MEASURING EFFICIENCY OF FOOTBALL TEAMS BY MULTI-STAGE DEA MODEL

Original scientific paper Using the nonparametric variable output-oriented DEA (Data Envelopment Analysis) model, this paper analyses technical efficiency of the observed national football team in the qualifications for 2010 FIFA World Cup. If DEA model has a two stage structure, the first stage uses inputs to generate outputs that then became the inputs to the second stage. The indexes of efficiency, previously generated for attack and defence, will be implemented as parameters in the second-stage DEA analysis in accordance with the initial aim of this paper, to present new results as the outcome of the comprehensive approach to both inseparable aspects of the football. This theoretical approach is structured as follows: the football teams are defined as DMU (DecisionMaking Units), then, the methodology is established, specifically, the stages of range and objectives, analysis of procedure and variables, and the description of the theoretical analysis used.


Introduction
Initial idea on application of production function [1] in sports economy (baseball league) defines output as a product of arithmetic means of income and number of visitors in the function of players' technical skills and number of players, as first input variable and second input variable which is represented by qualities of coach, standard of football field, technical skills of opponent team and transport.Later on, Scully [2] has done the first estimation of production function, again for baseball league, but unlike previous approach, monetary effect and operating efficiency represent the model's outputs.In other words, the first equation of the model defines income which is determined by the percentage of wins and geographic size of a market and the second one defines team's result whereas the percentage of wins is determined by aggregation of "hitter and pitcher".Relation between individual results of players, as input, and team's results, as output, is standardised specification of production function, and the idea has been extended for estimation of team efficiency in various sports disciplines: basketball [3,4], rugby [5] and for the first time -English Premier League [6].Contribution of these papers, published in journals dedicated only to economic issues, is in the first instance reflected in substitution of parametric techniques of assessment of production functions with nonparametric techniques of assessment, especially in application of DEA modelling [7].
Production activity, as a basic characteristic of football game, implies affiliation of football process to the set of production processes, which then can evidently be modelled, in football industry, with production function.For the past ten years, DEA applications in sports economics have been focused on issues of individual efficiency (for example, statistics of running in professional baseball) or they have been dealing with professional football in Europe, from institutional point of view, and therefore enlarged application in football (for the past few years) at the level of a team [8 ÷ 15] should show to technically oriented football professionals (and the media representatives) factors which determine efficiency and effectiveness, consistency of results, different combinations of input and output, degree of correlation between efficiency and effectiveness and position, for example, in final classification.
The paper is organized as follows.After the first, introductory part, the second part of the paper shows basis of DEA modelling and selected model from the set of DEA models.Then, tabular presentations show values of efficiency indices of offense and defense, obtained in separate analysis, which had output role in the second application of DEA method.By application of EMS (Efficiency Measurement System) software, the third part of the paper will essentially result in new conclusions in efficiency analysis of the observed national team during the qualifications for 2010 FIFA World Cup, derived from the initial assumptions that include comprehensive aspect of offense and defence.Finally, summary and discussion are presented in conclusion part.
and this is what independently determines efficiency of offense and efficiency of defence of a football team.
That is, variables of the first phase are input indicators of offense and defence capabilities, the signs of which can be positive and negative, leading to an estimate of efficient offense production limit, independently from the one for defence.
From the perspective of offense, possession of ball and its movement through passes, dribbling, hits on goal, etc., represents active actions of creating of the chances for scoring goals.Statistically, offenses, passes in penalty area and hits towards goal define offensive actions of a football team.For example, these three variables can be input variables containing relative significance of time during which a team possesses the ball, showing the level of team's offense.
Defensive actions of players are observed as taking away the ball from opponent team in order to, for example, perform new offense.These actions mean interception and prevention of opponent team from scoring goals, and therefore statistics of repossessing ball and goal-keeper's action are clear indicators of stated activities.Inputs can (but don't have to) be presented relatively, for example by time during which opponent team possesses ball, showing the level of team's defence.
Decomposition of analysis, output variables which facilitate triumph of a team, can be defined independently, as positive outputs and reverse negative outputs (for example, instead of result being defined in percentage of wins).Measurement of used outputs combines offense productivity (scoring goals), with defence efficiency (preventing opponent from scoring), and therefore scored goals (number of goals) and reverse number of received goals (own goals count as goals scored by opponent team) during cycle of a competition, can be used as outputs.
Indicated variables of the second phase are both indicators of technical efficiency of offense and defence, estimated in the previous phase, while the outputclassification achieved by a football team -is based on cumulative result obtained in each match individually.This analysis estimates importance of applied tacticsoffense, defence or combination thereof -which characterize teams depending on the final results achieved during a competition cycle.

Technical efficiency measurement of the national football team in qualification matches for 2010 FIFAWC
Analysis of efficiency of the national team which participated in qualification cycle for 2010 FIFA World Cup South Africa and which was successfully classified for final tournament is based on technical aspects of sports.Research will show whether the analysis of technical efficiency is of any help when it comes to explanation of performances of sport in this national team for the period that includes total of 17 matches, out of which 10 were part of qualification cycle, and 7 of them were friendly.Out of total number of games, national team in focus of this analysis (TEAM(*)) hosted 8, and in the remaining 9 played as a guest.The matches of the TEAM(*) were played with national teams of the other countries, 5 of 12 participating in the same qualyfing group: TEAM(1), TEAM(2), TEAM (6), TEAM (7), TEAM (8) and TEAM(3), TEAM(4), TEAM(5), TEAM (9), TEAM (10), TEAM (11), TEAM (12) which represent national teams of the countries national teams of countries selected as opponents in friendly matches that have significance in the tactical-technical preparation of the TEAM(*) for the main event.Indexing opponents was conducted in accordance with the matches' chronology.
Integration of DEA modelling and football in this case was facilitated by conclusion of agreement between The National Football Association to which it belongs TEAM(*) and Sport Universal Process SAS (SUP).SUP has developed global information system called AMISCO intended for assistance in management and professional sports sector.AMISCO measures correct positions of all players in the field during the entire game.Measures of players' positions and movements enable generating of 2D animation of a game, together with tactical and sports information on individual and team performance of players.SUP has developed additional software for analysis (Video Pro) which facilitates full tactical analysis of football game by including each contact of a player with ball and direct approach to digitally synchronised tactic analysis of both team and players.Therefore, data used in this paper (provided by Sport Universal thanks to copyright software for analysis of football games AMISCO, which represents leading programme for applied statistics in football), are based on detailed statistical analysis of technical and tactical elements of a game, according to records made by specific cameras which cover the entire field and overall activity of all players, regardless of ball position.Only a small subset of data included in this large database will be used in further analysis.
In accordance with the restrictions relating to the ownership of the data used in this analysis, publicly naming organizations and national teams is avoided, but the interested reader can get more details by contacting the authors.

Numerical values: Empirical results for the determination of technical efficiency
In football, the only production factor is a team with players, and therefore it is necessary to select such indicators which will precisely measure players' skills.
Football, during its development, has been improved through segments.Technical elements are considered one of the most important segments of a football game.Evolution of technical elements has developed new requirements for football game and football tactics.If some elements and sub-elements of football technique were thoroughly analysed, they would result in unambiguous conclusion that all of them are, in the end, some sort of tactics.In other words, technical element, alone or in combination with other elements, facilitates the development of targeted combinations representing a tactical idea.
For this reason, the following input and output variables have been selected as potentially important for achievement of results.

Output variables:
-Total number of recognized scored goals (NG).
Tab. 1 and Tab. 2 show data on identified factors.Except for previously stated observations, our selection of inputs for offense and defence is also based on analysis of correlation between inputs and an output variable.The selected inputs show positive correlations (generally of statistical importance) with relevant output.When it comes to defence, the idea is similar, even though additional assumptions should be introduced.In this case, events in the game achieved by opponent team will be taken as defensive inputs instead of defensive actions of a team that has been analysed.Therefore, meaning of defence efficiency can be expressed as capability to receive fewer goals with more chances of opponent team.

Numerical values: Reduction of model dimensions
In the first stage of problem observation, it may be concluded that it is impossible to keep all identified input variables due to incompliance with instructions for definition of appropriate DEA model.Having in mind that 14 input variables and one output variable have been identified, with 17 decision making units, it is clear that some reductions have to be done.

Reduction of inputs by compression of similar variables
The first suggested reduction of input dimension refers to introduction of new parameters which will represent effect of a team through relation of successful and total values of the same kind, in compliance with Tabs. 3 and 4.
New variables have been introduced: -Effect of successful ones out of total crosses (C), -Effect of successful ones out of total crosses in play (CP), Technical Gazette 22, 3(2015), 763-770 -Effect of successful ones out of total passes in play (PP), -Effect of successful ones out of total number of air duels (AD), -Effect of successful ones out of total number of ground duels (GD).
By application of this step, we have reduced number of inputs from 14 to 9.However, it is necessary to carry out additional reductions.

Factor analysis
The second step of reduction is considered an application of a factor analysis.The factor analysis model specifies that variables are determined by common factors (the factors estimated by the model) and unique factors (which do not overlap between observed variables).The computed estimates are based on the assumption that all unique factors are uncorrelated with each other and with the common factors [16].
Having in mind that application of factor analysis imposes that it is applicable only in problems with big sampling (over 300 observations), feasibility test for factor analysis application will be carried out.There are two such tests in SPSS programme package (IBM Statistical Package for Social Scinces): Bartlett's test of sphericity and Kaiser-Meyer-Olkin (KMO) measures of sampling adequacy.Bartlett's test of sphericity should be important (for p<0,005), in order to make factor analysis feasible.KMO indicator can have value between 0 and 1 and therefore 0,6 is recommended as the smallest amount of original set of inputs.
Firstly, it will be checked whether the set of original inputs is adequate for factor analysis (Tab.5).Having in mind that value of KMO indicators is equal to 0,602, and that value of Bartlett's test of sphericity indicators is significant (p=0,000), factor analysis is feasible.Pursuant to Kaiser criteria, only inputs with characteristic value 1or more will be selected for analysis.Results shown in Tab.6 will be needed to identify those components.
Header Initial Eigenvalues shows characteristic values of all components.Only the first three (crosses, crosses in play, passes in play) have characteristic values above 1 (2,709; 1,856; 1,137).Those three components explain total of 63,352 % of variance (see column Cumulative %).Having in mind that factor analysis is only a technique of data research, some "football suggestions" to interpretation of results has been added.Problem dimension allows us to add two more inputs.Isolated factors will be accompanied by: number of shots (NS) and number of shots in the target (NST), primarily because they represent elements of the game which are connected closer than any other with possible scores, that is, scored goals.
In Tab. 8, matches with inefficient offensive activities are shown, with real input and output and desired (targeted) input and output which would provide efficiency.Desired input values are generated as difference between the real and equalising values, while desired value of output is obtained as product of real value and efficiency index reduced for equalising variable.
Number of opponent's received goals (TG), including goal difference achieved by TEAM (*) in each game (GD), and preserving the nature of DEA model, should give a new overview of TEAM(*) efficiency analysis.
Modelling shall be carried out on sample of 10 qualification games, while values of offense and defence efficiency, generated pursuant to technical and tactical elements, including friendly games, due to overview TEAM(*) total "level" in all games during "qualification" period.Having in mind that output-oriented model was also used in the previous analysis, obtained indexes of efficacy represented as input in new observation, will be prepared in their reciprocal values, due to nature of influence on efficacy phenomenon considered in this model.Also, due to data negativity condition, output which represents goal difference was modified by simple transformation [7].

Conclusion
Isolating of tactical and technical elements in football game, given in numeric form as evaluation of TEAM(*) players' activity in the field (or in case of defence: through impossibility to prevent identical movements and activities of opponent) has contributed to creation of prerequisites for application of DEA model in efficacy analysis of individual or, as in this case, the whole circle of football events.Integral consideration of TEAM(*) efficiency in qualification tournament for the biggest football competition of national selections, taking in account previously obtained efficiency indexes in form of input values, in two contexts of achieved result set as outputs (TG, GD).
Taking results into consideration, it can be decidedly concluded that efficiency of total play has been changed, i.e. the presented play and achieved result of TEAM(*) has undergone transformation in integral model.Games where TEAM(*) was efficient with regards to both aspects, but separately, by putting together results from the previous analysis, became "inefficient" and vice versa.
Different decompositions of problems and difference in start-up strategies at selecting research direction indicate the fact that complex process of modelling exact situations represents to authors both great challenge and in exhaustible resource of inspiration and further work, despite all difficulties.
Input variables: -Total number of crosses (NC), -Number of successful crosses (NsC), -Total number of crosses in play (NCP), -Number of successful long through balls in play (NsCP), -Total number of passes in play (passing of a ball) (NPP), -Number of successful passes in play (NsPP), -Number of shots (NS), -Number of shots on target (NST), -Total number of fouls made (NFm), -Total number of air duels (NAD), -Number of won air duels (NADw), -Total number of ground duels (NGD), -Number of won ground duels (NGDw), -Number of successful dribbling (NsD).

Table 1
Data for TEAM(*) for the offense on the matches at home and away

Table 2
Data for TEAM (*) for the defense on the matches at home and away

Table 3
Data for TEAM(*) for the offense on the matches at home and away -reduction

Table 4
Data for TEAM (*) for the defense on the matches at home and away -reduction

Table 5
Measuring sample data adequacy for factor analysis

Table 6
SPSS: Communalities and Principal component analysis

Table 7
Measuring offense efficiency

Table 10
Real and desired values of inputs and outputs of inefficient DMU -defence analysis

Table 11
Measuring offence and defence efficiency in the second-staged DEA