Adding Context to Goals and Chance Creation.

One of the major problems when adding context to the generally collected counting stats that are used to quantify player quality, even in the broadest terms, is the constantly changing match environment in which a team is playing. 

Goals arrive at steadily increasing rates as we approach the later minutes, as trailing teams become more adventurous, leading to chances at both ends of the pitch and fatigue begins to provide more space for the creative players to operate. The steady accumulation of cards, more often by the defensive players also contributes to more frequent scoring. Short term patterns may appear to occasionally indicate otherwise, but over more representative timescales, 45 percent of goals appear before the break and 55 percent after.

As a consequence, a player, such as a regular substitute, playing predominately in the later, more goal laden minutes of a match, can see his scoring rate artificially inflated. If he also gains a positive premium from the more extreme fluctuations that can occur in smaller samples, he may be substantially over rated. 

There is also the likelihood that a player introduced after an hour will have a substantial fitness advantage over the majority of players that have already toiled throughout the game. An investigation of the high energy sprints made by a substitute in the final half hour compared to his usual output when he plays for the whole 90 minutes may be illuminating.

Small sample size can be dealt with by regressing such counting stats towards a league average, but accounting for playing environment also requires an awareness of the kind of figures a player's team mates were recording after he joined them on the pitch. In this post on the scoring rate of Dzeko, both as a sub and a starter, I illustrated than then apparent large difference could probably be accounted for by a combination of small sample size and a richer scoring environment.

One way to visualize the scoring impact a player is to record the percentage of each minute he has played, so we can see if he was more often present on the pitch when scoring was more likely to be at a high or low. 

In his final season at Arsenal, Ces Fabregas played in around around half of Arsenal's available matches and in the EPL his substitute appearances broadly balanced out the occasions where he left the field early, either due to injury, replacement or by way of red card. As a consequence his profile for the proportion of each minute played is relatively equally spread across each of the 90+ individual minutes. He played around half of the available time, with very little bias towards any phase of the match.

By contrast, in the 2012/13 season at Barcelona, he was regularly replaced around the hour mark and therefore played a much reduced percentage of the later minutes of a game compared to the earlier minutes, where universally longterm scoring rates tend to be reduced.

If we repeat the plot for this particular season the dip in the later minutes is apparent and an appreciation of Fabregas' scoring record at Barcelona should be seen in the context of his regular absence later in games. He played more frequently overall, than he had done in his final year at Arsenal, but spent a greater proportion of that time watching the latter stages from the bench in Spain.

Principally, Fabregas is a creator rather than a scorer of goals and while the rate at which chances are created across match time is much less readily available, it is likely to be broadly linked to scoring rate and game state. If chance creation, in general also accelerates in the later stages of a match, then Fabregas' absence in those later stages may under rate his actual performance. 

To attempt to eliminate this potential bias, I've looked at the proportion of total team chances created by Fabregas while he was on the pitch, firstly at Arsenal in 2010/11 and then at Barcelona in 2012/13. 

The rate at which a team creates chances will be influenced by many factors, from tiring defences, to overall team ability and any particular need due to the current game state. The distorting influence of these factors may be partly eliminated by quoting a player's raw counting stats as a percentage of those recorded by the team as a whole while he was on the pitch.

Team and Season.	Minutes per Chance Created by Fabregas.	Minutes per Chance Created by Team while Fabregas was on the Pitch.	Percentage of Total Chances Created by Fabregas while on the Pitch.
Arsenal, 2010/11.	26-50s	6-50s	25%
Barca, 2012/13.	48-20s	9-20s	19%

Fabregas' importance to Arsenal is readily seen even in an injury curtailed final season. When he was on the field, the Gunners created nearly a chance every seven minutes, with a quarter of them coming from the Spaniard. By contrast, Barcelona were tactically more measured, stretching to a chance created every 9 minutes. As a consequence, there was longer between Fabregas' bouts of creative talent, but he still accounted for nearly 20% of the chances created while he was present, in arguably a more competitively talented environment.