*Data that is discussed in this blog was described here.*

### Replicating Myers’ study to substitutions

###### WHY am i digging this up?

Although there is seemingly very little a manager can do during a game and substitutions seem to be their main way to have an impact on the game, there is very little discussion about them. When getting the data on Premier League subs, as discussed before*, *I remembered some statements from the book “The Numbers Game” where a study from Myers is mentioned. I wanted to see how much of his conclusions hold true over a larger time period in the Premier League (the dataset from Myers covers only one season).

###### His conclusions

According to the study the optimal moments to make a first, second and third substitution are before minutes 58, 73 and 79 respectively. Myers named this the “Decision Rule”. Anderson & Sally later made this rule more popular in their book “The Numbers Game” and also renamed it a bit to the “<58<73<79 rule” .

One of the main factors Myers mentions is that subs have to be made earlier (than is done now) in the game as a result of fatigue and the fact that “managers overvalue starters and undervalue substitutes”. An important note he makes it that his rule only applies to losing teams. And something that was not completely clear for me initially; although once he mentions “substitution patterns” nowhere does he make a relation between moments of the different subs (first, second and third).

In his results he shows that making a substitution before the stated minutes when trailing the game takes the chance of improving the game state **from 22% to 40%**.

###### what are we looking at?

First of all let’s take a look at the current distribution of substitutions. Having taken out first half subs and subs that involved a goalkeeper 7,830 changes remain. As we are only looking at situations in which the team bringing on a fresh player is losing we will filter for those changes: 2,945 subs.

Graphing those it becomes clear that the average Premier League manager does not religiously adhere to this rule: the first sub is generally (30%) made at half time but in total more often between minutes 60 and 70, later than the suggested 58′. The second substitute player is introduced between minuted 65 and 80, and the last 20 minutes (75 to 95) are used to bring on a third and last initially benched player:

In numbers the above looks like the following:

*Note that the second sub is most frequently made in the 45th minute, although from the distribution graph above that is no very clear (explanation: mode is taken per minute, the graph takes five minute intervals).*

###### But how did they work out?

When trailing in the second half 389 of the 894 (that 44% we saw above) teams made a substitution before minute 58. In 122 (31%) of those cases the team making that early substitution had improved the game state by the end of the match. Teams that subbed on a first player later that that improved the result in only 22% of the matches. The same patterns can be seen for the second (14% to 25%) and third substitutions (9% to 21%):

The 31% is a bit lower than the results from Myers (40%), but the 22% chance of coming back from trailing without making a substitution is the same. On the other hand, the chance of a worse result is also higher, but given that the team was already losing at the initial point in time and the games are played in a 38-game league format (and goal difference seldom is key) that should be negligible. All and all, we can confirm Myers’ conclusion that substituting before the given minutes is more likely to give you an improvement in the score than waiting longer to bring on subs.

*(as is already quite visible from the data, all differences are significant at the p<0.05 level)*

###### is that it?

For know yes. But knowing this two questions in particular remain: what kind of substitutions – defensive, neutral or attacking – led to the change in game state? And are 58, 73 an 79 indeed the optimal ‘make-a-sub-before-those-minutes’ values or should the subs be made even earlier? I’ll save this for another post.

## 2 thoughts on “Premier League substitutions: 2. The Decision Rule”