water polo is a sport with a growing worldwide interest , particularly for research purposes , with a noticeable increase in the number of recent publications ( prieto et al . , 2013 ) . the available research has tried to identify the performance characteristics of the game for both mens and womens competitions ( escalante et al . , 2011 , 2012 ; lupo et al . , 2010 , 2012 ) . the main aim of the performance analysis is to identify the strengths and weaknesses of the water polo teams in order to improve their performance ( carling et al . notational analysis is a method of registering and analysing the dynamics of a complex situation in a sporting context ( hughes and franks , 2004 ) , which collects performance indicators . a performance indicator is a selection , or combination , of action variables that aims to define some or all the aspects of a performance . clearly , to be useful , performance indicators should relate to a successful performance or outcome ( hughes and bartlett , 2002 ) . the performance characteristics of the water polo players are related to the success of the winning teams in short term competitions ( escalante et al . , 2011 , 2012 , 2013 ; lupo et al . , 2014 ) such as the olympic games , world championships and european championships . in mens games , eight game - related statistics i.e. shots , extra player shots , 5 m shots and assists ( offensive efficiency ) , blocked shots , goalkeeper - blocked shots , goalkeeper - blocked extra player shots and goalkeeper - blocked 5 m shots ( defensive efficiency ) were considered ( escalante et al . , 2011 ) to distinguish between winning and losing teams in the final phase of the 2008 olympic games held in beijing . in other studies , the offensive performance indicators i.e. centre goals , power play goals , counterattack goals , assists , offensive fouls , steals , blocked shots and won sprints , as well as defensive indicators i.e. goalkeeper - blocked shots , goalkeeper - blocked inferiority shots and goalkeeper - blocked 5 m shots ( escalante et al . , 2013 ) were used to distinguish between performances in international championships and their relationship with the phases of competition . the losing teams performed more even actions , whereas winning teams performed more counterattacks ( lupo et al . , 2012 ) . power play actions were more frequent in closed games than unbalanced games ( lupo et al . , 2012 ) . a margin of victory ( balanced with a  3 goals difference and unbalanced with a > 3 goals difference ) has been introduced to minimize the effect of the situational nature inherent in other teams ( lupo et al . , 2012 ) , as a measurement of match outcomes in performance analysis . on the other hand , performance in a regular water polo season has been analysed by different authors ( garca et al . , 2015 ; lupo et al . , 2010 ; prieto et al . , 2013 ) the favourable games had averages that were significantly higher for counterattacks and counterattack shots , goals and shots from zone 5 and 6 , whereas unfavourable games had significantly higher averages in even attacks and even shots , no goal shots and shots originated from zone 3 and 4 ( garca et al . , however , considering the joint influence of all variables in the same group on the match score would be problematic because the sum of the variables is 100% . penalties achieved  ,  offensive fouls  and  lost possessions  . since one and only one of these six  attack outcomes  can occur , the sum of the parts is 100% . compositional data are multivariate data in which the components represent some part of a whole . they are usually recorded in a closed form , accruing to a constant ( e.g. one if measured in parts per unit or 100 if measured in percentages ) . compositional data are characterised by particular and important numerical properties that have major consequences for any statistical analysis ( pawlowsky - glahn and egozcue , 2006 ) . in chemistry , compositional data are frequently used to reflect elemental concentrations as percentages , as the sum of the concentrations is 100% ( pesenson et al . , 2015 ) . considering that the variables of each group of offensive performance indicators in the present study had a constant sum which equaled 100% , we used appropriated statistical techniques for compositional data . to the best of our knowledge , very few references on compositional data in sport sciences may be found ( liu , 2004 ; li and liu , 2004 ; shimizu et al . , the aim of the present study was to identify the groups of offensive performance indicators that discriminated between match scores ( favourable , balanced or unfavourable ) , taking into account the nature of compositional data during a regular season in the spanish water polo league . these performance indicators can be of great interest for coaches and players due to their usefulness for training plans and the direction of competitions . a notational analysis was performed on 88 mens water polo matches ( 2 samples per match , in total 176 samples ) played by 10 teams from the first spanish water polo division during 3 seasons ( 2011 - 2014 ) . the results of 88 matches were : 41 favourable matches , 94 balanced matches and 41 unfavourable matches . these indicators had been used before by other researchers ( escalante et al . , 2012 ; hraste et al . , 2010 ; lupo et al . , moreover , they were clustered in five groups :  attack situation  ,  shot situation  ,  outcome  ,  zone  and  flakes  ( table 1 ) . the dependent variable in this study was the match score ( unfavourable , balanced and favourable ) . in relation to the match score , we considered a balanced score (  3 goals difference ) , and an unfavourable or favourable score ( > 3 goals difference ) . the matches were recorded by a video camera positioned at the side of the pool , at the level of the midfield line . a match analysis system ( longmatch , system version 0.20.8 , barcelona , spain ) was used for the notational analysis . the inspection and its related search were made by the same researcher to avoid bias . we randomly selected three matches which were observed with a lapse of time ; the kappa index was 0.97 . considering that the variables of each group ( table 1 ) had a constant sum which was equal to 100% , the data were compositional . based on previous studies with compositional data ( aitchison , 1986 ) , the variables of each composition were changed using log - quotient transformation between the parts ( specifically , additive log - ratio ( alr ) transformation ) . for example , in the group  attack situation  , the variables pati , pats , patco and patpe were transformed into : log(pats / pati ) , log(patco / pati ) and log(patpe / pati ) . discriminant analyses were applied to the transformed variables in order to identify the offensive performance indicators which discriminated between the favourable , balanced and unfavourable scores ; thus , two discriminant functions were obtained . indicators with structure coefficients ( sc1 , sc2 ) values 0.30 were considered relevant when their sc belonged to a significant discriminant function . the eigenvalue , the canonical correlation index , the wilks lambda and the percentage of right classification were used to measure discriminant power . the homogeneity assumption was evaluated with the boxs m test . to deal with the problem of essential zeros in the variable patpe , two analyses were conducted : for matches with penalties and for matches without penalties . for groups shot situation  and  outcome  , we followed the same methods to transform and to deal with the zeros . the groups  zone  and  flakes  needed only one analysis with all the matches due to the non - essential zeros . furthermore , a final discriminant analysis was applied to the independent variables with the sc > 0.30 obtained in the previous analyses . the log - transformation improved the normality of the variables , allowing the use of models that assumed normality , like the discriminant analysis . all statistical analyses were performed using spss software release 18.0 ( spss inc . , chicago , il , usa ) . a notational analysis was performed on 88 mens water polo matches ( 2 samples per match , in total 176 samples ) played by 10 teams from the first spanish water polo division during 3 seasons ( 2011 - 2014 ) . the results of 88 matches were : 41 favourable matches , 94 balanced matches and 41 unfavourable matches . these indicators had been used before by other researchers ( escalante et al . , 2012 ; hraste et al . , 2010 ; lupo et al . , moreover , they were clustered in five groups :  attack situation  ,  shot situation  ,  outcome  ,  zone  and  flakes  ( table 1 ) . the dependent variable in this study was the match score ( unfavourable , balanced and favourable ) . in relation to the match score , we considered a balanced score (  3 goals difference ) , and an unfavourable or favourable score ( > 3 goals difference ) . the matches were recorded by a video camera positioned at the side of the pool , at the level of the midfield line . a match analysis system ( longmatch , system version 0.20.8 , barcelona , spain ) was used for the notational analysis . the inspection and its related search we randomly selected three matches which were observed with a lapse of time ; the kappa index was 0.97 . considering that the variables of each group ( table 1 ) had a constant sum which was equal to 100% , the data were compositional . based on previous studies with compositional data ( aitchison , 1986 ) , the variables of each composition were changed using log - quotient transformation between the parts ( specifically , additive log - ratio ( alr ) transformation ) . for example , in the group  attack situation  , the variables pati , pats , patco and patpe were transformed into : log(pats / pati ) , log(patco / pati ) and log(patpe / pati ) . discriminant analyses were applied to the transformed variables in order to identify the offensive performance indicators which discriminated between the favourable , balanced and unfavourable scores ; thus , two discriminant functions were obtained . indicators with structure coefficients ( sc1 , sc2 ) values 0.30 were considered relevant when their sc belonged to a significant discriminant function . the eigenvalue , the canonical correlation index , the wilks lambda and the percentage of right classification were used to measure discriminant power . the homogeneity assumption was evaluated with the boxs m test . to deal with the problem of essential zeros in the variable patpe , two analyses were conducted : for matches with penalties and for matches without penalties . for groups  shot situation  and  outcome  , we followed the same methods to transform and to deal with the zeros . the groups  zone  and  flakes  needed only one analysis with all the matches due to the non - essential zeros . furthermore , a final discriminant analysis was applied to the independent variables with the sc > 0.30 obtained in the previous analyses . the log - transformation improved the normality of the variables , allowing the use of models that assumed normality , like the discriminant analysis . all statistical analyses were performed using spss software release 18.0 ( spss inc . , chicago , il , usa ) . the results of the discriminant analyses are presented in table 2 , for each group ( table 1 ) with alr transformed variables . taking into account all measures of discriminant power ( percentage of right classification is specifically shown ) , in matches with penalties , the group  outcome  qualified the most ( 60.4% for the original sample and 52.2% for cross - validation ) , followed by the group  shot situation  ( 51.1% for original sample and 46.8% for cross - validation ) ,  attack situation ( 49.5% for original sample and 40.4% for cross - validation ) ,  flakes ( 49.1% for original sample and 44.1% for cross - validation ) and  zone ( 43.7% for original sample and 37.3% for cross - validation ) . in matches without penalties , moreover , with regard to the first discriminant function ( the only significant one ) , table ii shows that out of 11 variables corresponding to groups  attack situation  ,  outcome  and  shot situation  , only 4 had the sc  0.30 in matches with penalties and out of 8 variables corresponding to groups  attack situation  ,  outcome  and  shot situation  , only 6 variables showed the sc  0.30 in matches without penalties . the groups  zone  and  flakes  had 7 variables with a relevant sc (  0.30 ) . table 3 presents the results of discriminant analysis of those variables with the sc  0.30 obtained in previous analyses in matches with penalties and without penalties . for the matches with penalties , the discriminant functions classified correctly 73.3% ( original sample ) and 63.3% ( cross - validation ) of the match score . in this discriminant analysis , only the first discriminant function was significant and the variables that had higher discriminatory power were log(pg / pls ) ( sc = .735 ) , log(plco / pli ) ( sc = .483 ) and log(patco / pati ) ( sc = .391 ) . for matches without penalties , the discriminant functions classified correctly 91.9% ( original sample ) and 43.2% ( cross - validation ) of the match score . in this discriminant analysis the variable that had higher discriminatory power was log(pg / pls ) ( sc = .576 ) . schema of the division of the court according to 6 zones ( lupo et al . , 2012 ) list of performance indicators clustered in five groups discriminant analyses between the match score for each group with log - transformed variables ( matches with penalties ( ) and without penalties ( ) ) * p < .05 ; * * p < .01 ; * * * p < .001 results of a discriminant analysis of the variables that were significant in the previous analysis * p < .05 ; * * p < .01 ; * * * p < .001 the aim of the present study was to identify groups of offensive performance indicators which best discriminated between the match scores ( favourable , balanced or unfavourable ) in water polo . although summative season long performance comparisons between teams may be considered limited by different authors ( lago , 2009 ; prieto et al . , 2013 ; tucker et al . , 2005 ) , studies of this type can provide general values that aid our understanding of water polo and could establish the norms according to which training sessions are structured ( garca et al . we used the offensive performance indicators transformed with alr transformation within each group , as their sum was 100% . to our knowledge , the present paper was the first to assess the impact of the offensive performance indicators on match scores in water polo using compositional data analysis . the results of the initial discriminant analyses identified  outcome  as the most discriminatory group , followed by the groups  shot situation  ,  attack situation  ,  flakes  and  zone  , respectively . in the final discriminant analysis , only three variables ( log(pg / pls ) , log(patco / pati ) , log(plco / pli ) ) were found to discriminate teams in relation to match scores in matches with penalties , and one variable ( log(pg / pls ) ) in matches without penalties . in line with the present results , that the variable  goal  , within the group  outcome  , had the greatest discriminatory power between the match scores , together with the variables related to a playing situation : counterattack , even and power play , within the groups  shot situation  and  attack situation  . the same variables ( goal and playing situation : even , counterattack and power play ) were also shown to be one of the most potent discriminant indicators between successful and unsuccessful teams in different championship phases ( escalante et al . , 2013 ) . equally , the winning teams made a higher number of counterattacks in womens international championships ( escalante et al . also the winning teams had low frequencies in even attacks and high frequencies in counterattacks during matches with an unbalanced match score ( > 3 goals of difference ) ( lupo et al . , 2014 ) . in other team sports such as handball , several papers have shown the importance of the counterattack ( bilge , 2012 ; foretic et al . , 2010 ) . moreover , the present study showed , as a further result , the importance of the exclusion and power play actions in matches without penalties . the effectiveness of power play shots was a performance indicator which discriminated between winning and losing teams in the final phase of the 2008 olympic games held in beijing ( escalante et al . , 2011 ) . again exclusion was a performance indicator that differentiated between women and men ( escalante et al . , 2011 ) , reflecting greater aggressiveness of men in competition . these performance indicators , i.e. goal and exclusion , belong to the group  outcome  , which was the most discriminating group between match scores in this study . although there are few studies that have analysed the discriminant power of different groups , the variables of the  outcome  group ( goal , exclusion ) were selected as relevant in other studies which compared water polo game - related statistics considering match scores ( escalante et al . , 2011 , 2013 ; the groups  shot situation  and  attack situation  reinforce the results obtained by other researches ( garca et al . , 2015 ) , however , the groups  flakes  and  zone  had the lowest discriminatory power , maybe because they are groups including more specific variables . some authors ( garca et al . , 2013 ) concluded that winning and losing teams had approximately the same opportunities to play with a numerical advantage , which is in line with our results regarding matches with penalties . nevertheless , our results of compositional data which selected matches without penalties , go one step beyond , showing the importance of performance indicators related to numerical inequality ( exclusion , power play attacks and shots ) . the number of penalties is a common performance indicator in strong teams with a favourable match score . therefore , by selecting only matches without penalties , we eliminated many of the matches with a favourable score ( a decreased number of goals , counterattacks , attacks and shots ) , so this may be a possible explanation of the increase of discriminatory power of other performance indicators ( exclusion and power play actions and shots ) considering scores in matches without penalties . taking into account the right percentages of classification , our results using compositional data ( 73.3% original sample and 63.3% cross - validation ) are more accurate compared to the results by garca et al . ( 2015 ) who used untransformed variables ( 72.2% original sample and 61.9% cross - validation ) . concerning the limitations of the current study , two aspects should be underlined . firstly , although the sample was the largest compared with other research conducted so far including analysis of these characteristics in water polo , the sample was not random , because of the difficulty of achieving the matches recordings . this generated an unbalanced design of the match scores ( favourable , balanced or unfavourable ) . secondly , despite the compositional data analysis is more appropriate statistically , its interpretation may be less understandable for coaches and water polo players . as practical implications , the coaches should enhance the skills of the counterattack and reinforce withdrawal in defense . in addition , they should take into account that the power play actions are a key factor in balanced matches . following this advice
abstractthe aim of the present study was to identify groups of offensive performance indicators which best discriminated between a match score ( favourable , balanced or unfavourable ) in water polo . 
 the sample comprised 88 regular season games ( 2011 - 2014 ) from the spanish professional water polo league . 
 the offensive performance indicators were clustered in five groups : attacks in relation to the different playing situations ; shots in relation to the different playing situations ; attacks outcome ; origin of shots ; technical execution of shots . 
 the variables of each group had a constant sum which equalled 100% . 
 the data were compositional data , therefore the variables were changed by means of the additive log - ratio ( alr ) transformation . 
 multivariate discriminant analyses to compare the match scores were calculated using the transformed variables . with regard to the percentage of right classification 
 , the results showed the group that discriminated the most between the match scores was  attacks outcome 
 ( 60.4% for the original sample and 52.2% for cross - validation ) . 
 the performance indicators that discriminated the most between the match scores in games with penalties were goals ( structure coefficient ( sc ) = .761 ) , counterattack shots ( sc = .541 ) and counterattacks ( sc = .481 ) . in matches without penalties , goals were the primary discriminating factor ( sc = .576 ) . 
 this approach provides a new tool to compare the importance of the offensive performance groups and their effect on the match score discrimination .