# What is the best way to model expulsions in a plus/minus model?

In my post a couple of weeks back on adjusted plus/minus, I said that I wasn’t comfortable with the idea of zeroing out the appearance index for players who were sent off in a football match, but hadn’t come up with a suitable alternative.  I’ve arrived upon an alternative now, so I’ll update the model and see if it makes any difference in the plus/minus results.

## Adjusted Plus/Minus Model with Player Expulsions

Let’s recall the adjusted plus/minus model:

$90\frac{\Delta G}{M_j} = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + \ldots + \alpha_i x_i + \ldots + \alpha_N x_n + e$

which has the following terms:

• $$\Delta G$$: Goal margin, $$G_{home} – G_{away}$$
• $$M_j$$: Length of time segment, the interval in which no substitutions or expulsions occurred, for $$j = 1, \ldots, R$$ segments [in minutes]
• $$\alpha_0$$: Average home advantage in competition
• $$\alpha_i$$: Influence of player $$i$$ on goal differential, for $$i = 1, \ldots, N$$ players in competition
• $$x_i$$: Player appearance index:
• +1: Player $$i$$ is playing at home
• 0: Player $$i$$ is not playing
• -1: Player $$i$$ is playing away

I model expulsions by extending conditions for the player appearance index.  The concept is this:

• An expelled player is treated as if he is on the field if his team is scored against in subsequent match segments, but ignored if his team manages to score without him.
• If a teammate is also sent off, the same treatment applies to both players.
• If a player on the opposing team is sent for an early bath, I ignore the earliest expulsion on the other team.

One way to visualize this in a way that makes sense (at least to me) is through two queues, which I call penalty queues. Here is a very rough schematic of what happens when a player from the home team is sent off:

Here’s what happens when the player’s teammate also sees red (bullet #2).  He is also pushed into the penalty queue:

Say this has been a really bad-tempered match, and a player from the away side gets a second booking (bullet #3).  That player is pushed into the away team’s penalty queue, but the first player in the home team penalty queue is removed (or ‘popped’ in queue terminology):

Players in these queues for any segment of the match are given appearance indices of +1 or -1 only if their team has a negative goal difference during that segment.  If their team has a zero or positive goal difference, they are given an appearance index of 0.

## Adjusted Plus/Minus Results: 2011-12 English Premier League

Let’s demonstrate the new adjusted plus/minus model on the same match data I used in my previous post: the 2011-12 English Premier League.  The modeling procedures are the same, so if you want to find more details consult that post at this link.

For reference, here are the top twenty players (minimum playing time 900 minutes) in terms of adjusted plus/minus — each player’s contribution to his team’s scoring margin over 90 minutes — using the original model:

 Name Default Position Mins APM/90 Jonny Evans Defender 2430 1.067 Leon Best Striker 1163 0.800 James Perch Defender 1359 0.714 Michael Williamson Defender 1916 0.689 Mikel Arteta Midfielder 2606 0.572 Thomas Vermaelen Defender 2506 0.539 Edin Džeko Striker 1513 0.518 Lucas Leiva Central Midfielder 1047 0.492 Alexandre Song Midfielder 3018 0.481 Ashley Young Midfielder 1572 0.463 Danny Murphy Central Midfielder 2731 0.444 Chris Smalling Central Defender 1301 0.442 Emmanuel Adebayor Striker 2844 0.429 David Vaughan Central Midfielder 1496 0.428 Gaël Clichy Left Full-back 2527 0.424 Ryan Giggs Midfielder 1480 0.422 James Milner Midfielder 1587 0.408 Steven Caulker Defender 2342 0.406 André Santos Left Full-back 1021 0.398 Ledley King Central Defender 1807 0.389

And here are the top twenty players (minimum playing time 900 minutes) in terms of APM augmented with the player expulsion model:

 Name Default Position Mins APM/90 Jonny Evans Defender 2430 1.070 Leon Best Striker 1163 0.814 James Perch Defender 1359 0.722 Michael Williamson Defender 1916 0.700 Mikel Arteta Midfielder 2606 0.559 Edin Džeko Striker 1513 0.534 Thomas Vermaelen Defender 2506 0.524 Lucas Leiva Central Midfielder 1047 0.491 Alexandre Song Midfielder 3018 0.468 Ashley Young Midfielder 1572 0.449 Gaël Clichy Left Full-back 2527 0.444 Danny Murphy Central Midfielder 2731 0.441 Chris Smalling Central Defender 1301 0.427 Emmanuel Adebayor Striker 2844 0.427 David Vaughan Central Midfielder 1496 0.426 James Milner Midfielder 1587 0.412 Ryan Giggs Midfielder 1480 0.407 Steven Caulker Defender 2342 0.401 André Santos Left Full-back 1021 0.399 Paul Scholes Midfielder 1170 0.395

The  effect of the augmented APM model on the test data RMSE is minimal: adding player expulsions adds about 0.5% to the variance in the goal difference data that is explained by the model.  The average home advantage is relatively unchanged from +0.390 goals per 90 minutes to +0.388 goals per 90 minutes.  About 80% of the players see their APM change by 0.01 goals per 90 minutes, which translates to a shift in rank of about four places.  So a majority of players don’t see any difference to their APM ratings.

That leaves about 20% — 60 players — whose APMs change by more than 0.01 goals per 90 minutes, which often results in significant changes relative to their peers.  Here are the 30 players whose APMs increase by 0.01:

 Name Default Position Mins APM/90 Rank APM/90 (Exp) Rank (Exp) Diff APM Diff Rank Joey Barton Midfielder 2855 0.061 141 0.126 102 0.065 39 Daniel Gabbidon Defender 1423 0.047 150 0.102 116 0.055 34 Shaun Wright-Phillips Right Winger 2214 -0.357 310 -0.310 302 0.047 8 Jamie Mackie Striker 2189 -0.236 293 -0.191 275 0.045 18 Anton Ferdinand Defender 3024 0.098 117 0.130 99 0.032 18 Bobby Zamora Striker 2386 0.189 74 0.221 61 0.032 13 Nedum Onuoha Defender 1461 0.043 153 0.074 134 0.031 19 Luke Young Defender 2172 0.095 119 0.124 104 0.029 15 Alejandro Faurlín Midfielder 1799 -0.159 271 -0.130 257 0.029 14 Samir Nasri Midfielder 2260 0.085 125 0.112 111 0.027 14 Taye Taïwo Left Full-back 1204 0.065 138 0.088 123 0.023 15 Gaël Clichy Left Full-back 2527 0.424 15 0.444 11 0.020 4 Nigel de Jong Midfielder 1081 -0.568 326 -0.548 324 0.020 2 Adel Taarabt Midfielder 2016 0.112 108 0.132 98 0.020 10 Micah Richards Right Full-back 2050 0.309 34 0.328 30 0.019 4 Joe Hart Goalkeeper 3430 0.151 93 0.168 83 0.017 10 Davide Santon Left Full-back 1672 0.270 41 0.287 38 0.017 3 Edin Džeko Striker 1513 0.518 7 0.534 6 0.016 1 David Silva Midfielder 2809 0.180 78 0.195 71 0.015 7 Clint Hill Central Defender 1709 -0.288 300 -0.273 296 0.015 4 Pablo Zabaleta Right Full-back 1555 -0.486 321 -0.471 320 0.015 1 Leon Best Striker 1163 0.800 2 0.814 2 0.014 0 Zak Whitbread Central Defender 1449 -0.109 249 -0.096 237 0.013 12 Adam Drury Defender 909 0.123 101 0.135 96 0.012 5 Vincent Kompany Central Defender 2779 0.299 35 0.311 35 0.012 0 Ákos Buzsáky Midfielder 902 0.041 155 0.053 145 0.012 10 Mario Balotelli Striker 1348 0.213 63 0.224 57 0.011 6 Paul Scholes Midfielder 1170 0.384 22 0.395 20 0.011 2 Michael Williamson Defender 1916 0.689 4 0.700 4 0.011 0 Jay Bothroyd Striker 1103 0.027 161 0.038 153 0.011 8

It’s probably more than a little amusing that the player most helped by incorporating expulsions into an adjusted plus/minus rating is Joey Barton.  But if you look at his disciplinary record, he was sent off two times during the 2011-12 season.  QPR’s goal difference in his absence wasn’t good — goal differential of -3 — but that doesn’t seem to be a difference-maker.  QPR had a man advantage just one time in 2011-12, at home to Chelsea in an exceptionally bad-tempered match on 23 October 2011, but they had already scored their goal when they went one and later two men up. Daniel Gabbidon, who also played for QPR that season, also has a large shift in his APM, but he was used sparingly during the season.  It’s not clear what accounts for the  large change in Joey Barton’s APM, and I will have to dig through the numbers to find out.

A look at the 31 players whose APMs decrease by 0.01 goals per 90 minutes is even more interesting:

 Name Default Position Mins APM/90 Rank APM/90 (Exp) Rank (Exp) Diff APM Diff Rank Shaun Derry Midfielder 2240 -0.025 194 -0.036 199 -0.011 -5 Kieran Richardson Left Winger 2291 -0.091 238 -0.102 240 -0.011 -2 David de Gea Goalkeeper 2618 0.206 65 0.195 70 -0.011 -5 Dean Whitehead Central Midfielder 2383 0.101 115 0.090 121 -0.011 -6 Ivan Klasnić Striker 1521 0.072 131 0.061 141 -0.011 -10 Fernando Torres Striker 1920 0.042 154 0.030 156 -0.012 -2 Antolín Alcaraz Central Defender 2198 -0.220 285 -0.232 289 -0.012 -4 Demba Ba Striker 2756 -0.091 237 -0.103 242 -0.012 -5 Nicky Shorey Left Full-back 2019 -0.126 259 -0.138 264 -0.012 -5 Alexandre Song Midfielder 3018 0.481 9 0.468 9 -0.013 0 Sebastian Larsson Midfielder 2712 -0.110 250 -0.123 254 -0.013 -4 Jason Lowe Midfielder 2689 -0.060 217 -0.073 223 -0.013 -6 Wayne Rooney Central Forward 2845 0.012 177 -0.001 178 -0.013 -1 Mikel Arteta Midfielder 2606 0.572 5 0.559 5 -0.013 0 Phil Jones Defender 2112 -0.354 309 -0.367 310 -0.013 -1 Ashley Young Midfielder 1572 0.463 10 0.449 10 -0.014 0 Javier Hernández Striker 1500 -0.215 283 -0.229 287 -0.014 -4 Ryan Giggs Midfielder 1480 0.422 16 0.407 17 -0.015 -1 Darren Pratley Midfielder 1289 -0.106 246 -0.121 252 -0.015 -6 Chris Smalling Central Defender 1301 0.442 12 0.427 13 -0.015 -1 Thomas Vermaelen Defender 2506 0.539 6 0.524 7 -0.015 -1 Jamie O’Hara Midfielder 1659 0.203 67 0.188 74 -0.015 -7 Steven Taylor Defender 1279 -0.292 301 -0.308 301 -0.016 0 Park Ji-Sung Midfielder 939 0.388 21 0.372 23 -0.016 -2 Stephen Kelly Right Full-back 2029 -0.542 324 -0.558 326 -0.016 -2 Ramires Central Midfielder 2480 0.175 82 0.159 91 -0.016 -9 Danny Welbeck Central Forward 2022 -0.223 287 -0.240 292 -0.017 -5 Luis Suárez Striker 2556 -0.088 235 -0.105 243 -0.017 -8 Kenwyne Jones Striker 986 -0.026 196 -0.044 203 -0.018 -7 Emile Heskey Striker 1454 0.029 158 0.011 173 -0.018 -15 Kyle Naughton Right Full-back 2736 -0.183 277 -0.201 276 -0.018 1 Per Mertesacker Central Defender 1830 0.253 48 0.234 52 -0.019 -4 Damien Duff Right Winger 2118 0.262 44 0.243 46 -0.019 -2 Gareth Barry Central Midfielder 2738 0.024 165 -0.225 285 -0.249 -120 Armand Traoré Left Full-back 1713 0.256 46 -0.311 303 -0.567 -257

For the large majority of players, a drop in APM by 0.01 goals per 90 minutes isn’t going to matter all that much; over a season with 2500 league minutes played the change in APM represents an impact of about -0.45 goals.  But there are a couple of players with significant changes in APM that moved them to opposite ends of the table.  Gareth Barry, who played for Manchester City in 2011-12, saw his rating drop from a mid-table position to the lower end of the APM table.  Armand Traoré, a left back for QPR, saw his rating drop by over half a goal per 90 minutes, which sent him almost to the bottom of the table.  Both players require further examination.

As I said in my previous post, it’s just one league season, so there is a lot of noise and multicollinearity that a Tikhonov regularization can’t reduce on its own.  It would be interesting and vital to examine how these ratings behave with two or three more years of match segment data.  As for the player expulsion model, it’s nice to have it so that I can say that I’ve accounted for expulsions, but the results don’t appear to say that it’s necessary for improved modeling.  We’ll just have to wait and see until we get more data.