Fixing ice hockey’s low scoring flip side? Just flip the sides

Abstract

The global sports industry is extremely competitive. We focus on ice hockey and propose a (largely) costless reform to its operations that can potentially enhance the sport’s popularity by increasing the number of goals. The reform optimizes a logistic convention on the use of on-ice sides and off-ice benches, which affects the amount of attacking play and within-game scoring dynamics through the players’ substitutions. Using several empirical approaches, we provide estimates of the resulting scoring rise for all 11 geographically diverse competitions in our unique dataset. They are sizeable and robust to a number of checks, making case for trialling the reform in some (lower-tier) ice-hockey league.

Appendix

1.1 Available seasons

To accompany our dataset’s summary statistics reported in Table 1, in Fig. 5 we plot the seasons for which we have data available. For each we also indicate the average number of goals, whereby more scoring is shown by a darker shade.

Fig. 5

Included seasons and per-game goal averages (combined for the regular season and playoffs)

1.2 Baseline model selection

Table 4 reports results (in the form of p values) for various tests suggesting that some factors of our larger models can be omitted. In particular, the p values implying that our baseline multiplicative model (2) is suitable are shown in bold, whereas the values possibly contradicting this model are in normal font.

Table 4 Estimated p values for omitting terms from the larger models

According to the p values in the table, we can distinguish three main groups of competitions. The first group consists of all competitions except NHL, FIN, SVK and SWE. For this large group of seven competitions, the results show that the minute factor as well as all interactions can be omitted from the full three-factor model. As the last column of panel (a) and the three columns in panel (b) report, the associated p values are all above 0.05. The second group includes FIN, SVK and SWE, where the interactions between the period and the minute are statistically significant, but not the minute factor. The third category is the NHL, where all interactions can be omitted, but the minute factor is statistically significant.

Table 5 The first two columns report values of the AIC (in normal font) and the BIC (in small font) for the baseline multiplicative model (2)

However, if we consider our baseline model (2) as a submodel of the full three-factor model featuring all interactions (panel d), the minute factor can be excluded even for the NHL, and the period-versus-minute interaction term can be excluded even for FIN, SVK and SWE.¹⁷

All these results back up the conclusion that scoring intensities differ across the three periods, column (p), and across the seasons, column (s). Nevertheless, the estimated differences across the three periods (and hence the Long Change effects) are identical in all seasons (column $p^{*}s$). This provides support for the use of our baseline model (2) for all competitions.

This is further validated by the AIC and BIC criteria. Table 5 reports their values for selected multiplicative models, whereby the AIC is in normal font and BIC in smaller font. In particular, the first column (p, s) reports their raw values, whereas the remaining columns offer the differences between the AIC or BIC of the various models and of our baseline model in (2). Negative values in the table imply a lower value of the particular criterion and thus a superior model.

The table confirms that according to the AIC our baseline model (2) is generally the best among all models without the playoff factor. This is the case for all competitions with two exceptions: the NHL, where the three-factor model with the minute is preferred, and the top German league, where a one-factor model (only featuring the period) is preferred. In most competitions the extended model (p, s, pl), i.e. our baseline model with the playoff factor included, features better AIC values than the baseline model, but the difference is usually small. The BIC penalizes a greater number of parameters more prohibitively, and therefore provides support for simpler models such as our baseline, or even more so for a one-factor model only containing the period (this model is however in contradiction with the p values in Table 4).

1.3 Parameter estimates

For completeness, Tables 6 and 7 report values and standard deviations of parameter estimates in models (1) and (2) respectively. As it is not practical to report all the seasonal parameters $\mu _{s}$ (there are up to 24 of them), the column $\Delta \mu _{s}$ provides the estimated difference between the seasons with the maximal and minimal $\mu _{s}$ estimate for each competition.

Table 6 Estimates of the parameters of the three-factor additive model in (1)

Table 7 Estimates of the parameters of the baseline two-factor multiplicative model in (2)