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Table 1 Fit parameters of Eq. 3 for the respective dataset curves displayed in Fig. 2 and according continuative calculations

From: Statistical influence of travelling distance on home advantage over 57 years in the men’s German first soccer divisionStatistischer Einfluss der Anreisedistanz auf den Heimvorteil über 57 Jahre in der ersten Fußball-Bundesliga der Männer

Fit parameter Dataset A (1964–1989) Dataset B (1990–2020) Explanation of parameters in Eq. 3
r0 0.689 ± 0.006 (±0.9%) *** 0.598 ± 0.029 (±4.8%) *** Maximal result value r(d) and thus maximal HA
r −0.076 ± 0.026 (±34%) ** −0.018 ± 0.025 (±139%) The distance-dependent contribution to HA (in contrast to r*)
d0 (41 ± 27) km (±66%) (283 ± 1131) km (±400%) A saturation distance for the HA
Calculation
r(0 km) = r0 + r 0.613 ± 0.032 (±5.2%) *** 0.580 ± 0.054 (±9.3%) *** Maximal distance-independent HA for 0 km travelling distance
r* := r(0 km) − 0.5 0.113 ± 0.032 (±28%) ** 0.080 ± 0.054 (±68%) Distance-independent contribution to HA (in contrast to r)
rHA := r0 − 0.5 = |r*|+|r| 0.189 ± 0.006 (±3.1%) *** 0.098 ± 0.029 (±30%) ** Combined HA (distance-dependent and distance-independent)
α := |r / r0| (11.0 ± 3.8) % (±35%) ** (3.0 ± 4.3) % (±144%) Relative share of the distance-dependent contribution on the maximal HA (α) and on the combined HA (β)
β := |r / rHA| (40 ± 15) % (±38%) * (18 ± 31) % (±172%)
  1. The given errors are the asymptotic standard errors s according to the least square fit procedure. The rightmost bracket gives the relative error in percent (%). The stars (*) denote that the respective value is significantly different from zero (*p < 0.05, **p < 0.01, ***p < 0.001). The values of all calculations for datasets A and B are significantly different (p < 0.01), except for the d0 parameter. These differences represent a significant change (decrease) of home advantage (HA) over the decades. The fit parameter r0 represents the maximal result value r(d) in the closed interval [0;1] regarding normalised match results (0 = loss, 0.5 = draw, 1 = win). Thus, the more r0 exceeds the value of 0.5 (or rHA the value of 0.0), the more HA is present. Furthermore, since r(0 km) > 0.5 is valid (or r* > 0), an HA also exists for negligible travelling distances, for example in local stadium derbies (d ≈ 0 km). The fit parameter r represents the distance-dependent contribution to HA, which can be contrasted to the distance-independent contribution r*. The parameter d0 is a measure for the travelling distance above which HA saturates (exponentially according to the model of Eq. 3). The variables α and especially β show the relative shares of the distance-dependent contribution (r) to the maximal result or HA (see α) as well as to the combined HA (see β), directly displaying the maximum possible impact of travelling distance on the normalized result or HA. A robustness test for this analysis with normalised result value has been executed relating to an analysis with 3 points per win (see Discussion)