Empiricalresult
Themodel developed in this research paper is meant to investigate theeffect of team rank, years of experience and age of a player on theincome earned. One or all of these factors may be used incontracting a player in a team. In this case the dependent variable(Y) is the income earned ($). Four independent variables consideredin his paper are team rank (X_{1}),years of experience (X_{2}),age of the player (X_{3})and points per game (X_{4}). A relationship between the variables is developed by making aregression of the data and using linear models [ CITATION Mat14 l 1033 ].
Hypothesis:H_{o}

Team ranking determines the income of a player.

Years of experience of a player determine the income of the player

Age of the player determines the income of the player.

Points scored per game determines the income of the player
Thenull hypothesis: H_{o}

Team ranking does not determine the income of a player.

Years of experience of a player does not determine the income of the player

Age of the player does not determine the income of the player.

Points scored per game do not determine the income of a player.
Theteam rank is given by a number between 1 and 30. Age of the player,and years of experience are given by a whole number. Point score pergame are given by numbers. It is expected that the linearrelationship between Y and X_{1}has a negative slope indicating that a drop in ranking willultimately decrease the income. It is also expected that the linearrelationships between Y and X_{2},Y and X_{3,}and Y and X_{4}havea positive slope. A more experienced player is expected to get moreincome and consequently an older player will also earn more.
Thetable below shows data collected from different teams involvingdifferent players in different positions and a combination of factorsrequired in our models.
Table1: Data
Season 20082009 

Player 
Team: 
Salary($) 
Position 
Rebound/ game 
Asissts/ game 
Points/ game 
Years of Exp. 
Age of Player 
Team Rank 
Maurice Williams 
Cavaliers 
8353000.00 
Guard 
3.8 
4.1 
17.8 
5 
26 
1 
Lebron James 
Cavaliers 
14410581.00 
Forward 
7.6 
7.2 
28.4 
5 
24 
1 
Zydrunas Ilgauskas 
Cavaliers 
10841615.00 
Center 
7.5 
1 
12.9 
10 
33 
1 
Kobe Bryant 
Lakers 
21262500.00 
Guard 
5.2 
4.9 
26.8 
12 
31 
2 
Pau Gasol 
Lakers 
15106000.00 
Forward 
9.6 
3.5 
18.9 
7 
28 
2 
Andrew Bynum 
Lakers 
2769300.00 
Center 
8 
1.4 
14.3 
3 
21 
2 
Jameer Nelson 
Magic 
7600000.00 
Guard 
3.5 
5.4 
16.7 
4 
26 
4 
Rashard Lewis 
Magic 
16447871.00 
Forward 
5.7 
2.6 
17.7 
10 
29 
4 
Dwight Howard 
Magic 
13041250.00 
Center 
13.8 
1.6 
20.6 
4 
23 
4 
Chauncey Billups 
Nuggets 
11050000.00 
Guard 
3 
6.4 
17.9 
11 
32 
5 
Carmelo Anthony 
Nuggets 
14410581.00 
Forward 
6.8 
3.4 
22.8 
5 
24 
5 
Johan Petro 
Nuggets 
1939893.00 
Center 
2.3 
0.4 
2.2 
3 
23 
5 
Tracy McGrady 
Rockets 
19614000.00 
Guard 
5 
4.4 
15.6 
11 
29 
9 
Ron Artest 
Rockets 
7400000.00 
Forward 
5.8 
3.3 
17.1 
9 
28 
9 
Yao Ming 
Rockets 
15070550.00 
Center 
9.9 
1.8 
19.7 
6 
29 
9 
Jason Kidd 
Mavericks 
21372000.00 
Guard 
6.2 
8.7 
9 
14 
35 
10 
Dirk Nowitzki 
Mavericks 
18077904.00 
Forward 
8.4 
2.4 
25.9 
10 
30 
10 
Erick Dampier 
Mavericks 
11553000.00 
Center 
7.7 
1 
5.7 
12 
33 
10 
Joe Johnson 
Hawks 
14232566.00 
Guard 
4.4 
5.8 
21.4 
7 
27 
13 
Josh Smith 
Hawks 
10000000.00 
Forward 
7.4 
2.5 
16 
4 
23 
13 
Zaza Pachulia 
Hawks 
4000000.00 
Center 
10.7 
1.4 
11.8 
5 
24 
13 
Allen Iverson 
Piston 
20840625.00 
Guard 
3.1 
4.9 
17.4 
12 
33 
17 
Rasheed Wallace 
Piston 
13680000.00 
Forward 
7.4 
1.4 
12 
13 
34 
17 
Kwame Brown 
Piston 
4000000.00 
center 
5 
0.6 
4.2 
7 
26 
17 
Vince Carter 
Nets 
14724125.00 
Guard 
5.1 
4.7 
20.8 
10 
32 
20 
Bobby Simmions 
Nets 
9920000.00 
Forward 
3.9 
1.3 
7.8 
6 
28 
20 
Brook Lopez 
Nets 
2098560.00 
Center 
8.1 
1 
13 
1 
20 
20 
Jose Calderon 
Raptors 
7348018.00 
Guard 
2.9 
8.9 
12.8 
5 
27 
22 
Chris Bosh 
Raptors 
14410581.00 
Forward 
10 
2.7 
22.7 
5 
24 
22 
Jermaine O`Neal 
Raptors 
21372000.00 
Center 
7 
1.6 
13.5 
12 
30 
22 
Monta Ellis 
Warriors 
11000000.00 
Guard 
4.3 
3.7 
19 
3 
23 
24 
Corey Maggette 
Warriors 
8600000.00 
Forward 
5.5 
1.6 
18.6 
9 
29 
24 
Andris Biedrins 
Warriors 
9000000.00 
Center 
11.6 
2 
11.9 
4 
22 
24 
O.J Mayo 
Grizzlies 
3875040.00 
Guard 
3.8 
3.2 
18.5 
1 
21 
26 
Hakim Warrick 
Grizzlies 
2119102.00 
Forward 
5 
0.8 
11.6 
5 
23 
26 
Hamed Haddadi 
Grizzlies 
1572221.00 
Center 
2.5 
0.4 
2.5 
1 
23 
26 
Baron Davis 
Clippers 
11250000.00 
Guard 
3.7 
7.7 
14.9 
9 
29 
28 
Marcus Camby 
Clippers 
10000000.00 
Forward 
11 
2 
10.3 
12 
34 
28 
Chris Kaman 
Clippers 
9500000.00 
Center 
8 
1.5 
12 
5 
26 
28 
Gilbert Arenas 
Wizards 
14653466.00 
Guard 
4.5 
10 
13 
7 
27 
30 
Antawn Jamison 
Wizards 
9923285.00 
Forward 
8.5 
1.9 
22.2 
10 
32 
30 
Brendan Haywood 
Wizards 
5500000.00 
Center 
7.3 
1.3 
9.7 
7 
29 
30 
Thedata was then analyzed using regressions. Descriptive Statisticalanalysis of the income gives the following information tabled below.
Table2: Descriptive statistics for the dependent variable Y (income $).
Income/ Salary $ 

Mean 
11046181.76 
Standard Error 
877506.5421 
Median 
10920807.5 
Mode 
14410581 
Standard Deviation 
5686892.36 
Sample Variance 
3.23407E+13 
Kurtosis 
0.681917666 
Skewness 
0.099271633 
Range 
19799779 
Minimum 
1572221 
Maximum 
21372000 
Sum 
463939634 
Count 
42 
Confidence Level (95.0%) 
1772160.394 
Thisshows the average income of a player is $11,046,182.
Aregression analysis done on the variables regressed against incomegives the graphs shown in the figures below.
Regressionon income against team ranking gives the results in figure 1 below.
Figure1: Graph of Y against X_{1}
Regressionon income against years of experience gives the results in figure 2below.
Figure2: Graph of Y against X_{2}
Regressionon income against player’s age gives the results shown in figure 3below.
Figure3: Graph of Y against X_{3}
Regressionon income against points per game gives figure 4 below.
Figure4: graph of Y against (X_{4})
Thetables below give an analysis of variability for the dependentvariable Y (income $) against the four variables or predictors X_{1},X_{2},X_{3}andX_{4}.
Table3: ANOVA test for variable X_{1}
SUMMARY OUTPUT 

Regression Statisticsteam ranking 

Multiple R 
0.24139196 

R Square 
0.05827008 

Adjusted R Square 
0.03472683 

Standard Error 
5587276 

Observations 
42 

ANOVA 

  
df 
SS 
MS 
F 
Significance F 

Regression 
1 
7.72644E+13 
7.72644E+13 
2.475023011 
0.123545169 

Residual 
40 
1.24871E+15 
3.12177E+13 

Total 
41 
1.32597E+15 
  
  
  

  
Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
13147478.1 
1589741.988 
8.270196176 
3.4139E10 
9934489.731 
16360466.48 
Team ranking 
139422.51 
88622.3506 
1.573220586 
0.123545169 
318534.956 
39689.94371 
Thisresults show that team ranking is not significant at 95% confidencelevel because the pvalue > 0.05. The intercept is negativeindicating that a drop in team ranking negates the salary of theplayer. The Fstatistic is 2.47.
Table4: ANOVA test for variable X_{2}
SUMMARY OUTPUT 

Regression Statistics years of experience 

Multiple R 
0.678616509 

R Square 
0.460520366 

Adjusted R Square 
0.447033375 

Standard Error 
4228871.296 

Observations 
42 

ANOVA 

  
df 
SS 
MS 
F 
Significance F 

Regression 
1 
6.1064E+14 
6.10636E+14 
34.14552377 
7.84644E07 

Residual 
40 
7.1533E+14 
1.78834E+13 

Total 
41 
1.326E+15 
  
  
  

  
Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 

Intercept 
3308479.811 
1476221.97 
2.241180445 
0.030631907 
324923.948 
6292035.673 

Years of experience 
1079679.342 
184768.487 
5.843417131 
7.84644E07 
706248.3039 
1453110.38 
  

Table5: ANOVA test for variable X_{3}
SUMMARY OUTPUT 

Regression Statistics age of player 

Multiple R 
0.586076186 

R Square 
0.343485296 

Adjusted R Square 
0.327072428 

Standard Error 
4665080.793 

Observations 
42 

ANOVA 

  
df 
SS 
MS 
F 
Significance F 

Regression 
1 
4.55451E+14 
4.55451E+14 
20.9278052 
4.54101E05 

Residual 
40 
8.70519E+14 
2.1763E+13 

Total 
41 
1.32597E+15 
  
  
  

  
Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Intercept 
11331119.05 
4944224.624 
2.291788887 
0.027259563 
21323769.7 
1338468.439 

Age of player 
817257.9427 
178647.6501 
4.574691815 
4.54101E05 
456197.5771 
1178318.308 
  
Table6:ANOVAtest for variable X_{3}
Regression Statisticspoints per game 

Multiple R 
0.44335508 

R Square 
0.19656373 

Adjusted R Square 
0.15427761 

Standard Error 
5140681.94 

Observations 
21 

ANOVA 

  
df 
SS 
MS 
F 
Regression 
1 
1.22842E+14 
1.22842E+14 
4.648422055 
Residual 
19 
5.02106E+14 
2.64266E+13 

Total 
20 
6.24948E+14 
  
  
  
Coefficients 
Standard Error 
t Stat 
Pvalue 
Intercept 
3499077.08 
3121868.534 
1.120827813 
0.276328577 
Points per game 
457373.108 
212137.6968 
2.156019957 
0.044110432 
Theresults obtained from the figures above show that the experience of aplayer and the age are significant at 95% confidence levels indetermining the income of a player because their pvalues are lessthan 0.05. Team ranking and points per game do not significantlydetermine income. A player with no experience is expected to have anincome of $3,308,480. Figures 1, 2, 3 and 4 confirm the expectationsand predictions made in this paper about how the variables arerelated. The expectations were correct and they are furthersupported by the ANOVA results in tables 3, 4 and 5. As for thehypotheses, the following can be concluded:

Hypothesis 1: accept the null hypothesis. Team ranking is not significant in determining the income of the player.

Hypothesis 2: reject null hypothesis. Experience of the player is significant in determining the income of a player.

Hypothesis 3: reject null hypothesis. Age of the player is significant in determining the income of the player.

Hypothesis 4: accept null hypothesis. Points per game do no determine the income of a player significantly.
Theprediction lines or models that confirm the expectations are:

For relationship between team ranking and income Y = 13147478 – 139423X_{1}. This is a negative relationship since the slope of the model line is negative.

For the years of experience and income Y = 3308479 + 1079679X_{2}. This is a positive relationship. It is also a model with the highest positive slope indicating that this is the strongest factor that affects the income of the player.

For the age and income Y = 11331119 + 817258X_{3}. This is a positive relationship. The slope is lower than that of years of experience.

For points per game Y = 3499077 + 457373X_{4}. This is a positive relationship and therefore an increase in points per game will result in an increase in income although the increase is not significant.
Fromthe Rsquare value, the effect of ranking on income is only 5.8%. The tvalue is negative hence showing a negative correlation betweenthe two variables and therefore team ranking can not affect incomesignificantly.
Yearsof experience has an Rvalue of 0.4605. This implies that 46% ofvariability in income of a player is determined by the years ofexperience. The tvalue is 5.84 showing that there is a strongcorrelation between the income of a player and years of experience.The Fstatistic is 34.
Agehas a positive correlation since the tvalue is 4.57 also showingthat age determines the income of a player. With an Rvalue of0.3434, it means that 34.34% of a players’ income is determined bythe age of the player. The Fstatistic is 21.
Pointsper game will determine 19.66% of variability in income as indicatedby the Rsquare value. The tvalue of 2.15 indicates a positivecorrelation and therefore points per game have a positive correlationwith income. The Fstatistic is 4.6.
Amodel that would combine the linear relations to fit the multipleregressions would be Y = B_{1}– B_{2}X_{1}+_{}B_{3}X_{2}+B_{4}X_{3}+_{}B_{5}X_{4}in which the constants represented by B will take certain values orproportions of income.
Themultiple regression model considered in this research paper will havesatisfactorily addressed the expectations of the researcher, andthere was enough evidence shown to conclude that the income of aplayer is strongly influenced and determined by the years ofexperience of the player, and the age of the player. However, a dropin team ranking results in a drop in the income of the player. Butan increase in points per game is likely to give an increase inincome although not significant.
Conclusion
Thisresearch paper reveals that the income of a player will depend on anumber of variable factors or predictors. The variables that wereconsidered in this case were team ranking, years of experience of theplayer, age of the player and points per game. In order to makeproper judgment, statistical tools were used in analyzing datacollected from several teams. The results clearly indicated that theincome of a player is determined by these factors in differentstrengths. Experience of the player was the strongest determiningfactor compared to team ranking, age of the player and points pergame. This implies that the contract of a player must be subjectedto a multiple regression model developed by management in order todecide on the salary or income of the player. The model willconsider all the necessary factors to arrive at the income of theplayer at a given time. This will remove complains that may arisefrom players about differing incomes, and it will also present anobjective way of paying the players of the team.
However,for the results to have a standard model that can be applied overtime, sampling of the teams should be wide, covering a large numberof teams which can give a proper representation of the population. In this paper, 14 teams were considered in the data collection andworking out the correlations. A more standard model can be developedif 30 or more teams are considered for statistical analyses. It isalso assumed that the data collected is correct and honest asconcerns variables like age of the player and years of experience. If the data is biased, the model developed will also be biased andthis will ultimately be a limitation to the application of the modelas a tool for determining the income of a player. Nevertheless,determining the income of a player requires the management to giveweight to a number of variables as indicated in this paper withoutsubjectivity, before deciding on the salary of a player.
Futureresearchers can also test whether the total number of games a playerhas undertaken can also be a significant factor in determining theincome of the player. But contemporary managers cannot afford to paya player without looking at several influencing factors such as theones considered in this paper.
References
Matlab. "Matlab and simulink." Matlab. 10 November 2014 <http://www.mathworks.com/help/stats/fstatisticandtstatistic>.
Summary.
Playersincome based on Team ranking, Years of experience, Age of player, andpoints per game.
Variabilityof income for many players is based on a number of factors whichinclude team ranking, years of experience age of the player andpoints per game. Questions often arise as to why some players arepaid less as compared to others either in the same team or adifferent one, but with the same roles. Construction of a model toanalyze data can help in getting some answers to such questions andhence help management to make decisions that are unbiased.
Inthis research paper, data was collected from 14 teams during the 2008and 2009 season and recorded as in table 1 of this research paper. ANOVA tests and regressions were carried out multiple linear modelsto address several expectations of the researchers. From the resultsof the models, conclusions were made to address the question ofvariability in income of players.
Thetests were done at 95% confidence levels and found out to agree withthe expectations of the research team. It was established that theexperience of the player, the age and points per game have a positiveeffect on variability in income. This implies that a contract signedbetween a player and a given team will definitely look into these andother factors modeled by the teams in order to determine the salaryof a given player. However, team ranking and points per game, thoughpositive in linear relation did not account for significantvariability in the income of a player.
Fromthese tests and results, a multiple regression model was developed todetermine the income of the player by considering the variables orpredictors. It is clear from this research paper that paying aplayer without making considerations of the factors one is subjectedto may result in under payment or over payment hence resulting incomplains. To avoid this problem, it is necessary to develop astandard model that would be used in determining the income ofplayers.