A Personal Example of the The Garbage-In/Garbage-Out Principle

Clearly, my problem was with a "garbage model".

The other night I was checking a few figures for an upcoming post that were based upon my binary logistic models (BLR) built from EPL match data, and I saw a number of counter intuitive trends.  That got me to check my data for a fourth time, and sure enough... I had fat fingered a formula!  It turns out that formula was used throughout the data set, and thus all of my analysis using DogFace's data had been using incorrect numbers.  In terms of my blog's header quote, my model was really wrong and wasn't really useful.  My post-match analysis of Arsenal/Blackburn and my quantification of Phil Dowd's officiating of Arsenal's matches were now both in question.  I had to re-run the analysis, which totaled several hours of statistical work and about twice as long checking my numbers.

It's a bit frustrating, as my blog relies on the accuracy of my numbers to drive its content.  I am very systematic in making sure they are right, and this is the first time I have found such an error.  However, more important than being right the first time is correcting mistakes when I find them.  This post is such a correction.

The Impact of Cards and Phil Dowd on Arsenal


After re-crunching the numbers, I found that the significant factors in the binary logistic regressions (BLR) turned out to be a bit better.  My erroneous calculations had resulted in my elimination of fantasy league points for cards from consideration in the BLR.  This was unfortunate because I had already done several posts on officiating at Arsenal's matches using that metric, and had hoped to re-use it here.  It turns out that when the numbers are correctly calculated, such a term is statistically significant.  Note that I have used Yahoo's Fantasy Premier League scoring system, which gives 3 points for a yellow card and 6 points for a red card.  All other terms from the BLR in the previous post - venues and differentials of shots, shots-on-goal, corners, and fouls - were included in the analysis.  The graphs below show the true relationship between those terms and fantasy point differential.

The updated models show that Arsenal is less impacted by cards at home than the average team in the Premier League, while they are impacted nearly identically as the average team when they are away from home.  Quite a different conclusion than my previous post that had erroneous data!  It turns out a number of the general conclusions regarding their overall odds, especially regarding their best away odds not even matching their odds of a home match where they experienced a fantasy point deficit of 5 points, didn't change much from the original post with bad data.

What about the affect of referees on Arsenal's matches?  The plots below show the impact each referee has on the odd's Arsenal wins a match compared to the odds of Arsenal winning the match had the referee handed out the average number of fouls and cards Arsenal saw over the five year period.  Only data from the 2006/07 through 2009/10 seasons was used as those were the four years where each of the eight referees had officiated at least one match.

Again, we see Phil Dowd lead the pack in odds differential penalty, although it is smaller than originally estimated (3% now vs. 4% with erroneous data).  Webb and Halsey's numbers round out those who have the greatest impact on Arsenal's odds of winning.  These corrected numbers will become more valuable in my next post on this topic, where I will compare the bias of Dowd, Webb, and Halsey to their records officiating other clubs' matches.

Finally, there's the small matter of Phil Dowd's officiating at Arsenal's recent match against Blackburn.  We Gooners still can't blame the loss on Phil Dowd - his officiating certainly helped the Gunners odds.  But their play didn't help nearly as much as my original post indicated, and in fact played into Balckburn's statistics a good bit.  The table below summarizes the match statistics, and shows the likelihood of winning by each club.

That's not a typo - Blackburn did actually have a higher odds of winning the match based upon the way the statistical models work.  Let me explain:

• Arsenal's coefficient for the constant term in the BLR is significant, and it is negative.  This means that before anything else is known about Arsenal's match, they start out with less than a 50% chance of winning.  This is not the case with Blackburn, whose constant term for their BLR is non-signficant and thus gives a coefficient of zero for no effect on their odds of winning.
• Arsenal's odds certainly increase playing at home and having a greater number of shots on goal, but their BLR coefficient for shots is not statistically significant so their is no significant contribution to their odds of winning from their 15 shot advantage.
• Blackburn, on the other hand, does have a statistically significant coefficient for the shots term of the BLR, but it is negative.  That means as the opposition's shot differential increases, Blackburn's odds of winning go up.  This can make some sense, if one thinks in terms of accuracy.  A shot doesn't really matter unless it is on target.  Blackburn's coefficient for SOG is statistically significant as well, and is positive, which makes sense.  This means the more SOG's the opposition gets, the more Blackburn's odds of winning go down.  However, they actually benefit when team's take wild shots with little likelihood of scoring a goal.  This is all reflected in Arsenal's need to take so many shots to get roughly the same percentage of shots on goal as Blackburn.  Had they had better accuracy, Blackburn's odds would have been much lower.
• Arsenal's odds suffer, and Blackburn's benefit, from Arsenal's corner differential.  This is a trend seen throughout the data set, both in the total league data and individual team data.  This possibly counter-intuitive trend will be explored in a later post.
• The foul differential is zero, but even if it weren't it would not matter.  Neither team's BLR coefficient for fouls is statistically significant.
• Clearly Arsenal benefits from only having a single yellow card vs. two and a red card for Blackburn.  This raises Arsenal's odds and lowers Blackburn's due to their statistically significant coefficients for fantasy points.

In all, it gives both teams high odds of winning the match.  Perhaps a draw was the most appropriate outcome according to the statistics.

Conclusion

The beautiful thing about blogging is that mistakes can be corrected instantly, unlike books where one must wait months or years until th enext edition is published or magazines where a retraction can be made next month.  The instantaneous nature doesn't negate the challenges associated with errors, but it does make the communication more honest and more open in a quicker manner.  I hope that in being honest with my mistake that it reassures you that I am always striving for quality data.  I'm redoubling my efforts to check all my numbers before I post any new material.  Catching the error and correcting it has turned out for the better when it comes to the flexibility of the data set and the conclusions that can be drawn.  Most importantly I have shown that fantasy points for yellow and red cards are statistically significant, which will enable the use of a single metric to capture the effect of two related, but one especially rare, events in a match.  Stay tuned for the follow up post...

Quantifying the Impact of the Bias of Arsenal's Referees, Part 1

Special thanks to Dog Face for the data (he and I will be collaborating on the second post in this series), and to Chris from Soccer By Numbers for help in dissecting the stats.

A little over a month ago I completed a post that quantified the different treatment Arsenal appeared to receive from various referees in the Premier League.  In that post I used statistics from Tim at 7AM Kickoff to show how shots taken, the ratio of shots-on-goal to shots taken, and Premier League fantasy points for yellow and red cards to show that Webb, Dean, and Dowd are the least favorable referees for Arsenal while Foy and Atkinson are the most favorable.  What I was unable to do at the time was to show how these different match statistics impacted the outcome of the match.  Luckily, a writer with Untold Arsenal that goes by the name of Dog Face contacted me and supplied data going back to the 2005/06 season and for every match in the Premier League for each of the seasons covered.  This data allowed for the analysis of the impact of such calls on match outcome.

The Data and Statistical Methods Used

Dog Face's data set contains key match statistics from every Premier League match from the 2005/06 season through the latest matches of this season.  To eliminate any error associated with using data from the incomplete 2010/11 season, I focused on the following attributes for the 2005/06 through 2009/10 seasons:

• Venue (home/away)
• Shots
• Shots-on-goal
• Corners
• Fouls
• Yellow cards
• Red cards

The data came to me paired - each row showing the data for both the home and away - so I broke it into unpaired team data.  I then calculated the differential for each statistic except venue, which was coded as a binary statistic (1 = home team, 0 = away team).

In attempting to assess the impact of play on the pitch and referee decisions, we have several options.  We could try and determine a relationship between goal differential and the inputs listed above, but this is problematic given the relative paucity of goals and resultant goal differential.  I've done enough analysis of soccer match data to know this is a fool's errand.  The better method is to determine the likelihood of winning a match given the differentials achieved by a team or dealt out by a referee.  To do this a binary logistic regression analysis was performed using all of the match statistics.  A set of dummy variables based upon the season the data point came from were created to observe any the affects of any overlooked variables in the analysis.  Such a regression analysis allowed the construction of a mathematical model to predict the likelihood of winning (earning 3 points), with (1-likelihood of winning) being equal to the likelihood of not winning (earning 1 point for a tie or 0 points for a loss).  Unfortunately, as it's name suggests binary logistic regression's output is binary in nature and thus cannot differentiate between a soccer match's three possible outcomes.  This is an compromise that must be made to use the analysis.

Like any other statistical analysis, binary logistic regression analysis allows statistical significance to be tested.  In this analysis, the general rule of thumb of p <= 0.05 was used to determine which terms in the analysis were significant (with allowances for slightly higher p-values in team data given the lower sample size).  Based upon this criteria, the following factors were significant in impacting the likelihood of winning a match:

• Venue (home/away)
• Shots-on-goal differential
• Yellow card differential
• Red card differential

The same criteria ended up also being significant when isolating for only the Arsenal data within the wider data set.  This allows for a comparison of the impact of various match attributes on the average Premier League team, and how Arsenal is impacted to a greater or lesser degree for the same match statistic.

The Effect of Yellow & Red Cards

A comparison of the effects of various match statistics could be completed once binary logistic models were created for the league and Arsenal over the five seasons.  The two of interest - yellow card and red card differential - are of most interest as the referee directly controls when a foul is simply a foul and when it is serious enough to warrant a card.  As noted by Chris at Soccer By The Numbers, binary logistic regression predictions present some challenges when trying to provide two dimensional plots of the likelihood of an event (in this case winning) versus a single variable (in this case yellow or red cards).  With Dog Face's data I used an approach of splitting the analyses into home and away games, and then set all other variables to their averages for each venue while sweeping through the max and min values of the variable of interest (either yellow or red cards).  The output generated by each sweep came in three forms: the nominal odds, the lower 95th percentile, and the upper 95th percentile.  Such an approach allows us to observe how sample size and the variability of outcome as the data set approaches its extremes (yellow card differentials of 7 or red card differentials of 2) impact the confidence in the model.

The plots below show the impact that yellow cards have on match outcome.  The first graph shows the impact at home, while the second graph shows the impact away.  The black lines represent the likelihoods based upon the full league data over the five seasons.  The red lines represent the likelihoods based upon Arsenal's data over the same five seasons.  Solid lines, and their associated equations, represent the nominal predictions from the model, while the dashed lines represent the upper and lower 95th percentile lines.

A few things are clear from the graphs above.

1. Playing at home clearly has its advantages.  Even with a six yellow card advantage at an away match while achieving their average away number of shots on goal, Arsenal's likelihood of winning an away match is only slightly better than a home when they are even on yellow cards playing to their average home form.
2. Clearly the reduction in data points for Arsenal (190 matches) versus the league wide data (1900 matches) contributes to the wide variation shown via the 95th percentile lines.  The relative obscurity of Arsenal matches with an absolute yellow card differential greater than 2 creates the uncertainty at the extremes - 88% of all Arsenal matches ended with an absolute yellow card differential of 2 or less.
3. A yellow card at home is only slightly less costly than a yellow card away - each yellow card away results in a 0.4% lower likelihood of winning versus a yellow card at home.  Clearly, the difference in home and away likelihoods of winning can't be chalked up to a difference in the impact of yellow cards when the yellow card differential home and away is even.
4. The non-parallel nature of the Arsenal and league average lines in both graphs indicates that the impact of yellow cards on Arsenal is more severe.  To be exact, it's nearly three times as severe.

Similar odds can be calculated for red cards.  The graphs below show such odds over the range of red cards in the data set, and follow the same conventions as the yellow card graphs above.

A few more conclusions can be drawn based upon the graphs above:

1. Playing at home has even bigger advantages when it comes to red cards.  In the case of Arsenal, even when they get a red card away their likelihood of winning with average away form is only 0.6, which is still 0.09 (or 15%) lower than the average home performance with no red card advantage or disadvantage.
2. While the Arsenal data set still shows greater variation than the league wide data due to decreased sample size, it does show greater separation in the data sets (especially at home).  It could be declared that the separation at home between the two data sets for 0 and +1 red card differential shows that Arsenal's improved chances of winning are statistically significant when compared with the league average.
3. Red cards are certainly a greater detriment to a team's likelihood of winning.  For an average Premier League team, they're 5 times as costly at home and away versus yellow cards.  For Arsenal, they're 4 times as costly at home and nearly 5 times as costly away.

The graphs above indicate the change in the likelihood of winning with each passing yellow or red card in a match, assuming Arsenal is playing at their average form for shots on goal.  They're very useful for illustrative purposes, but not very useful in assessing the impact of the referees identified in my previous posts.  For such an analysis, the individual likelihoods of winning each match are constructed from the match data, and a comparison between the referees is made.

The Impact of Referee's Decisions in Arsenal's Matches

From the graphs above, the impact of Arsenal's yellow and red cards are not the same as those on the average Premier League team.  Arsenal pays a much bigger penalty for their red and yellow cards compared to the average Premier League team, and thus the differentiation in referee statistics shown in my last post has a much bigger effect on Arsenal.

Now that a binary logistic regression has been created to predict the effects of various match statistics on the likelihood of an Arsenal win, the contribution from each statistic for each match can be measured.  In studying the referees, the match statistics have been broken into three categories:

1. Things neither team nor the ref can control - venue
2. Things the referee tangentially controls - shots on goal, corners, fouls, etc.
3. Things the referee directly controls - yellow and red cards

There certainly is some interplay between all three - a home team may sense a more lenient ref (see Scorecasting) and will likely achieve a higher number of fouls before a yellow card is thrown their way. Luckily, from a statistical point of view very few of these interactions matter.  The results from the binary logistic regression indicate a precious few variables are statistically significant: venue, shots-on-goal differential, yellow card differential, and red card differential.

To calculate the impact of each referee, a comparison was made between

1. Each match's likelihood of winning given the match statistics as called versus
2. How the likelihood of winning would have changed had Arsenal experienced their average number of cards (adjusted for whether the match was home or away).

A general linear model was then constructed with this data to observe the impacts that season and referee had on the difference to the expected average.  The results from the general linear model are presented below via the main effects and interaction effects plots.

The graphs above confirm that Phil Dowd provides the highest differential against Arsenal from their expected mean.  On average, he costs them 4% per match against their odds of winning a match if they had experienced their average card differential - equivalent to a little more than a yellow card per match officiated.  As mentioned in the previous post on this topic, this is especially odd given the high proportion of home matches that he has officiated (home matches should result in a lower number of cards and thus higher proportion of winning).  Howard Webb is the only other official of the eight with a negative differential.  Four of the remaining six officials are right at the average differential of zero, while Chris Foy and Mark Halsey provides the most beneficial treatment of Arsenal.

All of this demonstrates that of the referees who officiate the greatest number of Arsenal matches, Dowd and Webb are the most biased against the Gunners.  Is this due to them actually being biased against Arsenal, or are they simply "tougher" officials when it comes to every team?  The calculations to determine one theory over the other are a good bit more involved, and will have to wait until the second post in this series...

Quantifying the Bias of Arsenal's Referees

"Are you kidding me, Webb? I know you're biased against

us, but don't you know that I am Nicklas the Great?"

A few posts ago, I attempted to quantify the bias of Phil Dowd’s record over the last two years of officiating versus the bias of all other referees who have officiated at least three Arsenal matches over the same time period. Since then, I have received a good bit of constructive feedback, especially when it came to the sample sizes used in the study. None of the feedback indicated any major errors, but more regarding the subtleties associated with varying statistical theories that could be used as a substitute for my Mann-Whitney approach.

Nonetheless, the feedback indicated there was a good bit of demand for a more extensive analysis. Thus, I contacted Tim at 7AM Kickoff and made a deal – he would compiled the earlier seasons of data , and I would analyze the data using common statistical methods. This post is the output of that study.

Developing the Model and Collecting The Data

The approach I wished to pursue in this study was a general liner model (GLM) as it allows for the study of the impact of multiple factors (and their interactions) on various outcomes. In this case, Tim and I were interested in studying the effects of Premier League season, referee, and match venue on fouls, cards, and shots. This study grew out of two mutual interests: Tim’s desire to quantify perceived overall referee bias during this Premier League season when compared to previous ones, and our joint desire to understand the most- and least-favorable referees when it comes to our beloved Gunners. I decided to throw in the match venue impacts at the suggestion of a dedicated reader of my blog.

A GLM has one unique requirement that presents challenges when applying it to officiating data: it requires at least one sample of each unique combination of attributes. This means that to build an overall GLM, we would need a data point from each season where each referee officiated at least one home and one away game in each season analyzed. This immediately presents a challenge, because leagues purposefully randomize assignments to minimize the effects of officiating on match outcome, and thus have unbalanced officiating from year-to-year. Increasing the number of seasons in a desire to increase sample size ends up limiting the types of GLMs that can be created. Tim and I settled on pulling data from the 2006-2007 seasons through the current season – it provided enough balance in sample size and attribute combinations to allow a two-phase study of officiating of Arsenal’s matches.

A table displaying the count of each referee’s matches officiated over the last 4+ seasons is shown below – 178 matches in all (current through the Wolves game on February 12th). The columns across the top indicate the season, where the second half of each season is used to denote the full season (thus, the 2006-2007 data is found in the column labeled “2007"). Each referee has three rows associated with their name – a row indicating their count of home matches (1), away matches (-1) and total number of matches. The column on the far right, labeled “Grand Total”, shows the total number of home and away matches officiated by each referee. Click on table to enlarge it.

What becomes immediately clear is that a GLM of seasons 2007 through 2010 by official and by match venue would have an extremely limited data set – only Atkinson, Bennett, Webb, and Wiley have officiated at least one home and one away match during that time period. Most importantly, a study of Dowd’s officiating would be left out of such a model. Such a GLM can be useful in studying a few of the large effects and their interactions, but not in directly evaluating a wider set of referees.

What’s also interesting is that the some of the numbers in the “Grand Total” column are greatly skewed. Over time, Bennett, Dowd, Foy, and Riley seem to have officiated more Arsenal home matches than away matches, while Clattenberg, Dean, Marinner, Riley, and Webb have experienced the inverse in their assignments. If Scorecasting’s study on home pitch officiating bias holds true in the Premier League, we might have some confounding of individual referee performance with general bias against a visiting team.

Two GLM’s were constructed given the match count shown in the table above:

1. A wider GLM that looked at the effects of the 2007-2011 seasons and match venue. This will help confirm or deny Scorecasting’s general conclusion regarding referee bias against visiting teams as they apply to the Premier League.
2. A GLM of the 2007-2010 seasons using the data from Atkinson, Bennett, Dean, Dowd, Foy, Halsey, Webb, and Wiley while ignoring the aspect of match venue. This will help answer the question as to which referees are the most biased for and against Arsenal.

Each GLM will look at four main attributes that officiating can impact: shots taken, the ratio of shots-on-goal to shots taken, fouls, and Premier League points for yellow and red cards (see my last post on this topic for an explanation).  Each of these attributes is expressed as a differential. To be consistent with the direction of the differential in the first post, a negative differential in any attribute indicates Arsenal had the advantage (took more shots, had a higher ratio of shots-on-goal, fewer cards etc.), while the opposite indicates the opponent had the advantage in the match.

In the end, the two GLM’s should help us understand where the bias lies, and a few of its potential causes.

Addressing the Home Pitch Bias of Referees

The first GLMScorecasting, exists the Premier League.

The GLM used all data from the 2007 through 2011 seasons for all referees, categorizing it as home and away while ignoring the contributions of individual referees. It turned out that none of the interaction effects were statistically significant, so the analysis focused on the main effects.  Main effects plots for each of the four attributes are shown below, with commentary below each plot.

Note: The key to reading main effects plots is to look for the center line traveling across the middle of the graph. This indicates the overall average value for that metric, with the average values associated with the individual levels of the factors (x-axis) are indicated by the discrete points on the graph.

It is clear that while match venue has little impact on the foul differential, Arsenal’s beneficial foul differential has shrunk to nothing over the last 4+ seasons. In fact, this shrinkage is one of the rare statistically significant factors not aligned with match venue in any GLM in this study. Perhaps it’s Arsenal’s increasingly tough response to “kick them off the pitch” tactics that has generated this shift? Whatever the case, Arsenal has gone from being on of the cleanest teams to middle of the pack when it comes to fair play.

When it comes to throwing cards, 2011 does represent a new high point for Arsenal but the trend is not statistically significant. Clearly, though, match venue has a huge impact (statistically significant, in fact!). The average home match (value of 1) sees Arsenal acquiring one less Premier League fantasy soccer penalty point (essentially one less yellow card) than the opposition, while away from home (value of -1) they acquire a similar amount of penalty points as their opponents. Scorecasting’s biased referee observations are alive and well!

The behavior witnessed in the shots metric is similar to that seen in the cards category. Arsenal holds a pretty steady seven shots per game advantage over the competition over the last three years, but a statistically significant gap exists between home and away matches.

Finally, it seems Arsenal are improving their effectiveness at putting shots on target versus the competition. The last two seasons have seen the Gunners turn around what was a deficiency into a benefit – not only do they take more shots on average, but they also put a greater percentage of them on target. The fact that the difference between home and away performance is not statistically significant ensures that they are using their reduced shot advantage when away from the Emirates in a manner consistent with home performance.

Ultimately, Arsenal seems to be doing pretty well in the shots department, getting worse when it comes to fouls, and consequently suffering from expected referee bias against away teams when it comes to cards. This final conclusion is especially important given the skewed home/away officiating opportunities afforded several of the referees highlighted in the next section.

Identifying Referees Who Are Biased For and Against Arsenal

The second GLM involved using 2007 through 2010 data to observe any possible bias in the following referees: Atkinson, Bennett, Dean, Dowd, Foy, Halsey, Webb, and Wiley. These referees have officiated 56% of Arsenal’s matches over the last four seasons. Main effects plots for each of the four attributes are shown below, with commentary below each plot. In general, none of the factors studied is statistically significant per the standard GLM tests. What is interesting is that the average value for every metric shrinks compared to the full 2007 through 2011 study performed above. Perhaps the big name matches that guys like Webb officiate provide a better balance between the teams, or maybe it's an effect of the "better" referees getting more matches. Either way, the average gap between Arsenal and their opponents closes when these eight referees are involved.

In general, the average foul differential is tilted slightly in Arsenal’s favor when these eight referees officiate a match compared to the overall average seen in the previous section. This may be due to the exclusion of 2011 to achieve a balanced GLM in the first example, where we observed that overall they've seen their foul differential disappear. It also seems as if Dowd is one of the more generous referees when his 2011 performance is dropped and his 2007 through 2009 performance is added. This may indicate a shift in his average officiating in 2011, a subject I will explore later in this post. Conversely, Dean, Webb, and Wiley have their bias against Arsenal confirmed in the foul department. They also are the top three referees when it comes to the number of Arsenal matches officiated, with Wiley and Webb having a pretty even home/away split while Dean has a 2/13 home/away split that may be unduly influencing the results of his officiating in Arsenal matches.

While Wiley has the highest foul differentials against Arsenal, he has one of the lowest card differentials. On the other hand, Webb and Dean follow through on the fouls with the two highest card differentials. With an average card differential of nearly -0.50 with these eight referees, Dean’s and Webb’s nearly 0.5 card differential against Arsenal represents nearly 1 extra yellow card per match for the Gunners. Again, Dean’s results may be biased based upon his high ratio of away matches, but Webb’s officiating is nearly evenly split between home and away matches.

As with the other responses, average shot differential goes down compared to the total sample of 2007 through 2011 matches. Dean and Webb again lead the pack in terms of officials showing an anti-Arsenal bias. Bennett and Foy appear to be the most pro-Arsenal referees, with Atkinson and Dowd not far behind.

The shots-on-goal ratio shows perhaps the biggest average shift when compared to the full 2007 through 2011 GLM. Webb and Dean rank close to the average, with Dean even showing slight favor to Arsenal. Matches where Foy officiates show the biggest disadvantage for Arsenal, with nearly a 10% gap between their ratio of shots-on-goal versus the opponent’s.

The Increasing Bias of Phil Dowd

So where does this leave Phil Dowd in relation to the Gunners, especially given my last post?  It would appear from the data above that he's actually biased for, and not against, Arsenal!  A closer examination of the data suggests otherwise.

It seems as if there is a shift afoot in Mr. Dowd’s officiating when it comes to the Gunners.  We can see this shift when looking at the interaction plot of year vs. referee from the second GLM that was created.  The interaction plot shows the average value of each combination of factors - year and official.  The key element for which to look is non-parallel lines, which indicate bias by one of the officials.  The interaction plot is shown below (click it to enlarge).

Looking at the lower left graph shows how nearly every referee's average points based upon yellow and red cards went down from 2009 to 2010.  The only referee's score to go up was Dowd's.  Things have gotten worse in 2011 as well, with Dowd's average over the three games he's officiated coming in at a whopping 2.33.  Dowd is the most biased against Arsenal in the 2011 season and has seen his card point total increase by nearly three points (a full red card!) from last season.  In four seasons Dowd has gone from a -2.0 card point total in 2008 to a 2.33 total in 2011 (a full read and yellow card swing).  No other referee exhibits this kind of shift. Over that time period Dowd has officiated seven home matches and four away matches, clearly bucking the trend of biased officiating coming against away teams.

Conclusions

Clearly Webb and Dean are the referees that generally make more calls against Arsenal, with Webb the only one of the two with a reasonably balanced home/away officiating record that would eliminate away team bias from consideration.  Phil Dowd is rapidly approaching Webb's level of bias, with a massive shift in the way he's called matches since 2008, and a higher proportion of home matches officiated that eliminates away team bias as an excuse.

Foy and Atkinson may provide the most reliably pro-Arsenal calls.  Both provide the best combination of foul and red/yellow card advantages to Arsenal after Dowd is eliminated for his time-based shift in officiating results.

While Arsenal have consistently had a penalty and card advantage in years past, such an advantage is almost non-existent this season and last.  However, they have maintained their advantage in number of shots and percentage of shots on goal.

As a Gooner, I hope that we draw guys like Atkinson and Foy the last quarter of the season.  This gives the Gunners the best chance to close the gap to Manchester United.  I'll come back to this topic once the season is complete and we hopefully have a higher number of referees with home and away match officiating opportunities.  That would allow a more complete GLM of officials vs. venue and year.

Of immediate concern to Gooners is the fact that Dean will be officiating this Sunday's Carling Cup final.  Here's hoping he shows far less bias on the neutral pitch of Wembley than he has in Arsenal's away matches.