Around the goal: examining the effect of the first goal on the second goal in soccer using survival analysis methods

Appendix

A Detailed analysis

We now look for a more parsimonious model and compare it with the null model [Model (I)]. If we limit ourself to models in which all effects are at least weakly significant (p<0.1) we get two possible models,

and

The results for these models are displayed in Tables 4 and 5. Note that both models include TimeOfFirstGoal, though the coefficient and the standard error estimates are twice as big for Model (V).

The three models (including the null model) are nested, and hence a likelihood ratio test, could be used. Comparing Model (V) and Model (IV), separately, to Model (I) yields p-values of 0.013, 0.015 for Models (IV) and (V), respectively. By comparing Models (V) and (IV) a χ² statistic of 4.3 on 2 degrees of freedom is obtained (p=0.116), meaning Model (V) should not be preferred over Model (IV).

Note that Model (V) includes the time dependent term One could also consider some other function or even estimate it, although this matter is out of this paper’s scope. The linear function, that we have used so far, implies constant marginal effect over time. It is possible that the marginal effect is strongest in the minutes after the first goal scoring and declines during the rest of the game. We emphasize that the first goal effect is increases with time. Therefore, we also consider the natural log function, The log function seems to fit the data better since it preserves the significance of all covariates in the model and improves the likelihood of the model. The results for this model,

are displayed in Table 6 . If we repeat the likelihood ratio test to compare between this model and Model (IV), we get a 5.91 χ² statistic on 2 df, which yields a borderline p-value of 0.052.

In order to choose the most suitable model, we have tried to fit Models (VI) and (IV) using only HomeSecondGoalTime observations. Obviously, the variable Goal was eliminated from this study. All together we have 673 observations. The surprising result is that none of these models fit well, i.e., the effects related to the first goal are not significant.

Next, a stratified model with no interactions between the two goals was fitted to the data. That is,

and looking at the differences between the baseline cumulative hazard function estimators and (where ) may reveal which model is more adequate.

Figure 6 presents these baseline cumulative hazards. The baseline hazard function for the HomeFirstGoalTime seems to be linear. Linearity of the cumulative hazard function implies constant hazard rate, i.e., the probability of a immediate goal scoring at time t conditional on no goal scored yet is the same for all t. Constant hazard rate fits to the exponential distribution. This result is consistent with the usual model for Poisson distribution for the number of goals. The HomeSecondGoalTime baseline hazard, although close to linear, behaves differently at the edges. The hazard between the 20th and the 70th min is similar to the HomeFirstGoalTime hazard. Looking at the hazard ratio reveals that as time goes by, a second goal is more and more likely to be scored in comparison to a first goal. This result is consistent with Figure 1. These findings strengthen the need of a time dependent term for the HomeSecondGoalTime hazard.

Figure 6

Estimated cumulative baseline hazards for Model (VII). The linear HomeFirstGoalTime cumulative hazard function implies constant hazard rate for the first goal. The HomeSecondGoalTime cumulative hazard function is linear for the first half but then changes with time. Thus, it seems reasonable to consider time dependent covariates for the HomeSecondGoalTime.

The hazard for the first 15 min behaves differently. A goal is more likely to be scored if one has already been scored. Models (IV) and (VI) miss this phenomenon. We tried to include the appropriate effect in our models, but it was rejected. It should be noted that only 29 out of the 673 HomeSecondGoalTime observations were in the first 15 min and therefore it is harder to model them appropriately.

We summarize the model choice discussion by stating that the first goal may effect the HomeSecondGoalTime by a few small effects. When combining them together they are significant but each of them separately is not strong enough to be found significant. Moreover, we have presented evidence to time dependent changes for the HomeSecondGoalTime hazard. Therefore, we suggest that Model (VI) is more appropriate.