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t Can we measure how much more teams leave themselves open at the back when chasing down a lead and which teams tend to ‘shell’ when they go a goal up? Ben Woolcock #optaproforum

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Page 1: Woolcock opta pro analytics forum with links

t

Can we measure how much more teams leave themselves

open at the back when chasing down a lead and which teams

tend to ‘shell’ when they go a goal up?

Ben Woolcock

#optaproforum

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#optaproforum

Ben Woolcock

What’s it about?• A detailed study looking at how teams perform at different Game States (ie

tied, ahead/behind)

• Building on the great work carried out by a number of ‘fanalyst’ community, particularly:

• Sander IJtsma, creator of the 11tegen11 blog and writer for Volskrant

• Ben Pugsley, co-creator of the Statsbomb website and writer for Bitter & Blue

• Looking at the proportion of shots a team has when they are level, when they are leading or behind, and how that changes as score difference increases

• The rate at which those shots are converted at and the type of shots that we see

• What is Opta’s Big Chance (BC) stat

• Does the game state affect the creation and restriction Big Chances

• Do teams behave differently at each game state

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The Data• Granular data: 2012-13 season and 2013-14 up to and including 1/1/14

• 580 matches – 16,070 chances – 1,527 goals

• Non-granular data: 2010-11 season and 2011-12 seasons

• 640 matches – 22,093 chances – 2,049 goals

• Combined

• 1,220 matches – 38,163 chances – 3,576 goals

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Analysing Shots• Goals happen very rarely, which make them hard to analyse

• Following on from work carried out in the Ice Hockey analytics world, it was found that the number of shots a team takes has strong predictive power

• James Grayson pioneered this work on his site James’ Blog, go and take a look if you haven’t already!

• He has found that the metric with the best predictive power is Total Shot Ratio (“TSR”) is the proportion of shots a team takes compared to the opposition

• TSR = Total shots for/(Total shots for + total shots against)

• Can be considered as a proxy for the amount of control a team holds in its matches, their territorial advantage, and the ability to create chances

• TSR is the “go to” stat for comparing teams, there are a large number of observations, it is a strong predictor of goals/points, it correlates year on year, and it is very easy to calculate

• Year on year correlation important as it indicates that it is a ‘skill’ rather than ‘luck’

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TSR and Points• Teams that are able to control their matches in terms of shots tend to do

better, those that don’t are more likely to be relegated

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TSR and Repeatability• TSR shows a high level of repeatability year on year, which indicates that it is a

‘skill’.

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Game State Effects• Whilst TSR is becoming a widely accepted metric, its does have some

weaknesses

• Taking a team’s TSR taken in isolation, does not take into account what has happened in the games played and the effect that may have on TSR

• Although Score effects have only recently been measured, it has long been known that when a team takes the lead, they are more likely to attempt to keep possession and wait for better scoring opportunities

• Conversely, teams that go behind will attempt to increase the pressure on the opposition defence to create opportunities to score, and that as the game goes on their attempts often become more desperate

• Therefore a team that has spent a lot of time in the lead may see its TSR at a relatively low level compared to other teams due to not having the need to take as many shots, and its TSR may not give a true indication of their strength

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TSR and Game States in the Erevidisie• This graph by Sander IJtsma inspired the idea for this presentation

• It shows a general positive relationship between TSR and the score difference, but also the negative relationship between -1 to +1

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TSR and The Premier League (2013-14)• I don’t know about you, but not what I was expecting…

• We still see the general positive trend, but it looks more like a mountain range!

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TSR and The Premier League (2013-14)• Issue is ‘small’ number of observations at the more extreme Game States• About 85% of shots take place at Close Game States (ie -1, 0, +1)• The more detail we go into, the fewer Game States we’ll observe

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TSR and The Premier League (2012-14)• We still see the effect of this season’s numbers although there is only one

section of negative correlation

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TSR and The Premier League (2010-14)• So I went back to Opta to ask for more data!• Relationship now as expected. The average team that takes the lead does not

start to take over 50% of shots until 3 games ahead

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Goal Conversion by Game State• We see an uptick in conversion rate when teams are at +1, which continues to +2• Combination of the attacking team waiting for better opportunities and the losing

team leaving more space at the back

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How Does This Translate Into Goals?• The increase in conversion at +1 just about more than makes up for the drop in TSR• TSR dropped by 3.7%, conversion rate increased by 1.7%

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TSR by Venue and Game State• About a 12-13% differential between Home TSR and Away TSR at close Game States• Even when losing the average away team doesn’t take more than 50% of shots

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TSR by ‘Team Strength’ and Game State• Teams divided into the “Superior 7” and the “Threatened 13”• The average Superior 7 team performs stronger home and away than the

average home team

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Superior 7 vs Threatened 13• At tied game states, Superior 7 sides on average take double the amount of

shots as Threatened 13

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Threatened 13 vs Threatened 13• Bigger change in TSR when going a goal up/behind• At -1 away average away team takes over 50% of shots

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Superior 7 vs Superior 7• Change in TSR when going a goal up/behind is steeper still• At -1 away average away Superior 7 team takes a higher % than the average

team

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Some Chances are Easier than Others

• Another issue with TSR is that is treats all shots the same• However chance quality is obviously a factor that affects shot conversion rate• Over the past year there has been a large number of shot models which

calculate the Expected Goal probability of each shot based on the average conversion rate of similar shots.

• These models are all slightly different, and are based on a number of factors. These include:– shot location (most common), – where in the goal the shot was aimed at, – the part of the body the ball was shot with, – how the shot was created (eg cross, through ball etc), – distance from goal – angle of the shot

• Expected Goals can be compared with actual goals, and the efficiency/inefficiency of both teams and individuals can be measured

• Expected Goals can also be used to evaluate the quality of chances that teams create

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The Family Tree of Shots

Total Shots

15,948

Goal Conversion: 9.6%

Outside Box

6,828Proportion 39.4%

Conversion 3.4%

Save 21%

Block 31.9%

Off target 43.7%

Direct Free Kick

842Proportion 39.4%

Conversion 6.8%

Save 22.2%

Block 35%

Off target 36%

Penalty

127Proportion 0.8%

Conversion 80.3%

Save 17.3%

Off target 2.4%

Central Box

6,059Proportion 38%

Conversion 16.3%

Save 21%

Block 20.6%

Off target 42%

Wide Box

2,638Proportion 16.5%

Conversion 6.3%

Save 29.7%

Block 27.9%

Off target 36.1%

Head

2,427Proportion 40.1%

Conversion 11.1%

Save 19.9%

Block 11.6%

Off target 57.4%

Foot

2,538Proportion 96.2%

Conversion 6.4%

Save 29.8%

Block 28.6%

Off target 33.5%

Foot

3,603Proportion 59.5%

Conversion 19.8%

Save 21.8%

Block 26.8%

Off target 31.7%

Head

98Proportion 3.7%

Conversion 2%

Save 26.5%

Block 9.2%

Off target 60.2%

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Shot locations and Game States• Shots wide inside the box saw a large increase in positive game states• Shots central inside the box increased at negative game states!

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Shot Distance and Game States• Average distance of shots falls further from tied to -1 than to +1

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Shot Distance and Game States• Although the average distance of shots with foot and by headers both stay

stable

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How Players Shoot and Game States• This is because the percentage of headers increases when a team goes a goal

behind. Does the losing team launch more balls into the box as time ticks down?

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Player Positions and Game States• And defenders also take more shots when a team goes a goal behind

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Shot Outcome and Game States• Note the lower increase in conversion rate when a team is a goal up between

2010-14 data and 2012-14 data

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Location Location Location, or Not

• The main issue with the majority of the Expected Goal models is they don’t take into account defensive positioning or pressure

• Work by Colin Trainor which looked at why a smaller proportion of shots in the Premier League are on target than in the other major European leagues, found that it wasn’t due to worse shot locations, as locations were in fact better.

• Colin surmised that it could well be due to a higher level of defensive pressure in the Premier League compared to the other major leagues

• Ideally we need defensive positioning data to complement shot location data to improve Expected Goal models

• Whilst this is available to those who have access to systems like Prozone, those of us in the ‘fanalyst’ community have to make do with on ball events

• One method of attempting to measure the level of defensive pressure that has been discussed is to look at the number of shots blocked by a team

• I suggest that another method is to use Opta’s Big Chance stat

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What is a Big Chance?

• Also known as a Clear Cut Chance, it is one of Opta’s few subjective stats. Opta’s public description is

– “A situation where a player should reasonably be expected to score usually in a one-on-one scenario or from very close range”

• That doesn’t give us much to go on• What do we already know about Big Chances?

– about 13% of all shots are from a Big Chance– about 37.5% of Big Chances are scored (incl. penalties)– about 51% of goals are scored from Big Chances– Each team has on average about 2 Big Chances per Game

• These numbers are consistent year on year• I like to think that Big Chance stat as the inverse of having defensive

positioning. We still don’t know where the defenders are, but we know that they are not putting pressure on the attacker

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A Family Tree of Big Chances

Total Big Chances

2,066

Goal Conversion: 37.4%

Penalty

127Proportion 6.1%

Conversion 80.3%

Save 17.3%

Miss 2.4%

Central Box

1,603Proportion 77.6%

Conversion 38.2%

Save 25%

Block 5.8%

Miss 31%

Chance Missed

122Proportion 5.9%

Conversion 0%

Outside Box

31Proportion 1.5%

Conversion 25.8%

Save 32.3%

Block 6.5%

Miss 35.5%

Wide Box

183Proportion 8.9%

Conversion 27.9%

Save 38.8%

Block 7.7%

Miss 25.7%

Foot

1,176Proportion 73.4%

Conversion 40%

Save 26.2%

Block 6.5%

Miss 27.4%

Head

420Proportion 26.2%

Conversion 33.6%

Save 21.7%

Block 4%

Miss 40.7%

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Big Chance Ratio• The Big Chance Ratio correlates very highly with the amount of points a team

scores in a season, with an R2 of 0.73 over the last 3 seasons• This compares to an R2 of 0.63 for TSR over the same period

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Efficiency Metrics• Big Chances can be used to measure the ability of teams to create good chances.

The Creative Efficiency metric measures this

– measured as a proportion of Big Chances to Total Shots

• The Average Creative Efficiency is about 13%• A team with a high Creative Efficiency will, over time, create better chances and

convert shots at a higher rate• Can also be flipped to a defensive point of view with the Defensive Efficiency

metric

– measured as the proportion of Normal Chances (ie not Big Chances) conceded to Total Shots conceded

• The Average Defenisve Efficiency is about 87%• Can be viewed as a proxy for a lack of defensive pressure. • A team with a low Defensive Efficiency is unlikely to put much pressure on the

attacking team• A team that has high Defensive Efficiency does not necessarily mean that they do

pressure the opposition however, although it can be an indicator

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Combing Creative and Defensive Efficiency• The Creative Efficiency/Defensive Efficiency (“CEDE”) Score adds Creative and

Defensive Efficiency together to give a combined metric that looks at a team’s overall efficiency

• The average team has a CEDE Score of 100%. A team with a CEDE Score of over 100% is more efficient than the average team and vice a versa

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Creative Efficiency and Game States• Creative Efficiency increases markedly at a +1 Game State• Similar shape to the Conversion chart

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Defensive Efficiency and Game States• Defensive Efficiency we see the opposite as it drops when a team goes behind

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CEDE Score and Game States• More or less a straight line relationship with each change in Game State

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Big Chance Ratio and Game States• And we see a similar relationship wit the Big Chance Ratio

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Big Chance Ratio Vs TSR at Game States• Compared to TSR, we don’t see the negative correlation between -1 and +1

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And We Come Back to Goals… • And we see that the Big Chance Ratio pulls the Goal Ratio towards it at close

game states

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Team TSR and Game State• Each teams TSR at close game states for last season vs this season (to /1/1/14)

2012-13 -1 0 1 Total 2013-14 -1 0 1 Total

Arsenal 52.6% 61.9% 57.4% 59.8% Arsenal 55.9% 61.9% 54.8% 55.5%

Aston Villa 44.4% 42.7% 41.4% 40.8% Aston Villa 57.6% 44.7% 38.1% 46.9%

Chelsea 61.5% 59.2% 52.2% 57.4% Chelsea 72.0% 61.4% 56.6% 61.9%

Everton 73.9% 55.6% 54.8% 58.2% Everton 74.3% 55.4% 44.2% 56.3%

Fulham 44.5% 40.4% 34.5% 41.8% Fulham 40.3% 35.7% 38.0% 37.6%

Liverpool 58.3% 66.9% 60.7% 63.1% Liverpool 49.3% 61.1% 46.9% 57.3%

Manchester City 66.7% 63.0% 62.6% 62.9% Manchester City 85.7% 67.8% 57.2% 63.4%

Manchester United 68.1% 56.0% 50.9% 53.9% Manchester United 60.2% 50.4% 50.6% 52.7%

Newcastle United 57.9% 49.5% 41.9% 51.2% Newcastle United 59.1% 59.0% 46.7% 54.2%

Norwich City 43.7% 47.3% 37.1% 43.7% Norwich City 47.7% 48.2% 33.3% 43.5%

Southampton 55.0% 56.8% 47.5% 53.5% Southampton 66.0% 49.5% 47.1% 55.3%

Stoke City 50.3% 39.3% 38.8% 41.7% Stoke City 44.4% 46.9% 30.4% 41.3%

Sunderland 51.5% 36.7% 30.8% 39.7% Sunderland 42.3% 48.4% 25.2% 43.3%

Swansea City 55.6% 45.1% 44.8% 47.0% Swansea City 57.3% 46.5% 41.2% 51.4%

Tottenham Hotspur 68.5% 68.9% 58.5% 64.6% Tottenham Hotspur 68.3% 66.3% 51.6% 61.4%

West Bromwich Albion 48.1% 43.4% 44.7% 45.9% West Bromwich Albion 61.5% 45.0% 43.8% 49.6%

West Ham United 46.9% 42.8% 44.2% 44.1% West Ham United 46.7% 41.1% 50.0% 41.9%

Relegated Promoted

Wigan Athletic 52.5% 46.1% 41.1% 48.4% Cardiff City 34.7% 33.5% 42.2% 36.1%

Reading 37.6% 35.0% 33.9% 35.8% Hull City 48.6% 45.5% 45.2% 46.2%

Queens Park Rangers 49.4% 42.1% 30.5% 45.7% Crystal Palace 47.0% 42.0% 50.0% 45.6%

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Team TSR and Game State• Each teams TSR at close game states in each season

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Team TSR and Game State• TSR for Superior 7 teams in the current season (to 1/1/14)

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Team TSR and Game State• TSR for Selected Threatened 13 teams in the current season (to 1/1/14)

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Team Creative Efficiency and Game State• Each teams Creative Efficiency at close game states for last season vs this

season (to /1/1/14)2012-13 -1 0 1 Total 2013-14 -1 0 1 Total

Arsenal 11.3% 13.9% 13.3% 13.4% Arsenal 9.1% 17.4% 18.5% 18.0%

Aston Villa 14.5% 13.9% 12.1% 12.8% Aston Villa 12.3% 11.9% 12.5% 10.6%

Chelsea 10.9% 10.6% 11.7% 11.5% Chelsea 14.5% 7.8% 4.8% 9.5%

Everton 18.5% 14.4% 13.7% 14.4% Everton 14.7% 14.2% 25.4% 17.6%

Fulham 6.7% 14.0% 11.7% 12.4% Fulham 9.5% 11.2% 13.8% 14.8%

Liverpool 15.2% 11.0% 16.4% 13.3% Liverpool 11.5% 11.0% 8.6% 11.3%

Manchester City 22.0% 18.0% 16.3% 16.8% Manchester City 9.7% 9.1% 3.7% 8.9%

Manchester United 23.4% 16.0% 20.9% 20.0% Manchester United 9.1% 10.4% 12.5% 10.0%

Newcastle United 14.7% 12.4% 12.5% 11.8% Newcastle United 15.0% 13.4% 21.4% 14.1%

Norwich City 11.6% 15.7% 16.9% 13.8% Norwich City 5.7% 10.7% 18.5% 10.4%

Southampton 11.4% 11.0% 16.3% 13.0% Southampton 7.0% 5.0% 14.3% 8.7%

Stoke City 19.5% 12.0% 10.9% 12.8% Stoke City 10.7% 9.9% 23.1% 12.1%

Sunderland 13.7% 12.5% 9.6% 11.5% Sunderland 12.5% 14.2% 17.9% 13.7%

Swansea City 9.5% 12.1% 32.6% 13.3% Swansea City 6.1% 9.9% 21.4% 11.6%

Tottenham Hotspur 11.3% 7.2% 16.1% 9.5% Tottenham Hotspur 17.1% 9.2% 7.4% 11.0%

West Bromwich Albion 12.9% 12.0% 15.5% 13.6% West Bromwich Albion 11.4% 12.6% 5.8% 9.9%

West Ham United 11.5% 8.0% 17.5% 10.4% West Ham United 5.6% 12.0% 6.1% 9.0%

Relegated Promoted

Wigan Athletic 9.7% 12.6% 10.0% 10.8% Cardiff City 16.7% 11.1% 8.9% 11.6%

Reading 10.8% 12.8% 4.7% 11.2% Hull City 7.1% 11.4% 15.6% 10.6%

Queens Park Rangers 10.1% 10.0% 0.0% 45.7% Crystal Palace 16.1% 12.0% 21.1% 13.1%

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Team Creative Efficiency and Game State• Each teams Creative Efficiency at close game states in each season

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Team Creative Efficiency and Game State• The teams that created the highest proportion of Big Chances when a goal up

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Team Defensive Efficiency and Game State• Each teams Defensive Efficiency at close game states for last season vs this

season (to /1/1/14)2012-13 -1 0 1 Total 2013-14 -1 0 1 Total

Arsenal 71.9% 88.5% 85.8% 84.0% Arsenal 69.2% 94.1% 92.1% 88.7%

Aston Villa 94.2% 87.5% 84.5% 87.4% Aston Villa 88.1% 91.8% 90.4% 90.6%

Chelsea 90.0% 90.0% 89.4% 89.9% Chelsea 95.2% 88.5% 95.7% 91.7%

Everton 83.3% 84.5% 84.1% 83.6% Everton 78.9% 84.1% 86.8% 84.5%

Fulham 87.0% 88.8% 84.9% 87.1% Fulham 81.9% 91.7% 87.1% 88.4%

Liverpool 80.0% 90.3% 84.0% 87.6% Liverpool 91.4% 93.8% 89.5% 90.4%

Manchester City 85.4% 91.1% 92.9% 90.7% Manchester City 100.0% 79.4% 84.6% 85.1%

Manchester United 91.7% 94.1% 93.6% 92.0% Manchester United 83.8% 87.5% 90.9% 87.6%

Newcastle United 79.8% 81.1% 81.0% 81.4% Newcastle United 83.3% 84.1% 87.5% 85.2%

Norwich City 83.1% 87.5% 85.0% 85.5% Norwich City 92.6% 85.2% 92.6% 88.9%

Southampton 75.6% 85.0% 80.0% 82.4% Southampton 84.3% 83.7% 86.1% 85.3%

Stoke City 86.8% 89.5% 89.1% 88.2% Stoke City 94.0% 87.5% 87.5% 87.8%

Sunderland 83.8% 88.3% 90.6% 86.8% Sunderland 85.8% 92.7% 90.0% 89.6%

Swansea City 81.6% 89.5% 94.3% 88.1% Swansea City 81.1% 88.5% 86.7% 87.0%

Tottenham Hotspur 86.8% 80.0% 85.2% 82.7% Tottenham Hotspur 73.1% 91.8% 86.9% 84.7%

West Bromwich Albion 82.4% 88.6% 86.5% 86.3% West Bromwich Albion 92.0% 94.9% 86.1% 92.4%

West Ham United 88.4% 90.3% 89.1% 89.5% West Ham United 87.5% 88.3% 85.7% 88.3%

Relegated Promoted

Wigan Athletic 87.5% 86.2% 87.2% 86.0% Cardiff City 84.8% 89.8% 89.2% 88.2%

Reading 88.0% 84.4% 83.3% 85.0% Hull City 83.8% 90.2% 92.1% 90.5%

Queens Park Rangers 90.7% 86.6% 82.5% 87.2% Crystal Palace 83.6% 86.6% 97.0% 85.7%

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Team Defensive Efficiency and Game State• Each teams Defensive Efficiency at close game states in each season

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Team Defensive Efficiency and Game State• Which teams leave themselves open at the back when a goal behind?• Teams with the lowest Defensive Efficiency at -1. Teams that play with a high line?

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Team Defensive Efficiency and Game State• Which teams might shell when a goal up?• Teams with the highest Defensive Efficiency at +1

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Team Defensive Efficiency and Game State• Teams with normal shape that is just above average (and Arsenal) removed

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Team Defensive Efficiency and Game State• Which teams might shell when a goal up?• Teams with the highest proportion of blocked shots at +1

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How did Man Utd Win the League?• Manchester Utd won the league with a TSR of 53.9%, significantly lower than

any other title winners since the 2000-01 season

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What can Game State Analysis Tell Us?• Man Utd significantly ahead in terms of CEDE Score and Big Chance Ratio when a

goal behind• Man City actually had +91 shot differential and +17 Big Chances differential at tied

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How did Man Utd Win the League?• In the end, it was Man City’s inability to convert their chances and Man Utd’s

ability to convert theirs that made the difference• Man Utd converted about 5% more chances at both tied and -1, and scored

20 goals more over the season

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Conclusions• Looking at overall numbers in isolation may mean that we miss some of the

detail• Looking at score effects might help explain any ‘strange’ results that we see• Both TSR and conversion rates can be heavily influenced by score effects• Big Chances are perhaps under appreciated by some in the analytic

community, they can be very useful and explain a lot• Looking at on ball events can give us an indication if defensive pressure, even

if we don’t have the defensive positioning data• And finally, that Man Utd got lucky last season…

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Ben Woolcock

Suggested Reading• TSR

• TSR Primer http://grantland.com/the-triangle/what-is-total-shots-ratio-and-how-can-it-improve-your-understanding-of-soccer/

• TSR and points http://jameswgrayson.wordpress.com/2012/07/15/another-post-about-tsr/http://pena.lt/y/2013/04/02/understanding-total-shot-ratio-in-football/

• TSR and repeatability http://jameswgrayson.wordpress.com/2013/11/01/how-repeatable-are-total-shots/

• Game States• http://11tegen11.net/2013/03/16/the-next-step-in-football-analytics-

game-states/• http://11tegen11.net/2013/04/06/game-states-and-conversion/• http://www.statsbomb.com/2013/12/score-effects/• http://www.optasportspro.com/en/about/optapro-

blog/posts/2012/guest-blog-scoring-efficiency-and-current-score-by-mark-taylor.aspx

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Ben Woolcock

Suggested Reading• Home Advantage

• http://www.prozonesports.com/news-article-analysis-home-advantage.html

• The Superior 7 & the Threatened 13 • http://scoreboardjournalism.wordpress.com/2013/12/30/introducing-

the-superior-7-and-the-threatened-13/

• Shot Models• Paul Riley’s SPAM http://differentgame.wordpress.com/2012/12/29/shot-

position-average-model-spam/• Sander IJtsma’s Eredivisie model

http://www.statsbomb.com/2013/08/goal-expectation-and-efficiency/• Colin Trainor & Constantinos Chappas’ ExpG model

http://www.statsbomb.com/2013/08/goal-expectation-and-efficiency/• Michael Caley’s Shot Matrix

http://cartilagefreecaptain.sbnation.com/2013/11/13/5098186/shot-matrix-i-shot-location-and-expected-goals

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Ben Woolcock

Suggested Reading• Shot Models Continued

• Daniel Altman’s Shot Distance model http://www.bsports.com/statsinsights/football/shooting-skill-part-ii-shinji-kagawa-the-annie-oakley-of-the-premier-league

• Matin Eastwood’s Shot Distance model http://pena.lt/y/2014/02/12/expected-goals-for-all/

• Kickdex Angle of View model http://blog.kickdex.com/post/52303980749/angle-of-view

• Defensive Pressure• http://mixedknuts.wordpress.com/2013/06/08/positioning-is-everything/• http://statsbettor.wordpress.com/2013/06/21/shots-on-target-across-

the-big-5-leagues/

• Big Chances• Conversion • http://www.bsports.com/statsinsights/football/big-chance-conversion-

premier-league-liverpool-tottenham-southampton• http://www.wearepremierleague.com/2013/04/the-imapct-of-big-

chance-conversion.html

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Ben Woolcock

Suggested Reading• Big Chances Continued

• Shot models using Big Chances• http://www.wearepremierleague.com/2013_07_01_archive.html• http://woolyjumpersforgoalposts.blogspot.co.uk/2013/07/rate-of-attack-

and-creative-efficiency.html• http://thepowerofgoals.blogspot.co.uk/2014/02/twelve-shots-good-two-

shots-better.html

• Efficiency Metrics using Big Chances• http://woolyjumpersforgoalposts.blogspot.co.uk/2014/01/creating-some-

new-metrics-using-optas.html

• Manchester United’s TSR in the 2012-13 season• http://jameswgrayson.wordpress.com/2013/04/16/just-how-much-of-an-

outlier-has-manchester-uniteds-season-been/

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Any Questions?

Thanks!

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