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Extended ordered paired comparison models with application to football data from German Bundesliga

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Abstract

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. ...
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... 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. ...
Chapter
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). ...
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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. ...
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... 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. ...
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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). ...
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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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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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.
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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.
Article
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.
Article
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.
Article
. 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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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)