Injury Time, Substitutions & Goal Celebrations.

Injury time, stoppage time or " the fourth official has indicated there will be a minimum of four minutes time allowed" as it is usually described by the stadium announcer at The Britannia is one of the most contentious decisions made by the referee on match day. If your side is trailing, five minutes is never enough to compensate for the interminable time spent by the opposition while taking throws or goal kicks, while three minutes is a fantasy of an official's biased mindset if you are holding a tenuous one goal lead.

The Laws of the Game hand all of the cards to the referee in the stoppage time debate. Time can be added for time lost due to substitutions, assessment and removal of injured players, time wasting and any other cause. Just in case final point didn't adequately convey the free hand afforded the referee in deciding the amount of stoppage time, the law ends by informing it's readers that "time lost is at the discretion of the referee".

Substitutions are the most readily available data points and the least open to individual refereeing interpretation. Nowhere in the Laws of the Game is a specific amount of allowed time mentioned to compensate for a substitution, but 30 seconds has almost universally become an accepted figure. Similarly, the time added, post goal scoring, to allow the players time to perform a choreographed celebration has also semi officially been set at 30 seconds and presumably falls, along with streakers and escaped dogs under the final catch all reason of "any other cause".

We are therefore left with injury assessment and removal, and time wasting as two relatively unknowable causes of stoppage time. The first is difficult to record and is at the discretion of the official and the second is almost entirely at the referee's discretion and also appears to be score and team dependent. Has a player ever been booked for time wasting, late in game when his team is trailing in the match regardless of the time he dallies over a free kick ?

Creating models that describe footballing events where some important factors are missing isn't unusual, often perfectly useful constructions can be produced using limited inputs. So, using data from the MCFC data dump I've tried to predict the average amount of injury time a team could expect to experience during the  2011/12 season using only the number of goals and the number of substitutions that occurred in total during their matches.

Use of substitutes varied quite markedly across different teams in 2011/12, Newcastle, Norwich, Everton and Manchester City were among the sides which made full use of their replacements during the season, while Fulham and Blackburn were much more reluctant to ring the changes. Each of these sides featured at the extremes for total match substitutions,with higher overall numbers also being seen in games featuring Arsenal and Wigan and lower numbers involving Liverpool and Sunderland matches.

The amount of goal scoring is easier to categorize. Teams at either end of the table generally experience contests with more goals, partly as a result of the top sides recording wide margin wins against the strugglers. Mid table outfits, on average see less goal laden matches. During the present century games played by the top six have seen an average of 103 goals a season, compared to 102 for the bottom six and just 96 for the eight teams in between.

The correlation between average stoppage time and seasonal total match goals and total match substitutions is significant. Above I've plotted the average stoppage time per game against the predicted time derived from a regression for all other teams that uses the two inputs of goals and substitutions. The majority of teams congregate around the line of best fit, but an obvious outlier is Blackburn. They averaged just under 360 seconds of total injury time per game, despite their predicted total from the amount of goals and substitutions in their matches implying that they should have received considerably less.

The reason is fairly easy to spot. Junior Hoillet was stretchered off following an injury time clash of heads with Fulham's Mark Schwarzer in their 1-1 draw leading to 11 minutes of second half injury time compared a budgeted for three. Those "extra" eight minutes spent assessing and dealing with an injury contribute 13 seconds per game that aren't accounted for in the model. Once we start to account for another obvious contributing factor such as disproportionately large amounts of time lost due to treating injuries, the line of best fit moves more in line with reality and Blackburn cease to be such an extreme outlier.

Unsurprisingly, over a season, the number of game goals, substitutions and injury stoppages appear to be good predictors of stoppage time awarded to Premiership sides.

Too little, too much ?

Of more interest are the predictions spat out by the regression as the number of goals increase. Increased numbers of substitutions lead to more stoppage time, but increased goals over the season actually decreases the amount of stoppage time awarded with substitutions held constant. The data used in the regression has been aggregated over a season, so this trend may not carry over to future seasons or down to individual matches, but it's tempting to speculate that referees are implementing their version of the "mercy rule".

Games with lots of goals are more likely to be one-sided compared to games with fewer scores. Adding less stoppage time than is merited when one team reaches the 90th minute with a three or more goal advantage will hardly ever change the ultimate result. As a ballpark example, an underdog trailing 3-0 to the likes of Manchester United will take something from such a game under 1 time in 100,000 if offered 6 minutes of injury time, 1 in 38,000 if we extend it to 10 minutes. So a perhaps unconscious effort is being made by referees to put already beaten teams, prematurely out of their misery, knowing that the trailing side is almost certainly beyond hope.

The second half of United's 8-2 trouncing of Arsenal saw 6 goals, one dismissal and 5 substitutions, but just three minutes of stoppage time, although in reality a further 23 extra seconds were actually played. Similarly, Wigan 0 Arsenal 4 contained 2 second half goals, a full pack of substitutions and just 2 minutes of advertised stoppage time.

If the best and the worst sides are seeing games prematurely ended, this may explain why both Manchester clubs, along with relegated Bolton, finished in the bottom three for allotted stoppage time in 2011/12, although the often repeated conspiracy theory that the best get more time only when they need it will require more individual game data, hopefully in future posts.