In this article we suggest a beta regression model that accounts for the degree of preference in paired comparisons measured on a bounded metric paired comparison scale. The beta distribution for bounded continuous random variables assumes values in the open unit interval (0,1). However, in practice we will observe paired comparison responses that lie within a fixed or arbitrary fixed interval [-a,a] with known value of a. We therefore transform the observed responses into the interval (0,1) and assume that these transformed responses are each a realization of a random variable which follows a beta distribution. We propose a simple paired comparison regression model for beta distributed variables which allows us to model the mean of the transformed response using a linear predictor and a logit link function -- where the linear predictor is defined by the parameters of the logit-linear Bradley-Terry model. For illustration we applied the presented model to a data set obtained from a student survey of learning related emotions in mathematics. (authors' abstract)

Ranking a set of candidates or items from pair-wise comparisons is a fundamental problem that arises in many settings such as elections, recommendation systems, sports team rankings, document rankings and so on. Indeed it is well known in the psychology literature that when a large number of items are to be ranked, it is easier for humans to give pair-wise comparisons as opposed to complete rankings. The problem of ranking from pair-wise comparisons has been studied in multiple communities such as machine learning, operations research, linear algebra, statistics etc., and several algorithms (both classic and recent) have been proposed. However, it is not well under-stood under what conditions these different algorithms perform well. In this thesis, we aim to fill this fundamental gap, by elucidating precise conditions under which different algorithms perform well, as well as giving new algorithms that provably perform well under broader conditions. In particular, we consider a natural statistical model wherein for every pair of items (i; j), there is a probability Pij such that each time items i and j are compared, item j beats item i with probability Pij . Such models, which we summarize through a matrix containing all these pair-wise probabilities, have been used explicitly or implicitly in much previous work in the area; we refer to the resulting matrix as the pair-wise preference matrix, and elucidate clearly the crucial role it plays in determining the performance of various algorithms. In the first part of the thesis, we consider a natural generative model where all pairs of items can be sampled and where the underlying preferences are assumed to be acyclic. Under this setting, we elucidate the conditions on the pair-wise preference matrix under which popular algorithms such as matrix Borda, spectral ranking, least squares and maximum likelihood under a Bradley-Terry-Luce (BTL) model produce optimal rankings that minimize the pair-wise disagreement error. Specifically, we derive explicit sample complexity bounds for each of these algorithms to output an optimal ranking under interesting subclasses of the class of all acyclic pair-wise preference matrices. We show that none of these popular algorithms is guaranteed to produce optimal rankings for all acyclic preference matrices. We then pro-pose a novel support vector machine based rank aggregation algorithm that provably does so. In the second part of the thesis, we consider the setting where preferences may contain cycles. Here, finding a ranking that minimizes the pairwise disagreement error is in general NP-hard. However, even in the presence of cycles, one may wish to rank 'good' items ahead of the rest. We develop a framework for this setting using notions of winners based on tournament solution concepts from social choice theory. We first show that none of the existing algorithms are guaranteed to rank winners ahead of the rest for popular tournament solution based winners such as top cycle, Copeland set, Markov set etc. We propose three algorithms - matrix Copeland, unweighted Markov and parametric Markov - which provably rank winners at the top for these popular tournament solutions. In addition to ranking winners at the top, we show that the rankings output by the matrix Copeland and the parametric Markov algorithms also minimize the pair-wise disagreement error for certain classes of acyclic preference matrices. Finally, in the third part of the thesis, we consider the setting where the number of items to be ranked is large and it is impractical to obtain comparisons among all pairs. Here, one samples a small set of pairs uniformly at random and compares each pair a fixed number of times; in particular, the goal is to come up with good algorithms that sample comparisons among only O(nlog(n)) item pairs (where n is the number of items). Unlike existing results for such settings, where one either assumes a noisy permutation model (under which there is a true underlying ranking and the outcome of every comparison differs from the true ranking with some fixed probability) or assumes a BTL or Thurstone model, we develop a general algorithmic framework based on ideas from matrix completion, termed low-rank pair-wise ranking, which provably produces an good ranking by comparing only O(nlog(n)) pairs, O(log(n)) times each, not only for popular classes of models such as BTL and Thurstone, but also for much more general classes of models wherein a suitable transform of the pair-wise probabilities leads to a low-rank matrix; this subsumes the guarantees of many previous algorithms in this setting. Overall, our results help to understand at a fundamental level the statistical properties of various algorithms for the problem of ranking from pair-wise comparisons, and under various natural settings, lead to novel algorithms with improved statistical guarantees compared to existing algorithms for this problem.

Each year millions of dollars are wagered on the NFL during the season. A few people make some money, but most often the only real winner is the sports book. In this project, the effect of data transformation on the paired comparison model of Glickman and Stern (1998) is explored. Usual transformations such as logarithm and square-root are used as well as a transformation involving a threshold. The motivation for each of the transformations if to reduce the influence of blowouts on future predictions. Data from the 2003 and 2004 NFL seasons are examined to see if these transformations aid in improving model fit and prediction rate against a point spread. Strategies for model-based wagering are also explored.

As Internet communities such as question-answer (Q&A) forums and online social networks (OSNs) grow in prominence as knowledge sources, traditional editorial filters are unable to scale to their size and pace. This absence hinders the exchange of knowledge online, by creating an understandable lack of trust in information. This mistrust can be partially overcome by a forum by consistently providing reliable information, thus establishing itself as a reliable source. This work investigates how algorithmic approaches can contribute to building such a community of voluntary experts willing to contribute authoritative information. This work identifies two approaches: a) reducing the cost of participation for experts via matching user queries to experts (question recommendation), and b) identifying authoritative contributors for incentivization (authority estimation). The question recommendation problem is addressed by extending existing approaches via a new generative model that augments textual data with expert preference information among different questions. Another contribution to this domain is the introduction of a set of formalized metrics to include the expert's experience besides the questioner's. This is essential for expert retention in a voluntary community, and has not been addressed by previous work. The authority estimation problem is addressed by observing that the global graph structure of user interactions, results from two factors: a user's performance in local one-to-one interactions, and their activity levels. By positing an intrinsic authority 'strength' for each user node in the graph that governs the outcome of individual interactions via the Bradley-Terry model for pairwise comparison, this research establishes a relationship between intrinsic user authority, and global measures of influence. This approach overcomes many drawbacks of current measures of node importance in OSNs by naturally correcting for user activity levels, and providing an explanation for the frequent disconnect between real world reputation and online influence. Also, while existing research has been restricted to node ranking on a single OSN graph, this work demonstrates that co-ranking across multiple endorsement graphs drawn from the same OSN is a highly effective approach for aggregating complementary graph information. A new scalable co-ranking framework is introduced for this task. The resulting algorithms are evaluated on data from various online communities, and empirically shown to outperform existing approaches by a large margin. / text

Paired comparison models in loglinear form are generalised linear models and can be fitted using the IWLS algorithm. Unfortunately, the design matrices can become very large and thus a method is needed to reduce computational load (relating to both space and time). This paper discusses an algorithm for fitting loglinear paired comparison models in the presence of many nuisance parameters which is based on partition rules for symmetric matrices and takes advantage of the special structure of the design matrix in Poisson loglinear models. The algorithm is implemented as an R function. Some simple examples illustrate its use for fitting both paired comparison models and (multinomial) logit models. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

The following dissertation presents a new paradigm for improving the training of machine learning algorithms, oracle learning. The main idea in oracle learning is that instead of training directly on a set of data, a learning model is trained to approximate a given oracle's behavior on a set of data. This can be beneficial in situations where it is easier to obtain an oracle than it is to use it at application time. It is shown that oracle learning can be applied to more effectively reduce the size of artificial neural networks, to more efficiently take advantage of domain experts by approximating them, and to adapt a problem more effectively to a machine learning algorithm.

National soccer teams are currently ranked by soccer's governing body, the Federation Internationale de Football Association (FIFA). Although the system used by FIFA is thorough, taking into account many different factors, many of the weights used in the system's calculations are somewhat arbitrary. It is investigated here how the use of a statistical model might better compare the teams for ranking purposes. By treating each game played as a pairwise comparison experiment and by using the Bradley-Terry model as a starting point some suitable models are presented. A key component of the final model introduced here its ability to differentiate between friendly matches and competitive matches when determining the impact of a match on a teams ranking. Posterior distributions of the rating parameters are obtained, and the rankings and results obtained from each model are compared to FIFA's rankings and each other.

The purpose of this paper is to give a log-linear representation of a generalized Bradley-Terry (BT-) Model for paired comparisons which allows the incorporation of ties, order effects, concomitant variables for the objects and categorical subject specific covariates and interactions between all of them. An advantage of this approach is that standard software for fitting log-linear models, such as GLIM, can be used. The approach is exemplified by analysing data from an experiment concerning the ranking of European universities. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

This paper provides an alternative methodology for the analysis of a set of Likert responses measured on a common attitudinal scale when the primary focus of interest is on the relative importance of items in the set. The method makes fewer assumptions about the distribution of the responses than the more usual approaches such as comparisons of means, MANOVA or ordinal data methods. The approach transforms the Likert responses into paired comparison responses between the items. The complete multivariate pattern of responses thus produced can be analysed by an appropriately reformulated paired comparison model. The dependency structure between item responses can also be modelled flexibly. The advantage of this approach is that sets of Likert responses can be analysed simultaneously within the Generalized Linear Model framework, providing standard likelihood based inference for model selection. This method is applied to a recent international survey on the importance of environmental problems. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

This paper develops a new approach to the analysis of longitudinal paired comparison data, where comparisons of the same objects by the same judges are made on more than one occasion. As an alternative to other recent approaches to such data, which are based on Kalman filter- ing, our approach treats the problem as one of multivariate multinomial data, allowing dependence terms between comparisons over time to be incorporated. The resulting model can be fitted as a Poisson log-linear model and has parallels with the quadratic binary exponential distribution of Cox. An example from the British Household Panel Survey illustrates the approach. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics