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A general paired comparison model for the evaluation of sport competitions is proposed. It efficiently uses the available information by allowing for ordered response categories and team-specific home advantage effects. Penalized estimation techniques are used to identify clusters of teams that share the same ability. The model is extended to include team-specific explanatory variables. It is shown that regularization techniques allow to identify the contribution of explanatory variables to the success of teams. The usefulness of the methods is demonstrated by investigating the performance and its dependence on the budget for football teams of the German Bundesliga.

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... Recent studies have focused on regularization of ranking models with the aim of finding a group structure of ranks. Regularized ranking models [16,20,27,28] utilize a pairwise penalty function that groups the estimates of parameters with insignificant differences. In fact, the effective number of parameters is reduced to such an extent that the estimated model has a sparse structure [8]. ...

... In fact, the effective number of parameters is reduced to such an extent that the estimated model has a sparse structure [8]. According to [16,20,27,28], the sparsity enables an easier interpretation of the estimated model and provides a significant improvement in the quality of prediction, compared with non-regularization methods. However, the developed models have two drawbacks. ...

... The proposed model uses a modified penalty function that regularizes the differences between positive parameters in the Luce model. The proposed model is distinguished from [16,20,27,28] in two aspects: (1) the proposed model directly regularizes the loglikelihood function of the Luce model and (2) the penalty function of the proposed model is imposed on differences of positive parameters. Note that the models proposed by [16,20,27,28] are based on the pairwise comparison. ...

The Luce model is one of the most popular ranking models used to estimate the ranks of items. In this study, we focus on grouping items with similar abilities and consider a new supervised clustering method by fusing specific parameters used in the Luce model. By modifying the penalty function conventionally used in grouping parameters, we obtain a new method of grouping items in the Luce model without pairwise comparison modeling and develop an efficient algorithm to estimate the parameters. Moreover, we give an application of the proposed algorithm to the Bradley-Terry model with ties. In the real data analysis, we confirm that the proposed estimator provides an easier interpretation of ranks and an improvement in the quality of prediction.

... Similarly, Groll and Abedieh [15] , Groll et al. [6] , and Groll et al. [16] include many different variables in Poisson-type models for the FIFA World Cup or the EURO. Analogously, also in (ordinal) Bradley-Terry models, covariates can be incorporated, as, for example, demonstrated by Tutz and Schauberger [17] or Schauberger et al. [18] . When a large number of covariates is supposed to be incorporated into a model and/or if the predictive power of the single variables is not clear in advance, it can be sensible to estimate these models with regularized estimation approaches. ...

... Furthermore, in order to increase the interpretability and to reduce the complexity of the models, regularization approaches can be used to cluster teams with equal effects with respect to certain covariates. Approaches of that kind are, for example, proposed by Tutz and Schauberger [17] and Schauberger et al. [18] for Bradley-Terry models applied to data for the German Bundesliga. If prediction is the major purpose, Schauberger and Groll [19] show that approaches based on random forests (see Random Forests; Classification and Regression Tree Methods) are very promising. ...

We present the major approaches for the modeling and prediction of soccer matches. Two principal approaches can be distinguished, namely prediction of the scores of both teams and prediction of the match outcomes represented by the categories win, draw, and loss. The most important elements of these strategies are presented together with several different extensions and further developments.

... With respect to 3, a classical way to model soccer data focuses on the measure of teams' strength viewed as a latent variable, so that the observed result of a match is determined by this latent variable. In statistics, models based on this approach are known as paired comparison models and the most famous one is the Bradley-Terry (BT) model (Tutz and Schauberger, 2015). In its original specification, the probability that one team beats the opponent in a match only depends on the difference between the strength parameters of each of the two teams; the BT model can be extended in order to include both the possible results (win, draw and loss) and home team's advantage via home effect parameter. ...

... The original BT model allows for teams' strength estimation and ranking as well as for clustering teams; however, it does not explain why some teams are better than others. A standard way to explain the variation in performance is to include the difference in covariates between the two teams in the model (Tutz and Schauberger, 2015); in more complex models, different parameters for both the covariates of teams and matches can be specified (Cattelan et al., 2013;Schauberger et al., 2016). As the focus of our study is to assess whether team's performance indicators are able to predict the win of home team in a generic match, rather than estimating the strength of each team, a simple binomial logistic regression (BLR) model with only teams' difference in covariates has been adopted, assuming therefore that these predictors would capture the main effects on the result of interest (home team win). ...

This study explores a big and open database of soccer leagues in 10 European countries.
Data related to players, teams and matches covering seven seasons (from 2009/2010 to 2015/2016) were retrieved from Kaggle, an online platform in which big data are available for predictive modelling and analytics competition among data scientists. Based on both preliminary data analysis, experts’ evaluation and players’ position on the football pitch, role-based indicators of teams’ performance have been built and used to estimate the win probability of the home team with the binomial logistic regression (BLR) model that has been extended including the ELO rating predictor and two random effects due to the hierarchical structure of the dataset. The predictive power of the BLR model and its extensions has been compared with the one of other statistical modelling approaches (Random Forest, Neural Network, k-NN, Na¨ıve Bayes). Results showed that role-based indicators substantially improved the performance of all the models used in both this work and in previous works available on Kaggle. The base BLR model increased prediction accuracy by 10 percentage points, and showed the importance of defence performances, especially in the last seasons. Inclusion of both ELO rating predictor and the random effects did not substantially improve prediction, as the simpler BLR model performed equally good. With respect to the other models, only Na¨ıve Bayes showed more balanced results in predicting both win and no-win of the home team.

... Similar penalties have been used for the modelling of factors in GLMs by Bondell andReich (2009), Gertheiss andTutz (2010) and Oelker et al. (2014). More recently, penalties of this form have also been used in the modelling of paired comparison models, however, not for the modelling of heterogeneity by inclusion of covariates (Masarotto and Varin, 2012;Tutz and Schauberger, 2015). ...

... A big challenge with such an approach would be to find an appropriate penalty term to have a similar cluster effect as for the linear terms. Second, the model could be extended by object-specific covariates similar to Tutz and Schauberger (2015). For the application to the data from the GLES in this work, this would correspond to the inclusion of party-specific covariates, for example the popularity of the respective leading candidates. ...

In traditional paired comparison models heterogeneity in the population is simply ignored and it is assumed that all persons or subjects have the same preference structure. In the models considered here the preference of an object over another object is explicitly modelled as depending on subject-specific covariates, therefore allowing for heterogeneity in the population. Since by construction the models contain a large number of parameters we propose to use penalized estimation procedures to obtain estimates of the parameters. The used regularized estimation approach penalizes the differences between the parameters corresponding to single covariates. It enforces variable selection and allows to find clusters of objects with respect to covariates. We consider simple binary but also ordinal paired comparisons models. The method is applied to data from a pre-election study from Germany.

... For this reason, regularization methods are used for variable selection since they shrunk to zero coefficient estimates related to negligible covariates, reducing the parameters' variance. Among the others, Groll et al. (2015) and Tutz and Schauberger (2015) considered the LASSO framework, whereas the problem has not been tackled yet from the Bayesian perspective. A plethora of shrinkage priors for the regression coefficients are available (Bhadra et al. 2019), here we decide to adopt the regularized horseshoe prior by Piironen and Vehtari (2017): it easily allows to incorporate prior information about sparseness and can be interpreted as the continuous version of the popular spike-and-slab priors. ...

Passes are undoubtedly the more frequent events in football and other team sports. Passing networks and their structural features can be useful to evaluate the style of play in terms of passing behavior, analyzing and quantifying interactions among players. The present paper aims to show how information retrieved from passing networks can have a relevant impact on predicting the match outcome. In particular, we focus on modeling both the scored goals by two competing teams and the goal difference between them. With this purpose, we fit these outcomes using Bayesian hierarchical models, including both in-match and network-based covariates to cover many aspects of the offensive actions on the pitch. Furthermore, we review and compare different approaches to include covariates in modeling football outcomes. The presented methodology is applied to a real dataset containing information on 125 matches of the 2016–2017 UEFA Champions League, involving 32 among the best European teams. From our results, shots on target, corners, and such passing network indicators are the main determinants of the considered football outcomes.

... First, general works comparing different competition formats (Appleton, 1995;McGarry and Schutz, 1997;Marchand, 2002) or ranking methods (Mendonça and Raghavachari, 2000) avoid the use of specific prediction models. Second, while there exists a number of such models for football matches (Maher, 1982;Dixon and Coles, 1997;Koning et al., 2003;Tutz and Schauberger, 2015), handball seems to be a more difficult sport with respect to forecasting since it is a fast, dynamic, and high-scoring game. Significant differences can be observed between the total number of goals scored per match across the leading men's handball national leagues together with an increasing trend in all countries (Meletakos and Bayios, 2010). ...

This paper challenges the traditional seeding regime of round-robin tournaments that aims to create balanced groups. In particular, the design of the most prestigious European men’s handball club competition is compared to two alternative formats with equally strong groups via simulations. We find that it is possible to increase the quality of all matches played together with raising the uncertainty of outcome, essentially without sacrificing fairness. Our results have useful implications for the governing bodies of major sports.

... In the football data, the market value is an example for such an object-specific variable because the market values vary across teams but are constant across match days. Similar to the procedure proposed here, Tutz and Schauberger (2015) included an object-specific covariate in an analysis on the German Bundesliga. ...

In paired comparison models, the inclusion of covariates is a tool to account for the heterogeneity of preferences and to investigate which characteristics determine the preferences. Although methods for the selection of variables have been proposed no coherent framework that combines all possible types of covariates is available. There are three different types of covariates that can occur in paired comparisons, the covariates can either vary over the subjects, the objects or both the subjects and the objects of the paired comparisons. This paper gives an overview over all possible types of covariates in paired comparisons and introduces a general framework to include covariate effects into Bradley-Terry models. For each type of covariate, appropriate penalty terms that allow for sparser models and, therefore, easier interpretation are proposed. The whole framework is implemented in the R package BTLLasso. The main functionality and the visualization tools of the package are introduced and illustrated by real data sets.

... Further, the method of estimating the consistency of PCs was also given. Tutz and Schauberger [9] considered a general latent trait model for the assessment of sports' competitions. This model uses the consequences of playing at home, which can di er over teams. ...

Considering a number of Paired Comparison (PC) models existing in the literature, the posterior distribution for the parameters of the Rayleigh PC model is derived in this paper using the informative priors: Conjugate and Dirichlet. The values of the hyperparameters are elicited using prior predictive distribution. The preferences for the data of cigarette brands, such as Goldleaf (GL), Marlboro (ML), Dunhill (DH), and Benson & Hedges (BH), are collected based on university students' opinions. The posterior estimates of the parameters are obtained under the loss functions: Quadratic Loss Function (QLS), Weighted Loss Function (WLS), and Squared Error Loss Function (SELF) with their risks. The preference and predictive probabilities are investigated. The posterior probabilities are evaluated with respect to the hypotheses of two parameters comparison. In this respect, the graphs of marginal posterior distributions are presented, and appropriateness of the model is tested by Chi-Square.

... Stern (1990aStern ( , 1990b proposed the gamma models for PC. Thurstone-Mosteller model for PC was used to analyze volleyball data [7], Stern used a PC model to analyze sports datasets for the National League baseball season, and football data were analyzed by using Bradley-Terry model ( [7], [20], [13]). Neil and Jonathan (2015) investigated the use of Bradley-Terry models to analyze test match cricket, Abbas and Aslam (2009) showed that any group of individuals may be ranked using the Cauchy PC model via a Bayesian approach with an application on five top-ranked ODI cricket teams. ...

The analysis of sports data, especially cricket is an interesting field for the statisticians. Every year, a large number of cricket tournaments take place among the cricket playing nations. It is of interest to study their performance when they play with each other in a one-day international (ODI) match or a test match. In this study, we assess the performance of top ten cricket teams in the ODI cricket match and make a comparison among them. The abilities of teams change over time.
As a result, not a single team dominates the game over a long period. Therefore, a paired comparison method is more reliable and appropriate to compare more than two teams at the same time based on the outcomes of the matches they play. Arguably, a team’s performance also depends on whether they play at home or away. In this study, we consider Bradley-Terry model, a widely accepted model for pairwise comparison. In that, we consider home and away effect to demonstrate how the home advantages differ among these teams.

... Therefore, in high-dimensional settings as considered here (many teams and several covariates) regularization methods should be applied to reduce the complexity of the final models. Casalicchio et al. [5] presented a boosting approach while Tutz and Schauberger [27] and Schauberger and Tutz [24] use L 1 -type penalties. After all, the inclusion of subject-object-specific covariates is new and calls for a specific model and a suitable regularization method which will be elaborated in the following. ...

In modern football, various variables as, for example, the distance a team runs or its percentage of ball possession, are collected throughout a match. However, there is a lack of methods to make use of these on-field variables simultaneously and to connect them with the final result of the match. This paper considers data from the German Bundesliga season 2015/2016. The objective is to identify the on-field variables that are connected to the sportive success or failure of the single teams. An extended Bradley–Terry model for football matches is proposed that is able to take into account on-field covariates. Penalty terms are used to reduce the complexity of the model and to find clusters of teams with equal covariate effects. The model identifies the running distance to be the on-field covariate that is most strongly connected to the match outcome.

... In the football data, the market value is an example for such an object-specific variable because the market values vary across teams but are constant across matchdays. Similarily to the procedure proposed here, Tutz and Schauberger (2014) included an object-specific covariate in an analysis on the German Bundesliga. ...

In paired comparison models, the inclusion of covariates is a tool to account for the heterogeneity of preferences and to investigate which characteristics determine the preferences. Although methods for the selection of variables have been proposed no coherent framework that combines all possible types of covariates is available. There are three different types of covariates that can occur in paired comparisons, the covariates can either vary over the subjects, the objects or both the subjects and the objects of the paired comparisons. This paper gives an overview over all possible types of covariates in paired comparisons and introduces a general framework to include covariate effects into Bradley-Terry models. For each type of covariate, appropriate penalty terms that allow for sparser models and therefore easier interpretation are proposed. The whole framework is implemented in the R-package BTLLasso. The main functionality and the visualization tools of the package are introduced and illustrated by using real data sets.

... Additionally, unlike BTM's, state-space models would not typically suffer from identifiability problems were a team to win or lose all of its games in a single season (a rare, but extant possibility in the NFL). 1 For additional and related state-space resources, see Knorr-Held (2000), Cattelan et al. (2013), Baker and McHale (2015), and Manner (2015). Additionally, Matthews (2005), Owen (2011), Koopmeiners (2012), Tutz and Schauberger (2015), and Wolfson and Koopmeiners (2015) implement related versions of the original BTM. ...

Statistical applications in sports have long centered on how to best separate signal, such as team talent, from random noise. However, most of this work has concentrated on a single sport, and the development of meaningful cross-sport comparisons has been impeded by the difficulty of translating luck from one sport to another. In this manuscript, we use betting market data to develop a Bayesian state-space model that can be uniformly applied across sporting leagues to better understand the role of randomness in game outcomes. Our model can be used to extract estimates of team strength, the between-season, within-season, and game-to-game variability of team strengths, as well each team's home advantage. We implement our approach across a decade of play in each of the National Football League (NFL), National Hockey League (NHL), National Basketball Association (NBA), and Major League Baseball (MLB), finding that the NBA demonstrates both the largest dispersion in talent and the largest home advantage. Additionally, the NHL and MLB stand out for their relative randomness in game outcomes. We conclude by proposing a new metric for judging league competitiveness that works in absence of factors outside of team control.

... Basically, the model is a special case of a cumulative logit model and allows for the inclusion of so-called subject-object-specific covariates z ir . See also Tutz and Schauberger (2015) for a model including object-specific covariates z r and Schauberger and Tutz (2015) for a model including subjectspecific covariates z i . Y i(r,s) encodes an ordered response with K categories (including a category for draws) for a match between team a r and team a s on matchday i where a r played at its home ground. ...

A model for results of football matches is proposed that is able to take into account match-specific covariates as, for example, the total distance a team runs in the specific match. The model extends the Bradley-Terry model in many different ways. In addition to the inclusion of covariates, it considers ordered response values and (possibly team-specific) home effects. Penalty terms are used to reduce the complexity of the model and to find clusters of teams with equal covariate effects.

We present new statistical methodology for analysing rank data, where the rankings are allowed to vary in time. Such data arise, for example, when the assessments are based on a performance measure of the items, which varies in time, or if the criteria, according to which the items are ranked, change in time. Items can also be absent when the assessments are made, because of delayed entry or early departure, or purely randomly. In such situations, also the dimension of the rank vectors varies in time. Rank data in a time-dependent setting thus lead to challenging statistical problems. These problems are further complicated, from the perspective of computation, by the large dimension of the sample space consisting of all permutations of the items. Here, we focus on introducing and developing a Bayesian version of the Mallows rank model, suitable for situations in which the ranks vary in time and the assessments can be incomplete. The consequent missing data problems are handled by applying Bayesian data augmentation within Markov chain Monte Carlo. Our method is also adapted to the task of future rank prediction. The method is illustrated by analysing some aspects of a data set describing the academic performance, measured by a series of tests, of a class of high school students over a period of 4 years. Copyright

In the last two decades, regularization techniques, in particular penalty-based methods, have become very popular in statistical modelling. Driven by technological developments, most approaches have been designed for high-dimensional problems with metric variables, whereas categorical data has largely been neglected. In recent years, however, it has become clear that regularization is also very promising when modelling categorical data. A specific trait of categorical data is that many parameters are typically needed to model the underlying structure. This results in complex estimation problems that call for structured penalties which are tailored to the categorical nature of the data. This article gives a systematic overview of penalty-based methods for categorical data developed so far and highlights some issues where further research is needed. We deal with categorical predictors as well as models for categorical response variables. The primary interest of this article is to give insight into basic properties of and differences between methods that are important with respect to statistical modelling in practice, without going into technical details or extensive discussion of asymptotic properties.

Although each statistical unit on which measurements are taken is unique, typically there is not enough information available to account totally for its uniqueness. Therefore, heterogeneity among units has to be limited by structural assumptions. One classical approach is to use random effects models, which assume that heterogeneity can be described by distributional assumptions. However, inference may depend on the assumed mixing distribution, and it is assumed that the random effects and the observed covariates are independent. An alternative considered here is fixed effect models, which let each unit has its own parameter. They are quite flexible but suffer from the large number of parameters. The structural assumption made here is that there are clusters of units that share the same effects. It is shown how clusters can be identified by tailored regularised estimators. Moreover, it is shown that the regularised estimates compete well with estimates for the random effects model, even if the latter is the data generating model. They dominate if clusters are present.

This is a short overview of the R add-on package BradleyTerry2, which facilitates the specification and fitting of Bradley-Terry logit, probit or cauchit models to pair-comparison data. Included are the standard ‘unstructured’ Bradley-Terry model, structured versions in which the parameters are related through a linear predictor to explanatory variables, and the possibility of an order or ‘home advantage’ effect or other ‘contest-specific’ effects. Model fitting is either by maximum likelihood, by penalized quasi-likelihood (for models which involve a random effect), or by bias-reduced maximum likelihood in which the first-order asymptotic bias of parameter estimates is eliminated. Also provided are a simple and efficient approach to handling missing covariate data, and suitably-defined residuals for diagnostic checking of the linear predictor.

Ranking a vector of alternatives on the basis of a series of paired
comparisons is a relevant topic in many instances. A popular example is ranking
contestants in sport tournaments. To this purpose, paired comparison models
such as the Bradley-Terry model are often used. This paper suggests fitting
paired comparison models with a lasso-type procedure that forces contestants
with similar abilities to be classified into the same group. Benefits of the
proposed method are easier interpretation of rankings and a significant
improvement of the quality of predictions with respect to the standard maximum
likelihood fitting. Numerical aspects of the proposed method are discussed in
detail. The methodology is illustrated through ranking of the teams of the
National Football League 2010-2011 and the American College Hockey Men's
Division I 2009-2010.

The purpose of this paper is to propose an alternative log-linear representation of an adjacent categories (AC) paired comparison (PC) model. The AC model is well suited for modelling ordinal PC data by postulating a power relationship between the response category and the probability of preferring one object over another object. The model is applied to data collected on the motivation of Vienna students to start a doctoral programme of study.

The first part of this paper describes a series of loglinear preference models based on paired comparisons, a method of measurement whose aim is to order a set of objects according to an attribute of interest by asking subjects to compare pairs of objects. Based on the basic Bradley-Terry specification, two types of models, the loglinear Bradley-Terry model and a pattern approach are presented. Both methods are extended to include subject and object-specific covariates and some further structural effects. In addition, models for derived paired comparisons (based on rankings and ratings) are also included. Latent classes and missing values can be included. The second part of the paper describes the package prefmod that implements the above models in R. Illustrational applications are provided in the last part of the paper.

Shrinking methods in regression analysis are usually designed for metric
predictors. In this article, however, shrinkage methods for categorial
predictors are proposed. As an application we consider data from the Munich
rent standard, where, for example, urban districts are treated as a categorial
predictor. If independent variables are categorial, some modifications to usual
shrinking procedures are necessary. Two $L_1$-penalty based methods for factor
selection and clustering of categories are presented and investigated. The
first approach is designed for nominal scale levels, the second one for ordinal
predictors. Besides applying them to the Munich rent standard, methods are
illustrated and compared in simulation studies.

Classical categorical regression models such as the multinomial logit and proportional odds models are shown to be readily handled by the vector generalized linear and additive model (VGLM/VGAM) framework. Additionally, there are natural extensions, such as reduced-rank VGLMs for dimension reduction, and allowing covariates that have values specific to each linear/additive predictor, e.g., for consumer choice modeling. This article describes some of the framework behind the VGAM R package, its usage and implementation details.

We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, two-class logistic regression, and multi- nomial regression problems while the penalties include Ã¢ÂÂ_1 (the lasso), Ã¢ÂÂ_2 (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.

We study linear smoothers and their use in building nonparametric regression models. In the first part of this paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running-mean and running-line, kernel and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother, and several other topics such as the number of degrees of freedom of a smoother are discussed. In the second part of the paper we describe how linear smoothers can be used to estimate the additive model, a powerful nonparametric regression model, using the "back-fitting algorithm." We show that backfitting is the Gauss-Seidel iterative method for solving a set of normal equations associated with the additive model. We provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers.

This paper develops a predictive model for National Football League (NFL) game scores using data from the period 1988--1993. The parameters of primary interest, measures of team strength, are expected to vary over time. Our model accounts for this source of variability by modeling football outcomes using a state-space model that assumes team strength parameters follow a first-order autoregressive process. Two sources of variation in team strengths are addressed in our model; week-to-week changes in team strength due to injuries and other random factors, and season-to-season changes resulting from changes in personnel and other longer-term factors. Our model also incorporates a home-field advantage while allowing for the possibility that the magnitude of the advantage may vary across teams. The aim of the analysis is to obtain plausible inferences concerning team strengths and other model parameters, and to predict future game outcomes. Iterative simulation is used to obtain samples fro...

In statistical models of dependence, the effect of a categorical variable is typically described by contrasts among parameters. For reporting such effects, quasi-variances provide an economical and intuitive method which permits approximate inference on any contrast by subsequent readers. Applications include generalised linear models, generalised additive models and hazard models. The present paper exposes the generality of quasi-variances, emphasises the need to control relative errors of approximation, gives simple methods for
obtaining quasi-variances and bounds on the approximation error involved, and explores the domain of accuracy of the method. Conditions are identified under which the quasi-variance approximation is exact, and numerical work indicates high accuracy in a variety of settings.

The choice between fixed and random effects In prior chapters, group-specific effects were assumed to be drawn from a distribution, typically Gaussian. In applied research in the social and behavioral sciences, economics, public health, public policy, and many other fields, alternatives to this choice are often made, with the most common being the “fixed effects” approach. In its most basic formulation, group-specific intercepts are modeled using indicator variables, effectively making them open parameters in the model and not assigning them a distributional form. The choice between random and fixed effects has implications for the interpretation of parameter estimates and for estimation efficiency. 1 Specifically, the effect estimates for each group will be different depending on the modeling choice; the β estimates for the predictors in the model may be different depending on the modeling choice; the reliability with which one can make predictions for new groups differs; and there is ...

This article develops a predictive model for National Football League (NFL) game scores using data from the period 1988-1993. The parameters of primary interest - measures of team strength - are expected to vary over time. Our model accounts for this source of variability by modeling football outcomes using a state-space model that assumes team strength parameters follow a first-order autoregressive process. Two sources of variation in team strengths are addressed in our model; week-to-week changes in team strength due to injuries and other random factors, and season-to-season changes resulting from changes in personnel and other longer-term factors. Our model also incorporates a home-field advantage while allowing for the possibility that the magnitude of the advantage may vary across teams. The aim of the analysis is to obtain plausible inferences concerning team strengths and other model parameters, and to predict future game outcomes. Iterative simulation is used to obtain samples from the joint posterior distribution of all model parameters. Our model appears to outperform the Las Vegas "betting line" on a small test set consisting of the last 110 games of the 1993 NFL season.

Despite the enormous amounts of resources devoted to concept and product testing and the continued use of pretest market (PTM) modeling procedures, estimates of new product failures are still alarmingly high. The primary objectives of PTM modeling are to forecast the market share/sales volume of a new product and to determine the sources of new product share at the aggregate market level. The authors describe a new approach that is designed to provide a parsimonious description of competitive changes before and after a new product is introduced by identifying latent segments (i.e., groups of consumers) that vary in size and composition with respect to the relative preferences for a set of brands before and after a new product is introduced. Each latent segment represents a particular preference state characterized by a set of segment-level choice probabilities. The modeling framework is based on a class of dynamic latent class models that explicitly recognize two major types of preference heterogeneity: (1) heterogeneity caused by before-after changes in latent preferences for the brands (i.e., time-varying relative choice probabilities) and/or (2) heterogeneity caused by consumers changing their latent preference segment in response to a new product (i.e., time varying latent segment probabilities). As is demonstrated in the empirical application, the dynamic latent class models provide a comprehensive framework for understanding how a new product changes the competitive landscape.

A paired comparison technique is presented for fitting response surfaces. This is especially useful when subjective responses are involved, where it is often difficult to justify the basic assumptions of the classical procedure. This report discusses aspects of the estimation of parameters and their properties, tests of relevant hypotheses and the selection of experimental designs. The method is applied to an example in food testing.

A linear paired comparison model is proposed that allows a large number of draws and large variability of draw percentages among the players as is the case for chess or soccer matches. The model can also be extended to allow home ground advantage. When only summary results are available, maximum likelihood estimation is not feasible and we recommend a method based on matching the numbers of home wins, home draws, away wins and away draws for each team with their expected values. We also discuss the problem of obtaining estimated standard errors. As an illustration, the suggested model is fitted to the 1993-94 English Premier Soccer League results.

This book introduces basic and advanced concepts of categorical regression with a focus on the structuring constituents of regression, including regularization techniques to structure predictors. In addition to standard methods such as the logit and probit model and extensions to multivariate settings, the author presents more recent developments in flexible and high-dimensional regression, which allow weakening of assumptions on the structuring of the predictor and yield fits that are closer to the data. A generalized linear model is used as a unifying framework whenever possible in particular parametric models that are treated within this framework. Many topics not normally included in books on categorical data analysis are treated here, such as nonparametric regression; selection of predictors by regularized estimation procedures; ternative models like the hurdle model and zero-inflated regression models for count data; and non-standard tree-based ensemble methods, which provide excellent tools for prediction and the handling of both nominal and ordered categorical predictors. The book is accompanied by an R package that contains data sets and code for all the examples.

Two types of model are discussed for paired comparisons of several treatments using scales such as (A⋘B, A≪B, A<B, A=B, A⋙B, A≫B), where A≪B denotes strong preference for treatment B over treatment A,A≪B denotes moderate preference for B,A<B denotes weak preference for B,A=B denotes no preference, and so forth. For the binary scale (A<B,A>B), special cases of the models using logit transforms simplify to the Bradley-Terry model. When the same raters compare each pair of treatments, one can allow within-rater dependence by fitting the models with constrained maximum likelihood.

In the course of national sports tournaments, usually lasting several months, it is expected that the abilities of teams taking part in the tournament change in time. A dynamic extension of the Bradley-Terry model for paired comparison data is introduced to model the outcomes of sporting contests allowing for time-varying abilities. It is assumed that teams' home and away abilities depend on past results through exponentially weighted moving average processes. The proposed model is applied to sports data with and without tied contests, namely the 2009-2010 regular season of the National Basketball Association tournament and the 2008-2009 Italian Serie A football season.

Despite the enormous amounts of resources devoted to concept and product testing and the continued use of pretest market (PTM) modeling procedures, estimates of new product failures are still alarmingly high. The primary objectives of PTM modeling are to forecast the market share/sales volume of a new product and to determine the sources of new product share at the aggregate market level. The authors describe a new approach that is designed to provide a parsimonious description of competitive changes before and after a new product is introduced by identifying latent segments (i.e,, groups of consumers) that vary in size and composition with respect to the relative preferences for a set of brands before and after a new product is introduced. Each latent segment represents a particular preference state characterized by a set of segment-level choice probabilities. The modeling framework is based on a class of dynamic latent class models that explicitly recognize two major types of preference heterogeneity: (1) heterogeneity caused by before after changes in latent preferences for the brands (i.e., time-varying relative choice probabilities) and/or (2) heterogeneity caused by consumers changing their latent preference segment in response to a new product (i.e., time varying latent segment probabilities), As is demonstrated in the empirical application, the dynamic latent class models provide a comprehensive framework for understanding how a new product changes the competitive landscape.

When paired comparisons are made sequentially over time as for example in chess competitions, it is natural to assume that the underlying abilities do change with time. Previous approaches are based on fixed updating schemes where the increments and decrements are fixed functions of the underlying abilities. The parameters that determine the functions have to be specified a priori and are based on rational reasoning. We suggest an alternative scheme for keeping track with the underlying abilities. Our approach is based on two components: a response model that specifies the connection between the observations and the underlying abilities and a transition model that specifies the variation of abilities over time. The response model is a very general paired comparison model allowing for ties and ordered responses. The transition model incorporates random walk models and local linear trend models. Taken together, these two components form a non-Gaussian state-space model. Based on recent results, recursive posterior mode estimation algorithms are given and the relation to previous approaches is worked out. The performance of the method is illustrated by simulation results and an application to soccer data of the German Bundesliga.

An extension of the Bradley-Terry Luce model is presented which allows for an ordered response characterizing the strength of preference. The generalization includes models with ties as special cases. The model is derived from assumptions on underlying random utility functions. Necessary and sufficient conditions for the existence of scale values are given in a representation theorem and the uniqueness of the scale is considered. Estimation of parameters, goodness of fit tests, and tests of linear hypotheses are treated in the framework of a weighted least-squares method.

. In this article we propose an approach to study the effect of consumer-specific information on (complete) rank ordered preference
data by means of Bradley-Terry type models. The main idea is to transform the ranking data into paired comparison data, which
can be modelled within the Generalised Linear Model framework by means of a log-linear model for a corresponding contingency
table. Therefore, standard software can be used to estimate model parameters and a goodness-of-fit can be assessed in the
usual way. This approach allows to simultaneously estimate object-specific parameters which, in the marketing context, can
be interpreted as attractions of the analysed objects, as well as subject-object interaction parameters that represent the
effects of consumer-specific variables on the attractions. The interaction parameters offer a statistically motivated approach
for customer segmentation and market targeting. The outlined methodology is applied to preference judgements within a local
daily newspaper market. It is shown that certain socio-economic characteristics of the consumers have significant influences
on their preference structures.

Preference decisions will usually depend on the characteristics of both the judges and the objects being judged. In the analysis of paired comparison data concerning European universities and students' characteristics, it is demonstrated how to incorporate subject-specific information into Bradley-Terry-type models. Using this information it is shown that preferences for universities and therefore university rankings are dramatically different for different groups of students. A log-linear representation of a generalized Bradley-Terry model is specified which allows simultaneous modelling of subject- and object-specific covariates and interactions between them. A further advantage of this approach is that standard software for fitting log-linear models, such as GLIM, can be used.

The Bradley-Terry model for a paired-comparison experiment with t treatments postulates a set of t ‘true’ treatment ratings π1, π2, · · ·, πt such that πi ≥ 0, ∑ πi = 1 and the probability for preferring treatment i to treatment j is πi(πi + πj). Thus, according to this model, every comparison of two treatments results in a definite preference for one of the two. This is an unrealistic restriction since when there is no difference between the responses due to two treatments, any method of expressing preference for one over the other is somewhat arbitrary. This paper considers a modification of the Bradley-Terry model by introducing an additional parameter, called threshold parameter, into the model. This permits ‘ties’ in the model. The problem of estimation and tests of hypotheses for the parameters of the modified model is also dealt with in the paper.

We consider the problem of dynamically rating sports teams on the basis of categorical outcomes of paired comparisons such as win, draw and loss in football. Our modelling framework is the cumulative link model for ordered responses, where latent parameters represent the strength of each team. A dynamic extension of this model is proposed with close connections to nonparametric smoothing methods. As a consequence, recent results have more influence in estimating current abilities than results in the past. We highlight the importance of using a specific constrained random walk prior for time-changing abilities which guarantees an equal treatment of all teams. Estimation is done with an extended Kalman filter and smoother algorithm. An additional hyperparameter which determines the temporal dynamic of the latent team abilities is chosen on the basis of the optimal one-step-ahead predictive power. Alternative estimation methods are also considered. We apply our method to the results from the German football league Bundesliga 1996-1997 and to the results from the American National Basketball Association 1996-1997.

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the proposed model yields a reasonable representation of actual experimentation.

The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.

This paper presents two probabilistic models based on the logistic and the normal distribution for the analysis of dependencies in individual paired comparison judgments. It is argued that a core assumption of latent class choice models, independence of individual decisions, may not be well-suited for the analysis of paired comparison data. Instead, the analysis and interpretation of paired comparison data may be much simplified by allowing for within-person dependencies that result from repeated evaluations of the same options in different pairs. Moreover, by relating dependencies among the individual-level responses to (in)consistencies in the judgmental process, we show that the proposed graded paired comparison models reduce to ranking models under certain conditions. Three applications are presented to illustrate the approach.

When performing an analysis of variance, the investigator often has two main goals: to determine which of the factors have a significant effect on the response, and to detect differences among the levels of the significant factors. Level comparisons are done via a post-hoc analysis based on pairwise differences. This article proposes a novel constrained regression approach to simultaneously accomplish both goals via shrinkage within a single automated procedure. The form of this shrinkage has the ability to collapse levels within a factor by setting their effects to be equal, while also achieving factor selection by zeroing out entire factors. Using this approach also leads to the identification of a structure within each factor, as levels can be automatically collapsed to form groups. In contrast to the traditional pairwise comparison methods, these groups are necessarily nonoverlapping so that the results are interpretable in terms of distinct subsets of levels. The proposed procedure is shown to have the oracle property in that asymptotically it performs as well as if the exact structure were known beforehand. A simulation and real data examples show the strong performance of the method.

In statistical models of dependence, the effect of a categorical variable is typically described by contrasts among parameters. For reporting such effects, quasi-variances provide an economical and intuitive method which permits approximate inference on any contrast by subsequent readers. Applications include generalised linear models, generalised additive models and hazard models. The present paper exposes the generality of quasi-variances, emphasises the need to control relative errors of approximation, gives simple methods for obtaining quasi-variances and bounds on the approximation error involved, and explores the domain of accuracy of the method. Conditions are identified under which the quasi-variance approximation is exact, and numerical work indicates high accuracy in a variety of settings. Copyright Biometrika Trust 2004, Oxford University Press.

The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, their construction, and penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented. Keywords: Generalized linear models, smoothing, nonparametric models, splines, density estimation. Address for correspondence: DCMR Milieudienst Rijnmond, 's-Gravelandse...

ordBTL: Modelling comparison data with ordinal response

- G Casalicchio

A general family of penalties for combining differing types of penalties in generalized structured models

- M.-R Oelker
- G Tutz

Oelker, M.-R., Tutz, G.: A general family of penalties for combining differing types of penalties in generalized structured models. Technical Report 139, LMU, Department of Statistics (2013)

gvcm.cat: Regularized Categorial Effects/Categorial Effect Modifiers in GLMs. R package version 1

- M.-R Oelker

Oelker, M.-R.: gvcm.cat: Regularized Categorial Effects/Categorial Effect Modifiers in GLMs. R package
version 1.6 (2013)

Linear smoothers and additive models ordBTL: Modelling comparison data with ordinal response. R package version 7 (2013) Dynamic Bradley-Terry modelling of sports tournaments

- A Buja
- T Hastie
- R Tibshirani
- G Casalicchio
- M Cattelan
- C Varin
- D Firth

Buja, A., Hastie, T., Tibshirani, R.: Linear smoothers and additive models. Ann. Stat. 17, 453-510 (1989)
Casalicchio, G.: ordBTL: Modelling comparison data with ordinal response. R package version 7 (2013)
Cattelan, M., Varin, C., Firth, D.: Dynamic Bradley-Terry modelling of sports tournaments. J. R. Stat. Soc.
Ser. C (Applied Statistics) 62(1), 135-150 (2013)

Flexible smoothing with B-splines and Penalties Dynamic stochastic models for time-dependent ordered paired comparison systems

- P H C Eilers
- B D Marx
- L Fahrmeir
- G Tutz

Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and Penalties. Stat. Sci. 11, 89-121 (1996)
Fahrmeir, L., Tutz, G.: Dynamic stochastic models for time-dependent ordered paired comparison systems.
J. Am. Stat. Assoc. 89, 1438-1449 (1994)

Multivariate statistical modelling based on generalized linear models Quasi-variances Regularization paths for generalized linear models via coordinate descent

- L Fahrmeir
- G Tutz
- D Firth
- R De Menezes
- J H Friedman
- T Hastie
- R Tibshirani

Fahrmeir, L., Tutz, G.: Multivariate statistical modelling based on generalized linear models. Springer, New
York (2001)
Firth, D., De Menezes, R.: Quasi-variances. Biometrika 91, 65 (2004)
Friedman, J.H., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate
descent. J. Stat. Softw. 33(1), 1-22 (2010)