Inferring Attack Relations for Gradual Semantics

Peer reviewedPublisher PD

Reliable statistical modeling of weakly structured information

The statistical analysis of "real-world" data is often confronted with the fact that most standard statistical methods were developed under some kind of idealization of the data that is often not adequate in practical situations. This concerns among others i) the potentially deficient quality of the data that can arise for example due to measurement error, non-response in surveys or data processing errors and ii) the scale quality of the data, that is idealized as "the data have some clear scale of measurement that can be uniquely located within the scale hierarchy of Stevens (or that of Narens and Luce or Orth)". Modern statistical methods like, e.g., correction techniques for measurement error or robust methods cope with issue i). In the context of missing or coarsened data, imputation techniques and methods that explicitly model the missing/coarsening process are nowadays wellestablished tools of refined data analysis. Concerning ii) the typical statistical viewpoint is a more pragmatical one, in case of doubt one simply presumes the strongest scale of measurement that is clearly "justified". In more complex situations, like for example in the context of the analysis of ranking data, statisticians often simply do not worry about purely measurement theoretic reservations too much, but instead embed the data structure in an appropriate, easy to handle space, like e.g. a metric space and then use all statistical tools available for this space. Against this background, the present cumulative dissertation tries to contribute from different perspectives to the appropriate handling of data that challenge the above-mentioned idealizations. A focus here is on the one hand on analysis of interval-valued and set-valued data within the methodology of partial identification, and on the other hand on the analysis of data with values in a partially ordered set (poset-valued data). Further tools of statistical modeling treated in the dissertation are necessity measures in the context of possibility theory and concepts of stochastic dominance for poset-valued data. The present dissertation consists of 8 contributions, which will be detailedly discussed in the following sections: Contribution 1 analyzes different identification regions for partially identified linear models under interval-valued responses and develops a further kind of identification region (as well as a corresponding estimator). Estimates for the identifcation regions are compared to each other and also to classical statistical approaches for a data set on wine quality. Contribution 2 deals with logistic regression under coarsened responses, analyzes point-identifying assumptions and develops likelihood-based estimators for the identified set. The methods are illustrated with data of a wave of the panel study "Labor Market and Social Security" (PASS). Contribution 3 analyzes the combinatorial structure of the extreme points and the edges of a polytope (called credal set or core in the literature) that plays a crucial role in imprecise probability theory. Furthermore, an efficient algorithm for enumerating all extreme points is given and compared to existing standard methods. Contribution 4 develops a quantile concept for data or random variables with values in a complete lattice, which is applied in Contribution 5 to the case of ranking data in the context of a data set on the wisdom of the crowd phenomena. In Contribution 6 a framework for evaluating the quality of different aggregation functions of Social Choice Theory is developed, which enables analysis of quality in dependence of group specific homogeneity. In a simulation study, selected aggregation functions, including an aggregation function based on the concepts of Contribution 4 and Contribution 5, are analyzed. Contribution 7 supplies a linear program that allows for detecting stochastic dominance for poset-valued random variables, gives proposals for inference and regularization, and generalizes the approach to the general task of optimizing a linear function on a closure system. The generality of the developed methods is illustrated with data examples in the context of multivariate inequality analysis, item impact and differential item functioning in the context of item response theory, analyzing distributional differences in spatial statistics and guided regularization in the context of cognitive diagnosis models. Contribution 8 uses concepts of stochastic dominance to establish a descriptive approach for a relational analysis of person ability and item difficulty in the context of multidimensional item response theory. All developed methods have been implemented in the language R ([R Development Core Team, 2014]) and are available from the author upon request. The application examples corroborate the usefulness of weak types of statistical modeling examined in this thesis, which, beyond their flexibility to deal with many kinds of data deficiency, can still lead to informative substance matter conclusions that are then more reliable due to the weak modeling.Die statistische Analyse real erhobener Daten sieht sich oft mit der Tatsache konfrontiert, dass übliche statistische Standardmethoden unter einer starken Idealisierung der Datensituation entwickelt wurden, die in der Praxis jedoch oft nicht angemessen ist. Dies betrifft i) die möglicherweise defizitäre Qualität der Daten, die beispielsweise durch Vorhandensein von Messfehlern, durch systematischen Antwortausfall im Kontext sozialwissenschaftlicher Erhebungen oder auch durch Fehler während der Datenverarbeitung bedingt ist und ii) die Skalenqualität der Daten an sich: Viele Datensituationen lassen sich nicht in die einfachen Skalenhierarchien von Stevens (oder die von Narens und Luce oder Orth) einordnen. Modernere statistische Verfahren wie beispielsweise Messfehlerkorrekturverfahren oder robuste Methoden versuchen, der Idealisierung der Datenqualität im Nachhinein Rechnung zu tragen. Im Zusammenhang mit fehlenden bzw. intervallzensierten Daten haben sich Imputationsverfahren zur Vervollständigung fehlender Werte bzw. Verfahren, die den Entstehungprozess der vergröberten Daten explizit modellieren, durchgesetzt. In Bezug auf die Skalenqualität geht die Statistik meist eher pragmatisch vor, im Zweifelsfall wird das niedrigste Skalenniveau gewählt, das klar gerechtfertigt ist. In komplexeren multivariaten Situationen, wie beispielsweise der Analyse von Ranking-Daten, die kaum noch in das Stevensche "Korsett" gezwungen werden können, bedient man sich oft der einfachen Idee der Einbettung der Daten in einen geeigneten metrischen Raum, um dann anschließend alle Werkzeuge metrischer Modellierung nutzen zu können. Vor diesem Hintergrund hat die hier vorgelegte kumulative Dissertation deshalb zum Ziel, aus verschiedenen Blickwinkeln Beiträge zum adäquaten Umgang mit Daten, die jene Idealisierungen herausfordern, zu leisten. Dabei steht hier vor allem die Analyse intervallwertiger bzw. mengenwertiger Daten mittels partieller Identifikation auf der Seite defzitärer Datenqualität im Vordergrund, während bezüglich Skalenqualität der Fall von verbandswertigen Daten behandelt wird. Als weitere Werkzeuge statistischer Modellierung werden hier insbesondere Necessity-Maße im Rahmen der Imprecise Probabilities und Konzepte stochastischer Dominanz für Zufallsvariablen mit Werten in einer partiell geordneten Menge betrachtet. Die vorliegende Dissertation umfasst 8 Beiträge, die in den folgenden Kapiteln näher diskutiert werden: Beitrag 1 analysiert verschiedene Identifikationsregionen für partiell identifizierte lineare Modelle unter intervallwertig beobachteter Responsevariable und schlägt eine neue Identifikationsregion (inklusive Schätzer) vor. Für einen Datensatz, der die Qualität von verschiedenen Rotweinen, gegeben durch ExpertInnenurteile, in Abhängigkeit von verschiedenen physikochemischen Eigenschaften beschreibt, werden Schätzungen für die Identifikationsregionen analysiert. Die Ergebnisse werden ebenfalls mit den Ergebissen klassischer Methoden für Intervalldaten verglichen. Beitrag 2 behandelt logistische Regression unter vergröberter Responsevariable, analysiert punktidentifizierende Annahmen und entwickelt likelihoodbasierte Schätzer für die entsprechenden Identifikationsregionen. Die Methode wird mit Daten einer Welle der Panelstudie "Arbeitsmarkt und Soziale Sicherung" (PASS) illustriert. Beitrag 3 analysiert die kombinatorische Struktur der Extrempunkte und der Kanten eines Polytops (sogenannte Struktur bzw. Kern einer Intervallwahrscheinlichkeit bzw. einer nicht-additiven Mengenfunktion), das von wesentlicher Bedeutung in vielen Gebieten der Imprecise Probability Theory ist. Ein effizienter Algorithmus zur Enumeration aller Extrempunkte wird ebenfalls gegeben und mit existierenden Standardenumerationsmethoden verglichen. In Beitrag 4 wird ein Quantilkonzept für verbandswertige Daten bzw. Zufallsvariablen vorgestellt. Dieses Quantilkonzept wird in Beitrag 5 auf Ranking-Daten im Zusammenhang mit einem Datensatz, der das "Weisheit der Vielen"-Phänomen untersucht, angewendet. Beitrag 6 entwickelt eine Methode zur probabilistischen Analyse der "Qualität" verschiedener Aggregationsfunktionen der Social Choice Theory. Die Analyse wird hier in Abhäangigkeit der Homogenität der betrachteten Gruppen durchgeführt. In einer simulationsbasierten Studie werden exemplarisch verschiedene klassische Aggregationsfunktionen, sowie eine neue Aggregationsfunktion basierend auf den Beiträgen 4 und 5, verglichen. Beitrag 7 stellt einen Ansatz vor, um das Vorliegen stochastischer Dominanz zwischen zwei Zufallsvariablen zu überprüfen. Der Anstaz nutzt Techniken linearer Programmierung. Weiterhin werden Vorschläge für statistische Inferenz und Regularisierung gemacht. Die Methode wird anschließend auch auf den allgemeineren Fall des Optimierens einer linearen Funktion auf einem Hüllensystem ausgeweitet. Die flexible Anwendbarkeit wird durch verschiedene Anwendungsbeispiele illustriert. Beitrag 8 nutzt Ideen stochastischer Dominanz, um Datensätze der multidimensionalen Item Response Theory relational zu analysieren, indem Paare von sich gegenseitig empirisch stützenden Fähigkeitsrelationen der Personen und Schwierigkeitsrelationen der Aufgaben entwickelt werden. Alle entwickelten Methoden wurden in R ([R Development Core Team, 2014]) implementiert. Die Anwendungsbeispiele zeigen die Flexibilität der hier betrachteten Methoden relationaler bzw. "schwacher" Modellierung insbesondere zur Behandlung defizitärer Daten und unterstreichen die Tatsache, dass auch mit Methoden schwacher Modellierung oft immer noch nichttriviale substanzwissenschaftliche Rückschlüsse möglich sind, die aufgrund der inhaltlich vorsichtigeren Modellierung dann auch sehr viel stärker belastbar sind

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