3.1. Correlation and Descriptive Statistics
As PL is categorical data, the correlation between PL and salary was analysed using Spearman’s rho, and the continuous variables—salary, PTS, and EFF—were analysed using Pearson’s correlation coefficients in Table 3. The correlations between salary and PL, PTS, and EFF were significant at .65 (p < .001), .59 (p < .001), and .61 (p < .001), respectively. There was no multicollinearity problem because there was no high correlation. However, the correlation between PTS and EFF was very high at .93 (p < .001). Therefore, it is desirable to insert these two variables separately, as there was a multicollinearity problem in this context. Table 4 shows the descriptive statistics for levels 1 and 2.
Table 3
|
Salary
|
PL
|
PTS
|
EFF
|
Salary
|
1
|
|
|
|
PL
|
.65***
|
1
|
|
|
PTS
|
.59***
|
.43***
|
1
|
|
EFF
|
.61***
|
.45***
|
.93***
|
1
|
***p < .001.
a PL = performance level; PTS = points; EFF = efficiency
|
Table 4. Descriptive Statistics for Levels 1 and 2.
Level 1 descriptive statistics
(N = 3,247)
|
Level 2 descriptive statistics
(N = 30)
|
Variable Name
|
Skewness
|
Kurtosis
|
M
|
SD
|
Variable
Name
|
M
|
SD
|
Salary
|
1.38
|
1.59
|
5.66
|
5.17
|
Mslope
|
0.05
|
0.16
|
PTS
|
0.82
|
0.44
|
6.63
|
4.70
|
TM
|
4.72
|
0.59
|
EFF
|
0.64
|
-0.29
|
7.39
|
4.92
|
SDslope
|
0.07
|
0.20
|
|
|
|
|
|
TSD
|
4.58
|
0.90
|
a One million dollars per a unit for salary mean; one hundred scores per a unit for points and efficiency mean; PTS = points; EFF = efficiency; TM = Total mean across 11 years of mean salary for each team; TSD = Total mean across 11 years of Standard Deviation of salary for each team
3.2. Group Effect Analysis
The random-coefficient model in Table 5 shows that γ00 was 5.69 (t (29) = 48.08, p = .001), indicating that the mean salary of all participants at the individual level was similar to that shown in Table 3 (i.e. 5.66). At Level 1, after the level of performance explaining individual salarycurrent season, EFFprevious season, and EFFprevious season × level of performance was entered, the effect on individual salarycurrent season was examined. The results presented a positive fixed effect of EFFprevious season (β = 0.28, t (29) = 13.85, p = .001) and level of performance (β = 3.15, t (29) = 9.52, p = .001) on individual salarycurrent season. Therefore, players with higher EFFprevious season and level of performance earned higher salaries.
Table 5
Hierarchical Linear Model Analysis Results for Individual Salarycurrent season.
Type
|
Random-coefficient model
|
Conditional model
|
β
|
SE
|
β
|
SE
|
Fixed effect
|
|
|
|
|
Intercept, β0
|
Intercept, γ00
|
5.69***
|
0.11
|
5.69***
|
0.06
|
SDslope, γ01
|
|
|
0.19
|
0.36
|
TSD, γ02
|
|
|
0.61***
|
0.07
|
EFFprevious season, β1
|
Intercept, γ10
|
0.28***
|
0.02
|
2.86***
|
0.02
|
SDslope, γ11
|
|
|
0.05
|
0.11
|
TSD, γ12
|
|
|
0.01
|
0.02
|
PL, β2
|
Intercept, γ20
|
3.15***
|
0.33
|
3.13***
|
0.33
|
SDslope, γ21
|
|
|
3.23
|
1.88
|
TSD, γ22
|
|
|
-0.26
|
0.39
|
EFFprevious season × PL, β3
|
Intercept, γ30
|
0.23***
|
0.03
|
0.23***
|
0.03
|
SDslope, γ31
|
|
|
-0.25
|
0.19
|
TSD, γ32
|
|
|
0.10*
|
0.04
|
Random effect
|
|
|
|
|
Level 1, r
|
11.28
|
11.24
|
Level 2, u0
|
0.31***
|
0.02
|
Level 2, u1
|
0.00
|
0.00
|
Level 2, u2
|
1.40*
|
1.40*
|
Level 2, u3
|
0.01*
|
0.01
|
Total variance (i.e. ICC)
|
13.02 (13.3)
|
12.68 (11.3)
|
a PL = performance level; ICC = intra class correlation; SE = standard error; TSD = total mean across 11 years of standard deviation of salary for each team; EFF = efficiency. |
* p < .05 |
*** p < .001 |
The intra class correlation (ICC) in the random-coefficient model was 13.3%, suggesting that, of the total amount of variance in individual salarycurrent season, the proportion explained at the individual level (Level 1) by the three independent variables (level of performance, EFFprevious season, EFFprevious season × level of performance) was 86.7% (11.28/13.02 × 100), and the proportion explained at the team level (Level 2) was 13.3% (1.73 / 13.02 × 100). As u3 at 0.018 (χ2 (29) = 46.69, p = .020) was significant for the random effects in the random-coefficient model, the slope of the interaction variable (EFFprevious season × level of performance) on individual salarycurrent season was different for each of the 30 NBA teams, meaning that there were significant parts that could be explained by the Level 2 variables. Therefore, the conditional model analysis was conducted after entering the Level-2 variables, SDSlope and TSD.
The results showed that for the fixed effects in the conditional model, γ32 at 0.10 (t (29) = 2.49, p = .019) was significant. Therefore, the slope of the interaction variable EFFprevious season × level of performance was different for each of the 30 NBA teams. This difference was affected by a level-2 variable, namely TSD. To interpret this effect, the three-way interaction was graphed (Fig. 1) after identifying the top five teams with the largest TSD (high TSD group) and the bottom five teams (low TSD group).
In Fig. 1, the high performers, while selecting teams with greater TSD, that is, teams with greater variance among the team members’ performance, showed a stronger positive relationship between EFF and individual salary, compared to that while selecting teams with smaller TSD. This implies that the high performers, while selecting teams with larger performance gaps among team members, earned a higher salary than while selecting teams with lower performance gaps among team members. Interestingly, in the conditional model, where the Mslope and TM were entered, neither γ31 nor γ32 were significant. Therefore, assuming that salary is an objective measure of individual performance, the results can be interpreted to mean that social compensation, rather than the Köhler effect, was supported as the performance of high performers increase in NBA teams where the variance in PL among the members is large.
In Level 1 of the random-coefficient model, the level of performance explaining individual salarycurrent season, PTSprevious season, and PTSprevious season × level of performance was entered in order to examine the effect on individual salarycurrent season. The results indicated that, for fixed effects, PTSprevious season (β = 0.30, t (29) = 13.68, p = .001) and level of performance (β = 3.71, t (29) = 12.02, p = .001) positively affected individual salarycurrent season. Therefore, players with higher PTSprevious season and performance earned higher salaries. For random effect, the ICC was 11.7%, meaning that of the total variance in individual salarycurrent season (i.e. 12.97), predicted by three independent variables (level of performance, PTSprevious season, PTSprevious season × level of performance), 88.3% (11.46 /12.97 × 100) was explained at the individual level (Level 1) and 11.7% (1.51 / 12.97 × 100) was explained at the team level (Level 2). As u3 at 0.016 (χ2 (29) = 38.86, p = .104) was not significant, conditional model analysis was not conducted.