian and semicircular approximations. Our techniques are based on a fine combinatorial analysis of higher moments for homogeneous sums., Comment: 26 pages

We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring. Explicit determinantal formulae and multivariable extension of the Heine integral formula are stated. Moreover, a general family of covariants that includes transvectants is introduced. Such covariants turn out to be the average value of classical basis of symmetric polynomials over a set of roots of suitable orthogonal polynomials.

Non-Performing Loans (NPLs) are exposures in state of insolvency, i.e. loans whose collection by banks is uncertain in terms of the amount that will be returned and the time of return. In order to measure the recovery performance of a portfolio of NPLs, our idea is to draw a curve representing the recovery rates during time, here assumed discretized, for example, in years. In this way, the user can get simultaneously information about recovery rate and time to liquidate of the portfolio. However, the computation of this curve becomes infeasible when some elements of the portfolio are right censored, i.e. the recovery rate trajectory is known only until a particular year. In this talk, a new method of estimation of the recovery rate curve in case of right censored data is presented. It is studied from a theoretical point of view in a simulation study, where it is tested and compared with other methods. It is studied from a practical point of view, where it is applied to a real financial data set about some portfolios of Italian unsecured NPLs taken in charge by a specialized operator.

The literature on university-industry relations has gone beyond direct surveys by academic and industrial researchers, and now also focuses on co-patenting, co-working and research in coop eration. These activities represent specific channels of interaction, conveying knowledge flows of different “value”. Each forms of those interactions basically produce a different network and usually each network is analyzed and interpreted individually. In this study, we combine different collaboration relations among firms and among inventors (hired by individual firms/assignees). In particular, we use data on both patenting activities and research collaboration in projects among firms located in an Italian region, namely Friuli Venezia Giulia, which is characterized by a high concentration of research organizations and a lively sector of R&D small firms. Regional patenting activities give rise to a co-patenting networks (inventors- by-inventors), whereas collaboration in research projects leads to a firm-by-firm network. The combination of these two peculiar university-industry interactions when combined with affiliation network of investors and firms naturally leads to a multilevel network. The paper aims at exploring the underlying structure of the firms’ relationships in the region by means of a block- modeling approach suitable for linked/multilevel networks. Finally, we project all networks to the level of firms in order to verify the hypothesis that in- ventors’ cooperation on patents increases the probability of a tie between firms based on projects. The blockmodeling approach used allow us to show how cluster of inventors and firms at different levels are connected.

Preference data are a particular type of ranking data where some subjects (voters, judges, ...) give their preferences over a set of alternatives (items). It happens, in most of the real cases, that some items receive the same preference by a judge, giving raise to a ranking with ties. The purpose of our paper is to investigate on the consensus between rankings with ties taking into account the importance of swapping elements belonging to the top (or to the bottom) of the ordering (position weights). Combining the structure of the Taux proposed by Emond and Mason and the class of weighted Kemeny-Snell distances, we propose a position weighted rank correlation coefficient to compare rankings with ties. The one-to-one correspondence between the weighted distance and the rank correlation coefficient holds, analytically speaking, using both equal and decreasing weights.