Robust and Efficient Estimation in Ordinal Response Models using the Density Power Divergence

In real life, we frequently come across data sets that involve some independent explanatory variable(s) generating a set of ordinal responses. These ordinal responses may correspond to an underlying continuous latent variable, which is linearly related to the covariate(s), and takes a particular (ordinal) label depending on whether this latent variable takes value in some suitable interval specified by a pair of (unknown) cut-offs. The most efficient way of estimating the unknown parameters (i.e., the regression coefficients and the cut-offs) is the method of maximum likelihood (ML). However, contamination in the data set either in the form of misspecification of ordinal responses, or the unboundedness of the covariate(s), might destabilize the likelihood function to a great extent where the ML based methodology might lead to completely unreliable inferences. In this paper, we explore a minimum distance estimation procedure based on the popular density power divergence (DPD) to yield robust parameter estimates for the ordinal response model. This paper highlights how the resulting estimator, namely the minimum DPD estimator (MDPDE), can be used as a practical robust alternative to the classical procedures based on the ML. We rigorously develop several theoretical properties of this estimator, and provide extensive simulations to substantiate the theory developed

Finite mixtures for the modelling of heterogeneity in ordinal response

Die Modellierung von Heterogenität ist ein entscheidender Aspekt in jeder statistischen Analyse. Um ein geeignetes Modell zu finden, ist es notwendig, möglichst alle relevanten Strukturen und Einflussgrößen einzubeziehen. Die meisten statistischen Modelle können leicht beobachtete Strukturen einbinden, jedoch haben sie oft Schwierigkeiten latente Strukturen abzubilden. Misch-Modelle können Heterogenität berücksichtigen, die aus zugrunde liegenden latenten Strukturen entstehen, wie etwa die unbeobachtete Zugehörigkeit zu verschiedenen Gruppen oder unterschiedliches Antwortverhalten. Mit dieser Doktorarbeit möchte ich einen Beitrag für die Verwendung von Misch-Modellen zur Modellierung von Heterogenität bei ordinalen Zielgrößen leisten und Variablen Selektion in diesem Kontext durchführen. Zuerst konzentriere ich mich auf Heterogenität, die bei Umfragen auftritt, wenn beispielsweise die Befragten bei der Wahl einer bestimmten geordneten Kategorie unsicher sind. In diesem Fall bestehen die Misch-Modelle üblicherweise aus einer Präferenz-Komponente und einer Unsicherheits-Komponente. Ein Gewicht bestimmt die Neigung jeder Person zu einer dieser beiden Komponenten zu gehören. Das existierende CUB Modell verwendet eine verschobene Binomialverteilung für die erste und eine Gleichverteilung für die zweite Komponente. Im vorgeschlagenem CUP Modell wird die Präferenz-Komponente mit einem beliebigen ordinalen Modell wie dem kumulativen Logit Modell ersetzt, um eine höhere Flexibilität in der Präferenz-Komponente zu erreichen. Im BetaBin Modell wird das Konzept der Unsicherheit als zufällige Wahl einer Kategorie so erweitert, dass Unsicherheit auch die Tendenz zu der zentralen Kategorie und extremen Kategorien erfasst. Auf diese Weise wird die Gleichverteilung des CUP Modells durch einer flexiblere, beschränkte Beta-Binomial Verteilung ersetzt. Als zweites zeige ich, wie diskrete Cure Modelle verwendet werden können, um in der Survival-Analyse für diskrete Zeit mit Heterogenität umzugehen, die aus der unbeobachteten Zugehörigkeit zu verschiedenen Gruppen entsteht. "Cure" bezeichnet dabei den Umstand, dass eine Gruppe von Beobachtungen "geheilt ist" oder als sogenannte Langzeit-Überlebende charakterisiert ist, während die andere Gruppe dem Risiko des Ereignisses wie zum Beispiel "Eintritt von Arbeitslosigkeit" ausgesetzt ist. Die Zugehörigkeit zu dieser Gruppe ist unbekannt. Cure Modelle schätzen die Wahrscheinlichkeit zur Nicht-geheilten Population zu gehören und die Form der Survival Funktion für die Beobachtungen unter Risiko. Drittens führe ich Variablen Selektion für das CUB, CUP und das Cure Modell mit Hilfe von Penalisierung und teilweise schrittweise Selektionsverfahren durch. Die Herausforderung liegt insbesondere darin zu entscheiden, welche Variablen in welche Komponente des Misch-Modells aufgenommen werden sollen. Variablen können hier zum einen für die Schätzung der Gewichte der Komponenten und zum anderen für die Form einer oder zwei Misch-Komponenten verwendet werden. Es werden dafür spezifische Bestrafungsterme vorgestellt, die für das jeweilige Modell geeignet sind. Alle Modelle werden mit dem EM-Algorithmus geschätzt, der die unbekannte Zugehörigkeit zu einer der Komponenten als fehlende Daten behandelt. Es werden auch einige computationale Aspekte besprochen wie etwa mit der Initialisierung und der Konvergenz umzugehen ist. Die penalisierte Likelihood wird mit dem sogenannten FISTA Algorithmus geschätzt, da die Ableitungen der penalisierten Likelihood nicht existieren. Es werden sowohl Simulations-Studien als auch reelle Daten verwendet, um die Nützlichkeit der neuen Ansätze aufzuzeigen.Modelling heterogeneity is a crucial aspect of every statistical analysis. To find a reasonable model, it is necessary to include all relevant structures and explanatory variables. Most statistical models can easily include observed patterns but have often difficulties in dealing with latent structures. Mixture models can account for heterogeneity which arise from latent underlying structures, for example, the unobserved membership to different groups or different response styles. In this thesis, I contribute to the use of mixture models to model heterogeneity in ordinal response and perform variable selection in this context. First, I focus on heterogeneity, which occurs in surveys when, for instance, respondents are uncertain about choosing a certain ordered category. In this case, the mixture model traditionally consists of a preference component and an uncertainty component. A weight determines the propensity of each person belonging to one of these components. The traditional CUB model uses a shifted binomial distribution for the first and a uniform distribution for the later component. In the proposed CUP model, the preference component is replaced by any ordinal model, such as the cumulative logit model or the adjacent category model, to achieve more flexibility in the preference component. In the BetaBin model, the concept of uncertainty, understood as a random choice of a category, is extended in such a way that uncertainty can also capture the tendency to the middle and extreme categories. Thus, the uniform distribution of the CUP model is replaced by a more flexible restricted beta-binomial distribution. Second, I show how discrete cure models can be used for dealing with heterogeneity in the survival analysis for discrete time arising from the unobserved membership to different groups. "Cure" refers to the fact that one group of observations is "cured" or characterized as long-term survivors, while the other group is exposed to the risk of the event such as the "occurrence of unemployment". The membership to this group is unknown. Cure models estimate the probability for belonging to the non-cured population and the shape of the survival function of the observations under risk. Third, I perform variable selection for the CUB, the CUP and the cure model using penalization techniques and to some extend stepwise selection procedures. In particular, the challenge is to decide which variables should be included in which component of the mixture model. On the one hand, variables can be used to estimate the weights of the components and on the other hand, for the shape of one or two mixture components. Therefore, specific penalty terms are presented which are appropriate for the particular model. All models are estimated with the EM-Algorithm which treats the unknown membership to the components as missing data. I also address some computational issues, for instance, how to deal with initialization and convergence. The penalized likelihood is estimated with the so-called FISTA algorithm since the derivatives of the penalized likelihood do not exist. Both simulation studies and real data applications are used to demonstrate the usefulness of the new approaches

An Evaluation of a Community-Based Children’s Bereavement Group Using a Resilience Model

The death of a loved one can be an extremely painful process that can have detrimental consequences on the emotional well-being of individuals if they don’t receive grief intervention support. When it specifically comes to the emotional eudaemonia of children and teens experiencing the death of a loved one, research is limited on empirical-based bereavement intervention support to enhance their coping skills. Utilizing the theoretical framework of resilience theory, the purpose of this quantitative secondary research study was to evaluate the Sutter Sacramento Children’s Bereavement Art Group (CBAG) ten- week intervention for children and teens ages 5-12. The first question focused on the determination of the significant effect of the CBAG intervention to grief behavioral symptoms in participants ages 5-12. The second question centered on one subsystem of the research sample and addressed the question of if the effects of CBAG’s resilience model framework on coping skills on grief behavioral symptoms differed by gender in participants ages 5-12. This research study utilized secondary research with existing data collected from the CBAG program’s bibliotheca. The data were analyzed by a two-tailed signed Wilcoxon rank test. Moreover, the overall findings from this research determined if program participants from the CBAG program coping skills were influenced after bereavement intervention support and participant grief behavioral symptoms declined. The positive implications of social change occurring from this research study can provide further opportunities to promote and further advocacy and intervention services for bereaved children and teens

An Evaluation of a Community-Based Children’s Bereavement Group Using a Resilience Model

The death of a loved one can be an extremely painful process that can have detrimental consequences on the emotional well-being of individuals if they don’t receive grief intervention support. When it specifically comes to the emotional eudaemonia of children and teens experiencing the death of a loved one, research is limited on empirical-based bereavement intervention support to enhance their coping skills. Utilizing the theoretical framework of resilience theory, the purpose of this quantitative secondary research study was to evaluate the Sutter Sacramento Children’s Bereavement Art Group (CBAG) ten- week intervention for children and teens ages 5-12. The first question focused on the determination of the significant effect of the CBAG intervention to grief behavioral symptoms in participants ages 5-12. The second question centered on one subsystem of the research sample and addressed the question of if the effects of CBAG’s resilience model framework on coping skills on grief behavioral symptoms differed by gender in participants ages 5-12. This research study utilized secondary research with existing data collected from the CBAG program’s bibliotheca. The data were analyzed by a two-tailed signed Wilcoxon rank test. Moreover, the overall findings from this research determined if program participants from the CBAG program coping skills were influenced after bereavement intervention support and participant grief behavioral symptoms declined. The positive implications of social change occurring from this research study can provide further opportunities to promote and further advocacy and intervention services for bereaved children and teens

SIS 2017. Statistics and Data Science: new challenges, new generations

The 2017 SIS Conference aims to highlight the crucial role of the Statistics in Data Science. In this new domain of ‘meaning’ extracted from the data, the increasing amount of produced and available data in databases, nowadays, has brought new challenges. That involves different fields of statistics, machine learning, information and computer science, optimization, pattern recognition. These afford together a considerable contribute in the analysis of ‘Big data’, open data, relational and complex data, structured and no-structured. The interest is to collect the contributes which provide from the different domains of Statistics, in the high dimensional data quality validation, sampling extraction, dimensional reduction, pattern selection, data modelling, testing hypotheses and confirming conclusions drawn from the data

A generalized framework for modelling ordinal data

In several applied disciplines, as Economics, Marketing, Business, Sociology, Psychology, Political science, Environmental research and Medicine, it is common to collect data in the form of ordered categorical observations. In this paper, we introduce a class of models based on mixtures of discrete random variables in order to specify a general framework for the statistical analysis of this kind of data. The structure of these models allows the interpretation of the final response as related to feeling, uncertainty and a possible shelter option and the expression of the relationship among these components and subjects’ covariates. Such a model may be effectively estimated by maximum likelihood methods leading to asymptotically efficient inference. We present a simulation experiment and discuss a real case study to check the consistency and the usefulness of the approach. Some final considerations conclude the paper