Gravity: Part II. The Pressure of Gravity on Penalty Kickers

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Gravity: Part II. The Pressure of Gravity on Penalty Kickers


In my last post I introduced the idea of Gravity (Γ), a metric reflecting the probabilistic effect of an event on the outcome of a match.  Events - whether or not they actually occur - having a big potential influence on a match are “important” or “High Gravity” events.

In this post I will examine the effects of Gravity on penalty taking.  The hypothesis is that high Γ events are high pressure events; a high Gravity penalty - i.e. one that could have a big influence on the game - puts the penalty taker under more psychological pressure than a low Gravity penalty.

The idea that psychological pressure affects penalty taking is not new. Geir Jordet has made an extensive study of the effects of pressure in penalty shootouts. In these sudden death situations, a penalty kick that could win the match is converted significantly more often than a kick that could lose the match.  In one study of international football tournaments, Geir  found that the conversion rate for penalty kicks which could win a shoot-out, but not lose it, was 92.0%, while conversion rate for penalty kicks which could lose a shoot-out, but not win it, the was only 61.8%. There is less pressure on the first type of penalty than the second, because missing the first type does not lose the match, but missing the second type does.

But rather than looking at shoot-outs, this post looks at in-game penalties. In-game penalties are rarely as decisive as the kicks in a shoot-out, where a missed kick can result in a loss of the match with no chance to recover from the mistake.  Nevertheless, we can conjecture that pressure still might have some effect.

I used match data from three seasons (2014/15 - 2016/17) of Ligue 1, the Premier League, La Liga, the Bundesliga and Serie A. Because of missing data I had to discard eight matches.  There were 1,585 penalty kicks in the dataset.  I calculated goal Gravity for all the kicks using the procedure described in my previous post, where Γ is calibrated in expected points.  Home and away scoring intensities differed somewhat (for instance in France 2014 the away scoring intensity was 1.07 goals/90, and in the Bundesliga 2015 it was 1.27 goals/90), so I used different home and away scoring intensities for each competition-season.

Effects of Gravity on in-Game Penalty Scoring

Figure 1 shows the relationship between the probability of scoring and goal Gravity. Because there were only 71 penalty-kick events with gravity between 1.5 and 2.0 points, I assigned a gravity of 1.75 points to all the data in that range for plotting purposes.

Figure 1. Probability of Scoring a Penalty vs Goal Gravity

The effect of Gravity looks quite striking.  The conversion rate for ‘unimportant’ or low Γ penalties  is over 80%; however, as Γ increases, the conversion rate drops dramatically, reaching about 65% when Γ exceeds 0.75 points.  Pressure on in-game penalties seems to be as disruptive as pressure on shoot-outs.

Next, I examined the relationship statistically using logistic regression.  The outcome variable was the result of the penalty kick (coded 1 = goal 0 = miss).   The key independent variable was Γ, goal Gravity; because the relationship in Figure 1 is clearly non-linear, I used a “quadratic” form of  Γ, in other words, I included the square of Γ as a predictor as well as Γ itself.  This allows the regression line to take the form of a curve instead of a straight line.  I also included  indicators for competition-season to control for differences in penalty scoring base-rates between leagues. (Actually these didn’t make much difference.  I initially included a random effect for players, but it made no difference to the results, and was an unnecessary complication so I left it out. )

The regression coefficients for both the linear and squared Gravity term were both strongly significant (p < .001), indicating the relationship between Gravity and scoring is non-linear, and cannot be attributed to chance; the effect of Competition-Season was not however significant.  The regression effects are visualized in Figure 2. The top panel shows the effect of Gravity, and the bottom panel shows the effect of competition and season, small compared with the standard error bars.

Figure 2. Effects of Gravity and Competition-Season on probability of scoring a penalty.

Effect of Goal-Scoring Ability

Next I looked at the effect of goal-scoring ability.  We might expect that players used to scoring goals would be better penalty takers, and less prone to pressure.  Goal-scoring ability was measured by goals/90 minutes (excluding penalties), and to ensure reasonable estimates, players with fewer than 540 minutes (equivalent to six complete matches) were excluded from this part of the analysis. Table 1 shows the results of the logistic regression with goals/90 as a predictor in addition to Gravity and Competition-Season; to see whether skilled players were less affected by Gravity, I also included an interaction between goals/90 and goal Gravity.

Coefficient Std. Errorz valueSignificance level
(Intercept)2.100.307.02p <.001
Gravity-2.690.46-5.82p <.001
Gravity squared0.920.243.91p <.001
Gravity * Goals/901.460.652.23p=.03
Competition-season--LRT=11.1, df=14p=.674

As we can see in Table 1, Gravity and Gravity-squared are still significant, and the interaction between goals/90 and Gravity is also significant.  Figure 3 shows what this means in practice.

Figure 3. Effetcs of Gravity and Ability on scoring penalties

Once again the upper panel shows the effect of Gravity. The lower panel shows how the effects of Gravity depend on Ability; at low Γ (the leftmost panel) the downward sloping line indicates that skilled players perform somewhat worse than  unskilled ones, but at high Γ (rightmost panel), skilled players outperform their less skilled counterparts.  The first effect is counter-intuitive; why should better goal-scorers perform worse in low Γ situations?  It could be they are over-confident, or it could be that less-skilled players present more of a challenge to goal-keepers because they are unpredictable. But at the moment, I would not put too much weight on this particular result until it is replicated. What does seem clear though is that the contribution of high skill become progressively more pronounced as the pressure on the kicker increases.  Goal-scorers seem to be able to hold their nerve better.

The Bottom Line

The key finding here is that the importance of in-game penalties in a match puts considerable pressure on the penalty-taker, and this has a substantial impact on performance.  However, regular goal scorers are less affected this pressure.

This has implications for arranging penalty shootouts.   I haven’t yet worked out the details mathematically yet, but I think Γ would tend to increase as the  shoot-out progresses.  So instead of ‘wasting’ the best penalty takers early on in the shootout when the pressure is comparatively low and their skills are not needed, I suspect it might make sense to introduce them later on, when the pressure is greater and when their skills are most useful.

In a future post, I will look at the effects of Gravity on regular attempts on goal.


2017-09-04T10:56:06+00:00 September 4th, 2017|Recent Posts|0 Comments

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