Journal of Digital Life 2022,2,14

Review Article

A statistical examination into the structure of plays
and its relationship to wins/losses in basketballgames played by excellent players from thestandpoint of STATSKen Nagamine1*,Osamu Aoyagi1
1Faculty of Health and Sports Science, Fukuoka University, 8-19-10 Nanakuma, Jyonan-ku, Fukuoka 814-0180 JAPAN*Correspondence: knaga@fukuoka-u.ac.jp; Tel.: +81-93-871-6631ffiliation

Abstract
We reviewed and summarized the findings from some kinds of perspectives. This allowed us to realize thatno consistent findings can be obtained in terms of what plays/factors contribute most to wins/losses and thesequence of plays leading to scoring, that discriminant analysis is commonly used in the studies, and that forstructural analysis, no rotational technique was used. Based on these findings, we devised the following new
proposals: 1) Researchers need to exert more effort to analyze various kinds of games in the future, 2) We
can recommend logistic regression analysis, but not only discrimination analysis. 3) For structural analysis,
to make the research findings useful, the rotation of axes or principal components should be actively utilized.
Keywords: Basketball ; Game-statistics ; Wins/Losses.

1. Introduction
In major international or nationwide basketball games, not only game outcomes but also various plays or
violations/fouls as well as the playing times of players have been published. They have been utilized not only for press
releases, the selection of VIP players, and countermeasures against teams in upcoming games but also for studies tograsp the tendencies/characteristics of general basketball games. They are referred to as STATS or box scores and are
well known among basketball-related people. In the STATS of basketball games, many items are employed to provide
information not only on the success of the shot but also on the play during the shooting process, such as shot failures(shot attempts) and pass cuts (steals) leading up to the shot. For this reason, in the research field of game analysis,
many researchers examine the relationship between these STATS and the wins/losses of games or the association
among STATS using data centering on these STATS. Thus, this study reviews studies examining the relationshipamong plays or between plays and wins/losses of games by statistically analyzing these STATS or similar statisticsof the plays, summarizes their findings, and proposes new directions for further studies by pointing out some problems.
2.The relationship between individual plays and game outcomes
Tamaki (2007) reported that the analysis of 2-point and 3-point shooting in games in the quarterfinals or later at the
Kanto Women’s University Championships indicated that winning teams made more 2-point shots attempts or
successful 2-point shots than losing teams and that center players predominantly made many more scores. However,he avoided generalizing this conclusion because the center players in winning teams were taller than those in losing
teams which influenced the team’s strategies. Again, after comparing the successful shot rates of 2-point and 3-pointshooting between entire games and an individual game in the WJBL (Women’s Japan Basketball League), Yaita andNodera (2007) defined the games in which the latter excelled the former as “Good condition” and the games in which
the latter was worse than the former as “Bad condition,” as well as examined the association between these and gameoutcomes. As a result, they reported that although no relationship was found in free throws, a significant relationshipwas detected among 2-point and 3-point shooting. They proposed that coaches should consider making a flexible
Received: 24 May 2022, Revised: 2 July 2022, Accepted: 24 August 2022, Published: 13 October 2022
* Correspondence: knaga@fukuoka-u.ac.jp
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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagichange in their own team’s strategies in the event of a “Bad condition” where the successful shooting rates of theirown teams are low even against weak teams, or in event of a “Good condition” where the successful shooting rates ofthe opponents’ teams are high. Also, Kozar et al. (1994) found that the scores by free throws accounted forapproximately 20% of all the scores in 490 games of the NCAA (National Collegiate Athletic Association) Division
1. They did not find any significant difference in field goals between winning and losing teams, and discovered thatthe winning teams got more scores by free throws than the losing teams. They also asserted that when comparing thefirst 35 minutes and last 5 minutes, the winning teams got significantly more scores by free throws in both close(slight-difference-score) and unbalanced (large margin) games, indicating that the scores resulting from the last-5minute free throws are a decisive factor in determining a win or loss.
By examining the association between the success rates of assists and the rates of plays leading to getting scores ingames of the NBA (National Basketball Association), Maeyama (1996) found a significant correlation between them(r=0.777, p<0.05). They added that in order to win the game, not only must the shooter have excellent shooting ability,
but also required are assists to enable the shooters to effectively make scores.
Goto and Iwaki (2006) reported that in the official games at the Kansai University Championships, teams with highrates of acquiring rebounds won with a probability of 88.8% and a 94.4% probability, particularly in defensive
rebounds. They also found the relationship of y=2.52x-15.9 (r=0.74) between the difference of the rates of getting
rebounds (x) and the difference of scores (y), of which the relationship was significant even after separately tallying
them up by defensive (r=0.74) and offensive (r=0.73) rebounds. Moreover, they pointed out the relationship of
y=1.64x+7.80(r=0.71) between the total scores (y) and the number of rebounds obtained (x) and that of y=0.65x-15.4(r=0.75) between the number of rebounds obtained (x) and the scores from successful rebounds obtained (x), fromwhich successfully acquiring rebounds equates to achieving scores of 0.65 points. Oida and Shimada (1969) studied
the relationship between rebounds and game outcomes in the West-Japan University Championship and reported thatthe rates of acquiring offensive and offensive rebounds were 49% and 63% in winning teams, and 37% and 51% in
losing teams, indicating that winning teams were superior to losing teams in regard to rebounds and that acquiring
rebounds greatly contributed to wins/losses. Again, Okamoto (1989) found the effectiveness of acquiring rebounds
from the studies of wide-ranging finals for high schools, company teams, nationwide professional teams, and ininternational friendship games and cited the reason why acquiring rebounds enables even teams with low successful
shooting rates to have opportunities to shoot again so that the teams can make more scores.
Other studies asserted the usefulness of fast breaks. Matsubara et al. (1973) studied the fast break in games of the AllJapan University Championship and recognized the effectiveness of it because the success rates of the formation of
three-to-two that is advantageous in the final phase were very high, though it is not as significant as the establishedtheory that the success rate of the fast break is high and the three-line fast break, in particular, is well-balanced andthe preferable formation.
In addition, Okamoto (1989) concluded that fouls/violations are an important factor for wins/losses of games from thefact that many violations or fouls were found to be committed by the losing teams in a variety of games involving
players ranging from unskilled to skilled.
In short, we reviewed studies investigating the association between individual plays or fouls/violations and gameoutcomes to date. Although it is natural that the number of successful 2-point and 3-point shots is directly related to
the game outcomes, it can be considered meaningful to compare the ratios of both in shots attempted or both insuccessful shots. We realized that, as important factors influencing the win/loss of games, many researchers havelisted effective assists and fast breaks as measures leading to a shot, getting rebounds as measures that increase thenumber of shots, and fouls/violations giving the opponents the opportunity to shoot.
However, since any of these can be examined individually with the game outcomes, nothing mentions how different
these are from the others in contributing to wins/losses. Thus, in order to compare the contribution to wins/losses withother plays, they needed to be analyzed while simultaneously including plays that are thought to be related.
3. The relationship between plays and wins/losses from a comprehensive perspective
Miyazoe et al. (2007) conducted a comprehensive study using a variety of plays. They converted and subdivided theordinal STATS of games in the Men’s Kanto University Championship into 50 plays according to team and obtainedcorrelation coefficients with final score differences between winning and losing teams. The converted and subdivided
plays were subtotaled into the number of plays in the first and second halves, as well as the total number of plays inan entire game, their ratios, and the number of scores according to plays leading to successful shots (e.g., successful
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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi
shots from turnovers and successful shots from 3-point shots). They reported that significant correlations were foundin “Score;” “Successful shot rates in field goals;” “Score/offensive times;” “Successful rebound rates;” and “DEF EFF([the number of defensive rebounds acquired + opponents’ turnovers]/defensive times)” in greater than six teams.
Based on these plays, they proposed that in order for one’s own team to win, the team needs to achieve greater than80.3 points in “Score;” greater than 45.2% in “Successful shot rates in field goals;” greater than 0.85 in“Score/offensive times;” greater than 52.6% in “Successful rebound rates;” and greater than 45.9% in “DEF EFF,” aswell as ensure the opponent’s team makes less than 74.1 points in “Score;” less than 43.5% in “Successful shot rates
in field goals;” less than 0.77 in “Score/offensive times;” less 47.4% in “Successful rebound rates;” and less than
40.7% in “DEF EFF.”
This study implicitly suggests the need to pay attention to the association between plays. Namely, although theyexamined comprehensive plays from various points of view, their methodology is not considered to be a mutual
association, but rather the examination of individual plays independently. It is reasonable to consider that the plays inbasketball games are mutually related. For example, strategies aimed to steal the ball naturally cause fouls to increase;the frequent use of fast breaks forces a slow-paced offense because of reduced stamina; and frequent 3-point shooting
makes layup shooting through the gap between the defenders effective because the defense line extends. Additionally,
from the standpoint of statistical procedure, in order to statistically examine the substantial relationship among plays
after keeping the influence of the other plays constant, multivariate analysis that simultaneously deals with two or
more variables can be proposed as a measure as will be described in the next section.
4. The relationship between plays and wins/losses using multivariate analysis
Karipidis et al. (2001) examined the association between the following 14 plays and game outcomes in the European
Championship using canonical discriminant analysis (Dillon and Goldstein, 1984): Successful 2-point shots;
Unsuccessful 2-point shots; Successful 3-point shots; Unsuccessful 3-point shots; Successful free throws;
Unsuccessful free throws; Defensive rebounds; Offensive rebounds; Turnovers; Steals; Ball-possession times; Fast
breaks; Fouls; and Assists. As a result, the discriminant function consisting of Defensive rebounds (Partial regression
coefficient of discriminant function = 0.541); Unsuccessful 3-point shots (-0.335); Unsuccessful 2-point shots (0.288); Successful 3-point shots (0.270); Steals (0.057); Fast breaks (0.043); Turnovers (-0.040); and Offensive
rebounds (0.031) out of 14 items had a significant canonical correlation coefficient with 0.704 (chi-square =68.49,
df=8, p<0.0001) of which the correct discriminant ratio was 84.9%, indicating that these eight items were effective indetermining wins/losses.
Pojskic et al.(2009) tallied the following 22 items in 37 games performed in the Olympic Games by winning andlosing teams: 2-point shots made; 2-point shots attempted; Successful 2-point rates; 3-point shots made; 3-point shots
attempted; Successful 3-point-shot rates; Free throws made; Free throws attempted; Successful free-throw rates;
Offensive rebounds; Defensive rebounds; Total rebounds; Assists; Fouls; Turnovers; Steals; Block shots; Total
number of members participating; Fouls received; Scores made by non-starters; Scores made by starters; and Total
scores. Then, they conducted a canonical discriminant analysis using game outcomes as a dependent variable andthese 22 STATS as independent variables. They then reported that the canonical correlation coefficient was 0.809(chi-square =64.926, df=11, p<0.0001), which was extremely significant, sensitivity (the rate that correctly determines
winning teams) was 91.9%, and specificity (the rate that correctly determines losing teams) was 89.2%, indicating ahigh correct determination rate. Also, they claimed that since the obtained structure coefficients (correlation
coefficients between independent variables and discriminant scores) of a discriminant function were Assists (0.505);
Successful 2-point rate (0.481); 2-point shots made (0.410); Defensive rebounds (0.370); Scores made by non-starters(0.317); Steals (0.284); Successful 3-point-shot rate (0.274); Block shots (0.274); Total rebounds (0.268); 3-pointshots made (0.212); The total number of members participating (0.197); Scores made by starters (0.179); Total scores(0.178); 2-point shots attempted (0.139); Free throws attempted (0.121); Free throws made (0.102); Fouls
received(0.053); Successful free-throw rate (0.043); and Offensive rebounds (0.029) in descending order, Assists, 2point shots made, and Defensive rebounds are extremely vital for game outcomes.
Using the 15 items of 2-point shots made; Successful 2-point rates; 3-point shots made; Successful 3-point-shot rates;
Free throws made; Successful free-throw rates; Defensive rebounds; Offensive rebounds; Assists; Steals; Turnovers;
Block shots; Block shots received; Fouls; and Fouls received, Garcis et al.(2007) also analyzed 183 games in the EBA(a Spanish basketball championship that is the fourth tier level in the Spanish basketball league system) and obtained
a discriminant function differentiating winning and losing teams. As a result, significant structure coefficients were
found in Fouls (structure coefficient = 0.38); Assists (0.34); and Defensive rebounds (0.30), and they pointed out thatthese plays were indisputable plays to predict game outcomes.

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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi
In summary, the common method of these studies is that they all use discriminant analysis. The overall fit isdetermined by the correct discriminant rate, and the strength of the association of each play with wins/losses isdetermined by partial regression coefficients and structural coefficients.
5. The relationship between plays and wins/losses under various game conditions
Taking into account the fact that consistent results have not necessarily been obtained in comprehensive studies, somestudies attempting to obtain general results under various game conditions can be found.
First of all, as studies categorizing game conditions from a score difference perspective, we can find Ibanez et al.
(2003). After categorizing games in the Junior World Championship into the three groups of “Blowout,” “Close,” and“Very close,” Ibanez et al. (2003) analyzed them using discriminant analysis. Utilizing independent variables were
Assists; Fouls; Fouls received; Turnovers; Defensive rebounds; Offensive rebounds; Steals; Block shots; 2-point shots
made; 2-point shots failed; 3-point shots made; 3-point shots failed; Free throws made; and Free throws failed. In the
study, the association between plays and game outcomes under various game conditions was examined using thesignificance of structure canonical coefficients (SCC) of the discriminant function obtained. As a result, the following
plays were significant: Defensive rebounds (SCC = 0.50); 2-point shots made (SCC=0.36); and Free throws made
(SCC=0.31) in blowouts, and Defensive rebounds (SCC=0.31) in Close, but no plays in Very close.
Lorenzo et al. (2010) examined 132 games in the Junior European Championship according to score differences, i.e.,
the three categories of “Close,” “Slightly close,” and “Large margin,” using discriminant analysis. For this, gameoutcomes were a dependent variable and independent variables were the 13 items of 2-point shots made; 2-point shots
failed; 3-point shots made; 3-point-shots failed; Free throws made; Free throws failed; Defensive rebounds; Offensive
rebounds; Assists; Steals; Turnovers; Block shots; and Fouls. As it turned out, the canonical correlations in the threegroups of “Close,” “Slightly close,” and “Large margin” were 0.52(chi-square = 31.7; p<0.05); 0.84(chi-square =139.3,
p<0.01); and 0.97 (chi-square =52.2, p<0.05), respectively. Significant structure coefficients (SC) were found in
Assists (SC=0.33) and Turnovers (SC=0.47) in close games; 2-point shots made (SC = 0.34); and Defensive rebounds(SC = -0.36) in slightly-close games, and 2-point shots made (SC = 0.37) in large-margin games, in which plays
affecting wins/losses were different depending on game patterns.
Some studies examined the relationship between plays and rankings that put entire games in a league together, but not
individual game outcomes, because each game outcome is influenced by various factors other than ability.
Ibanez et al. (2008) also obtained a discriminant function to distinguish between higher- and lower-ranking teams thatwere obtained from game outcomes through all of the seasons of the Spain Professional League, Division 1. The plays
used were 2-point shots made; 2-point shots failed; 3-point shots made; 3-point-shots failed; Free throws made; Free
throws failed; Defensive rebounds; Offensive rebounds; Assists; Steals, Turnovers; Block shots; Block shots received;Fouls; and Fouls received. Consequently, a function significant at the 1% level was able to be obtained and it could
correctly predict both groups with an 82.4% probability. Then, the structure coefficients (SC) that are the correlation
coefficients between each item and discriminant scores were Assists (SC=0.47); Steals (SC=0.34); Block shots(SC=0.30); Fouls (SC=-0.24); Free throws made (SC=0.22); Defensive rebounds (SC=0.20); 3-point shots made
(SC=0.16); Offensive rebounds (SC=0.16); 2-point shots made (SC=0.08); 2-point shots failed (SC=-0.08); Fouls
received (SC=0.07); Block shots received (SC=-0.06); Turnovers (SC=-0.05); 3-point-shots failed (SC=-0.04); andFree throws failed (SC=0.02). They asserted that among them, Assists, Steals, and Block shots -- the top three plays-- were effective in differentiating between higher- and lower-ranking teams.
Sampaio et al. (2003) conducted a more comprehensive analysis, combining the three factors of “Regular seasons andplay-off;” “Home and Away;” and “Game-score difference.” Targeted games were 409 games (353 games in regularseasons and 56 in play-off) held in the Portugal Professional League, which were categorized into Very close games(less than 8-point difference), Close games (greater than a 9- and less than an 18-point difference), and Blowout games(greater than 19 points) according to Home and Away games. After checking the 13 items of Assists; Block shots;
Fouls; 2-point shots made; 2-point shots failed; 3-point shots made; 3-point-shots failed; Free throws made; Free
throws failed; Turnovers; Defensive rebounds; Offensive rebounds; and Steals, they conducted a canonicaldiscriminant analysis on them. The results indicated that all the functions were significant at the 5% level because all
the Wilks’ lambdas were less than 0.6. Significant structure coefficients were detected in 2-point shots made in
Blowouts and Home games during the regular season; Free throws made and Defensive rebounds in the Close andAway games in regular seasons; Fouls; 2-point shots failed and Free throws made in Blowouts and Home games
during the regular season; Free throws made, 3-point shots made and Defensive rebounds in Blowouts and Away
games during the regular season; Fouls and Offensive rebounds in Blowouts and Home games in the play-off; andCopyright: © SANKEI DIGITAL INC.

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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi
Free throws made, Offensive rebounds, and Free throws failed in Blowouts and Away games during the play-off. That
is to say, whereas only a small number of plays showed a significant relationship with the wins/losses of games in
Close and Blowout games, many various kinds of plays did so in Very close games.
As has been noted, an overview of the results of these studies shows that they have not succeeded in obtaining common
findings, because the findings obtained are diverse, depending on the situation, even among studies that attempt toobtain more general findings under detailed conditions.
6. Games of unskilled junior players
We have reviewed studies of games in which international or national skilled players performed. Since skilled players
have the most desirable characteristics for basketball players, many studies have used the games they played. However,some researchers studied the games in which junior players performed, who were still unskilled and did not display
desirable physical fitness or other basketball characteristics.
Focusing on Offensive time, The number of players participating; The number of passes; Group skills; Offensive
strategies; and Efficacy, Ortega et al. (2006) analyzed 34 games in the Andalusia-Region Championship in which 16
teams, composed of 184 boys aged 15 to 16 years old, competed. The results of this analysis pointed out that thecharacteristics of the winning teams usually showed less dribbling and fast breaks with only two players.
Koh et al. (2011) examined the association between that were performed by 80 male players aged from 16 to 17 yearsof age in 20 countries in the FIBA 33 (irregular 3-on-3 games) games of the Junior Olympic Games as independent
variables and wins/losses using discriminant analysis. The plays/fouls were the Successful rates of the 3-point shots;
Team fouls; Successful rates of field goals; Successful rates of 2-point shots; Mean standing height according to team;Team personal fouls; Assists; The number of free throws granted; and The number of rebounds acquired. As a result,the canonical correlation coefficient was significant at the 5% level with 0.86, and significant structure coefficients(SC) were found, in descending order, in the Successful rate of the 3-point shot (SC=0.53); Team personal fouls(SC=0.53); Successful rate of field goals (SC=0.49); Successful rate of 2-point shots (SC=0.43); Mean standing heightaccording to team (SC=0.39); and Team personal fouls (SC=0.34). Based on these facts, they focused on aggressive
offensive play styles as well as high standing height and high rates of successful shots and pointed out that juniorplayers received many fouls from opponents because of their high aggressiveness and they themselves committed
many fouls.
Summing up, in the case of junior players, characteristics not seen in skilled players were found, specifically dribbling
less, fast breaks involving only two players, and fouls. This fact allows us to suggest the need to conduct further
research because when the subjects are less skilled players, it is easily assumed that different results are likely to beobtained.
7. Structure of plays/factors
Studying comprehensive plays/factors allows researchers to handle many plays/factors, but sometimes makes it hardfor them to concisely summarize the findings obtained. For instance, if a certain type of play shows a strong
relationship with another play, which also shows a strong relationship with wins/losses, it is expected that the firstplay would also show a strong relationship with wins/losses. In a situation like this, it is reasonable to examine theassociation with wins/losses after factoring in plays/factors with high correlations to each other in advance. For thisreason, some studies examined the structure of plays/factors in advance before examining the association with gameoutcomes.
Sakai et al. (1998) examined the structure of plays in the games between two specific teams in the Kanto-University
League. The plays used were 2-point shots made; 3-point shots made; Free throws made; Defensive rebounds;
Offensive rebounds; Assists; Steals; Successful acquisition of lost balls; and Successful acquisition of held balls asplays contributing to a win; as well as 2-point shots failed; 3-point-shots failed; Free throws failed; Failed acquisitionsof held balls; The number of missed plays; Defensive fouls; and Offensive fouls as not contributing to a win. Moreover,they conducted a principal component analysis on 19 items adding the three composite scores such as “Contribution
to scoring” (obtained by weighing the frequency of plays contributing to scoring); “Contribution without scoring” thatis plays without scoring; and “General contribution,” and extracted three components. The first component seemed to
be a factor directly related to scoring because “Contribution to scoring” and “General contribution” were included.
The second principal component did not contribute to scoring because only “Contribution without scoring” was loaded.
The third principal component was not used in the analysis because it comprised only Defensive fouls and Offensive
rebounds and no obvious interpretation could be given. Then, they examined the individual characteristics of eachplayer using the first principal component and the second principal component that directly and indirectly contributed
to scoring, respectively.
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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi
Oga et al. (1990) applied factor analysis to the following 17 items: 3-point shots made; 3-point shots attempted; 2point shots made; 2-point shots attempted; Free throws made; Free throws attempted; Total scores; Personal fouls;
Offensive rebounds; Defensive rebounds; Recovery; Assists; Steals; Violations; Lost balls; Block shots; and Good
defense in games in men’s competitions at the Olympic Games. Then, they extracted the four factors of “2-point shots,”“3-point shots,” “Defensive rebounds,” and “Free throws.”
Using the 17 items of 3-point shots made; 3-point shots attempted; 2-point shots made; 2-point shots attempted; Free
throws made; Free throws attempted; Total scores; Personal fouls; Offensive rebounds; Defensive rebounds;
Recoveries; Assists; Steals; Violations; Lost balls; Block shots; and Good defense, Kodama (1989) factored games inmen’s competitions at the Olympic Games and derived “Free throw by receiving fouls;” “2-point shots;” “3-pointshots;” and “Aggressiveness of defense.” Then, he concluded that the offensive system is built upon centering on thetwo factors of “2-point shots” and “3-point shots” and that consistent competency to acquire the ball, represented by“Aggressiveness of defense,” is related to game outcomes.
Again, there are other studies employing non-metric multidimensional scaling as a technique to extract structure.
Kodama and Suzuki (1990) applied TORSCA, one method of non-metric multidimensional scaling, into the malecompetitions at the Universiade Games (World University Games) and extracted a three-dimensional space,interpreted as “Under-basket shooting after the competition in inside plays;” “Short- and middle-range shooting;” and“Aggressive passing and 3-point shooting.” The items used were the following 17 items: 3-point shots attempted; 3point shots made; 2-point shots attempted; 2-point shots made; Free throws attempted; Free throws made; Total scores;
Offensive rebounds; Defensive rebounds; Turnovers; Assists; Steals; Personal fouls; and Block shots. In the sameway, Suzuki and Kodama (1988) applied TORSCA to 14 plays (3-point shots attempted; 3-point shots made; 2-pointshots attempted; 2-point shots made; Free throws attempted; Free throws made; Total scores; Offensive rebounds;
Defensive rebounds; Turnovers; Assists; Steals; Personal fouls; and Block shots) in the Universiade Games andextracted the following three axes: “Basic offensive power through tug-of-war under the basket;” which relates to thescore-getting pattern by 2-point shots; “3-point shots” which leads to the score-getting pattern by 3-point shots; and“Opponents’ fouls and free throws” which is the score-getting pattern by free throws resulting from the opponents’fouls.
Again, Kodama (2009) applied ALSCAL, non-metric multidimensional scaling, to 16 items performed in the WorldChampionship, i.e., Playing times by player; 2-point shots attempted; 2-point shots made; 3-point shots attempted; 3point shots made; Free throws attempted; Free throws made; Offensive rebounds; Defensive rebounds; Assists;
Turnovers; Steals; Block shots; Personal fouls; Fouls received; and Total scores, and extracted a three-dimensional
space composed of “Inside/outside plays involved with scoring;” “Accuracy of plays and athletic ability;” and “Thelength of participating games.” He discussed the association between these axes and game outcomes.
By applying ALSCAL to the games played by the top eight teams in the World Championship, Kodama et al. (2010)
and Kodama (2009) derived a three-dimensional space, which comprised “Inside/outside plays involved with scoring;”“Accuracy of plays and athletic ability;” and “Air superiority (Seiku-ken in Japanese).” From the results, they reported
that 2-point shots, 3-point shots, and Free throws, which are measures used to score, were involved with the first,
third, and second dimensions, respectively.
To sum up, in terms of methodology, these studies mainly used multidimensional scaling or principal component
analysis to clarify the factors that explain the relationship among the various plays based on the interpretation of theaxes. Although factor analysis usually involves the rotation of axes (factors) to make them easier to interpret, thesewere not commonly used in these studies.
8. Discussion
8.1 Findings of studies of plays/factors involved with game outcomes using multivariate analysis
Up till now, we have reviewed the research findings of studies examining the association between plays/factors andwins/losses mainly using discriminant analysis. Research findings based on studying skilled players playing in the
Olympic Games, European Championship, and Nationwide Top Leagues, in particular, pointed out that Defensive
rebounds; 2-point shots made; Free throws made; Fouls; Assists; and others are important plays determining gameoutcomes. However, consistent agreement is not necessarily seen with regard to what plays/factors among them are
the most decisive ones. With junior players, in particular, the unique characteristics that winning teams display -fewer dribbling plays, only two-men fast breaks, and frequent fouls -- were seen. This fact enables us to actually

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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi
attempt to study various levels of players, because different plays/factors were found as crucial factors depending on
the player’s skill/physical fitness level when studying less skillful players.
8.2 Findings of studies of the structure of plays/factors using multivariate analysis
We have reviewed the findings of the studies about the structure of plays/factors in order to examine the relationshipbetween plays/factors and game outcomes while considering the mutual relationships among them. Studies using
factor analysis or non-metric multidimensional scaling yielded a relatively small number of dimensions, e.g., three orfour factors/axes. Especially when limiting the targeted games to the Olympic Games or equivalent international
games, common factors/axes, such as 2-point-shot related plays, 2-point-shot related plays, and free-throw related
plays from fouls, were commonly derived. That is to say, a series of plays leading to final shooting was structured.
However, in university-level games, no structure that was differentiated by shooting method could be obtained. Only
undifferentiated structure, categorized into the two groups of plays common to all shooting methods and plays notleading scoring, was able to be derived. This fact allows us to assume that the same undifferentiated structure will be
extracted in games with junior players play because the players are also unskillful.
8.3 Statistical methods
From the viewpoint of analyzing procedures using multivariate analysis, (canonical) discriminant analysis has beencommonly used in studies examining the association between plays/factors and game outcomes. The reason that ttests according to items individually was not used, but discriminant analysis was used instead, can be explained byexamining the relationship between plays/factors and game outcomes while statistically keeping the influence of othervariables constant. Furthermore, structure coefficients were utilized to find the effective independent variables. Since
plays/factors are not performed nor occur independently through the run of the games, but mutually relate to eachother, a certain relationship will be found. Thus, since a relationship like this is worth considering, discriminant
analysis can be said to be one of the preferable methods used for analysis. Incidentally, discriminant analysis is used
not only to differentiate only two groups but can also be used to differentiate two or more groups. However, in all thestudies reviewed, it was used only for the categorization of two wins and losses. When limiting game outcomes toonly two wins and losses, the application of logistic regression analysis based on binomial distribution can beconsidered. Although the normality and homoscedasticity of independent variables in each group are required asassumptions to their application, no assumptions like this can be made in logistic regression analysis (Hair et al.,2010).
Again, the studies reviewed used mainly factor analysis (principal component analysis) and non-metricmultidimensional scaling, in which the factors obtained or dimensions were not so many, with only three to four
factors/axes. Additionally, the considerable variance was explained by the first and second dimensions/axes, andalmost all of the plays/factors that were included in the two dimensions/axes and the successive third and/or fourth
were less convincing. This prevented researchers at times from clearly interpreting factors (in particular, principal
components) and axes. This is thought to have been caused by not rotating principal components or axes. Doing sowould make the variances of principal components or axes even and enables their interpretation. In order to understandthe structure obtained more deeply, rotating the principal components or axes, such as in Varimax rotation (Harman,
1976), is recommended.
9. Conclusion: our new proposals
We reviewed and summarized the findings from the perspectives of a) the relationship between individual kinds ofplays and wins/losses; b) the comprehensive association between plays/factors and wins/losses; c) the relationshipbetween plays/factors and wins/losses using multivariate analysis; d) the association between plays/factors andwins/losses in various situations; e) games involving junior basketball players; and f) the structure of plays/factors.
This allowed us to realize that no consistent findings can be obtained in terms of what plays/factors contribute mostto wins/losses, the sequence of plays leading to scoring (e.g., two-point shooting, three-point shooting, and free
throwing) by skilled players, and that discriminant analysis is commonly used in the studies. Based on these findings,
we devised the following new proposals:
1) No consistent findings about the relationship between plays/factors and wins/losses were able to be obtained andthat was dependent upon the games. This is because conditions are not able to be controlled such as experiments or
measurements, and the plays themselves will change depending on the opponents. For this reason, researchers needto exert more effort to analyze various kinds of games in the future.
2) The feature of an alternative to a win or loss in games allows us to eliminate the need to use only discrimination
analysis because it can discriminate three or more. Additionally, the normality and homoscedasticity of independent
variables are assumed for discriminant analysis, but not for logistic regression analysis. For this reason, we can also
recommend logistic regression analysis.

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Statistical examination into the structure of plays and its relationship wins/losses in basketball games played by excellent
players from the standpoint of STATSKen Nagamine and Osamu Aoyagi3) When investigating the structure of plays/factors, a rotation that easily interprets axes in MDS or principal
components has not been applied to the derived structure, which has made the research outcome ineffective. To makethe research findings useful, the rotation of axes or principal components should be actively utilized.
Notes
Note.1) Although different expressions/words are used depending on the researchers, the discriminant function andcanonical discriminant function were the same. Similarly, the canonical correlation and multiple correlation
ratios were also identical.
Note.2) Whereas canonical correlation indicates the goodness-of-fit, Wilks’ lambda indicates the badness-of-fit (Hairet al., 2010).
Note.3) TORSCA (TORgerson's SCAling) is one of the programs used in computing non-metric multidimensional
scaling and was developed by Young and Torgeson (1967) at an early stage. With it, configuration is obtained
while optimization and monotonic regression are repeated mutually.
Note.4) ALSCAL (Alternating Least Squares SCALing) is one of the programs used in computing both metric andnon-metric multidimensional scaling and was developed by Takane, Young, and de Leeuw (1977). Its solution
is obtained so that the squared sum of the difference between the squared dissimilarities and distances in thespace becomes the least. Its computation is conducted using the alternating least square method.
Author Contributions
Conceptualization, K.N.,O.A; methodology, K.N., O.A; validation, K.N.,A.O.; investigation, K.N, A.O; writing—original draft preparation, K.N.,A.O; writing—review and editing, K.N.,A.O; visualization, A.O.; supervision, K.N.,
A.O.; project administration, A.O.; funding acquisition, A.O;
All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported (in part) by funding from Fukuoka University (Grant No. 196008).
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