# The Relevance of General Spatial Anticipation Skills for Basketball Referees
## 3 Results  
Assumptions for normal distribution and homoscedasticity were checked with graphical measures for each group (Q-Q plots) and additional Shapiro-Wilk and Levene tests. Results partly reported significant violations of the normal distribution assumption and homoscedasticity assumption. The values of both tests for the five tasks can be found in Table S2 of the Supplementary Materials. The preconditions for multiple ANOVAs were not completely fulfilled; however, previous research has shown that ANOVAs are robust against violations of normal distribution [38]. Nevertheless, we performed additional non-parametric procedures (Kruskal-Wallis test) to confirm the validity of our results. Statistically significant effects between the different groups and the Vienna Test System results do not differ between the ANOVAs and the Kruskal-Wallis tests. In the following section, only the results of the ANOVAs are reported; the results of the Kruskal-Wallis tests can be found in the Supplementary Materials, Tables S3 and S4.  
We conducted five separate ANOVAs with one independent variable (3-factor levels: expert referees, amateur referees, and novices) for all five test results. Additionally, we performed Bonferroni-corrected multiple comparisons. Descriptive statistics, \(p\)-values, and effect sizes for the five ANOVAs are reported in Table 1. Only the difference between test results in spatial anticipation is statistically significant, \(F(2,83)=14.49\), \(p<0.001\), partial \(\eta^{2}=0.259\). Multiple comparisons between groups for spatial anticipation scores reveal a significant difference between the elite referees and amateur referees, \(p<0.001\) (\(M_{\text{diff}}=-22.79\), 95%-CI [\(-33.96\), \(-11.63\)]), and elite referees and novices, \(p<0.001\) (\(M_{\text{diff}}=-19.99\), 95%-CI [\(-31.16\), \(8.83\)]), but not between amateur referees and novices, \(p>0.999\) (\(M_{\text{diff}}=-28.0\), 95%-CI [\(-13.56\), \(7.96\)]). A graphical representation of the effect is reported in Figure 1. Thus, the elite referees performed better than the amateur referees and the novices. However, the results of the amateur referees and the novices do not significantly differ.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Expertise Level**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Elite Referees**} & \multicolumn{2}{c}{**Amateur Referes**} & \multicolumn{2}{c}{**Novices**} & \\ \hline  & **M** & \(SD\) & **M** & \(SD\) & **M** & \(SD\) & \(\eta^{2}\) & \(p\) \\ \hline \multicolumn{9}{l}{_The Vienna Test System_} \\ Spatial working memory & 10.27 & 2.31 & 8.80 & 2.63 & 10.57 & 2.19 & 0.090 & 0.020 \\ Visual orientation & 16.15 & 1.69 & 15.83 & 1.62 & 16.20 & 1.63 & 0.011 & 0.648 \\ Perceptual speed & 23.67 & 4.58 & 22.83 & 3.22 & 24.37 & 3.62 & 0.040 & 0.261 \\ Time anticipation & 91.27 & 44.80 & 88.80 & 73.82 & 101.93 & 32.86 & 0.016 & 0.522 \\ Spatial anticipation & 62.04 & 12.84 & 84.83 & 18.22 & 82.03 & 18.92 & 0.259 & \textless{}0.001 \\ \hline \hline \end{tabular} Note: Professional referees (\(n=26\); \(n=9\) for perceptual speed), amateur referees (\(n=30\)), and students (\(n=30\)). Time anticipation and spatial anticipation lower scores represent comparatively better performance. Scores for spatial anticipation are measured in media deviations in millimetres, scores for time anticipation are measured in median deviation in hundredths of a second, and scores for spatial working memory, visual orientation, and perceptual speed are measured as correct responses. Significant results: \(p<0.01\).  
\end{table}
Table 1: Effect size differences between elite referees, amateur referees, and novices for visual tests of the Vienna Test System.