Last night I was thinking about the results we found in the previous blog post about upsets and mismatches and wondered if the historical pattern of expected game margins was borne out in the actual results. On analysing the data I found that there were a lot more victories of 10 Scoring Shots or more in magnitude than MoSSBODS had predicted. In most seasons, at least one-third of the games finished with a victory margin equivalent to 10 Scoring Shots or more, which was usually two or three times as many as MoSSBODS had predicted.Read More
In the previous blog on this topic I posited that the Scoring Shot production of a team could be modelled as a Poisson random variable with some predetermined mean, and that the conversion of these Scoring Shots into Goals could be modelled as a BetaBinomial with fixed conversion probability and theta (a spread parameter).Read More
This week, thanks to Amazon, who replaced my unreadable Kindle copy of David W Miller's Fitting Frequency Distributions: Philosophy and Practice with a dead-tree version that could easily be used as a weapon such is its heft (and assuming you had the strength to wield it), I've been reminded of the importance of motivating my distributional choices with a plausible narrative. It's not good enough, he contends, to find that, say, a Gamma Distribution fits your data set really well, you should be able to explain why it's an appropriate choice from first principles.Read More
In response to my earlier post on the explained and unexplained portions of game margins, Friend of MatterOfStats, Michael, e-mailed me to suggest that variability in teams' points-scoring per scoring shot - or, equivalently, teams' conversion rates - might usefully be explored as a source of unexplained variability.Read More
Some blogs almost write themselves. This hasn't been one of them.
It all started when I read a journal article - to which I'd now link if I could find the darned thing again - that suggested a bias in NFL (or maybe it was College Football) spread betting markets arising from bookmakers' tendency to over-correct when a team won on line betting. The authors found that after a team won on line betting one week it was less likely to win on line betting next week because it was forced to overcome too large a handicap.
Naturally, I wondered if this was also true of AFL spread betting.
What Makes a Team's Start Vary from Week-to-Week?
In the process of investigating that question, I wound up wondering about the process of setting handicaps in the first place and what causes a team's handicap to change from one week to the next.
Logically, I reckoned, the start that a team receives could be described by this equation:
Start received by Team A (playing Team B) = (Quality of Team B - Quality of Team A) - Home Status for Team A
In words, the start that a team gets is a function of its quality relative to its opponent's (measured in points) and whether or not it's playing at home. The stronger a team's opponent the larger will be the start, and there'll be a deduction if the team's playing at home. This formulation of start assumes that game venue only ever has a positive effect on one team and not an additional, negative effect on the other. It excludes the possibility that a side might be a P point worse side whenever it plays away from home.
With that as the equation for the start that a team receives, the change in that start from one week to the next can be written as:
Change in Start received by Team A = Change in Quality of Team A + Difference in Quality of Teams played in successive weeks + Change in Home Status for Team A
To use this equation for we need to come up with proxies for as many of the terms that we can. Firstly then, what might a bookie use to reassess the quality of a particular team? An obvious choice is the performance of that team in the previous week relative to the bookie's expectations - which is exactly what the handicap adjusted margin for the previous week measures.
Next, we could define the change in home status as follows:
- Change in home status = +1 if a team is playing at home this week and played away or at a neutral venue in the previous week
- Change in home status = -1 if a team is playing away or at a neutral venue this week and played at home in the previous week
- Change in home status = 0 otherwise
This formulation implies that there's no difference between playing away and playing at a neutral venue. Varying this assumption is something that I might explore in a future blog.
From Theory to Practicality: Fitting a Model
(well, actually there's a bit more theory too ...)
Having identified a way to quantify the change in a team's quality and the change in its home status we can now run a linear regression in which, for simplicity, I've further assumed that home ground advantage is the same for every team.
We get the following result using all home-and-away data for seasons 2006 to 2010:
For a team (designated to be) playing at home in the current week:
Change in start = -2.453 - 0.072 x HAM in Previous Week - 8.241 x Change in Home Status
For a team (designated to be) playing away in the current week:
Change in start = 3.035 - 0.155 x HAM in Previous Week - 8.241 x Change in Home Status
These equations explain about 15.7% of the variability in the change in start and all of the coefficients (except the intercept) are statistically significant at the 1% level or higher.
(You might notice that I've not included any variable to capture the change in opponent quality. Doubtless that variable would explain a significant proportion of the otherwise unexplained variability in change in start but it suffers from the incurable defect of being unmeasurable for previous and for future games. That renders it not especially useful for model fitting or for forecasting purposes.
Whilst that's a shame from the point of view of better modelling the change in teams' start from week-to-week, the good news is that leaving this variable out almost certainly doesn't distort the coefficients for the variables that we have included. Technically, the potential problem we have in leaving out a measure of the change in opponent quality is what's called an omitted variable bias, but such bias disappears if the the variables we have included are uncorrelated with the one we've omitted. I think we can make a reasonable case that the difference in the quality of successive opponents is unlikely to be correlated with a team's HAM in the previous week, and is also unlikely to be correlated with the change in a team's home status.)
Using these equations and historical home status and HAM data, we can calculate that the average (designated) home team receives 8 fewer points start than it did in the previous week, and the average (designated) away team receives 8 points more.
All of which Means Exactly What Now?
Okay, so what do these equations tell us?
Firstly let's consider teams playing at home in the current week. The nature of the AFL draw is such that it's more likely than not that a team playing at home in one week played away in the previous week in which case the Change in Home Status for that team will be +1 and their equation can be rewritten as
Change in Start = -10.694 - 0.072 x HAM in Previous Week
So, the start for a home team will tend to drop by about 10.7 points relative to the start it received in the previous week (because they're at home this week) plus about another 1 point for every 14.5 points lower their HAM was in the previous week. Remember: the more positive the HAM, the larger the margin by which the spread was covered.
Next, let's look at teams playing away in the current week. For them it's more likely than not that they played at home in the previous week in which case the Change in Home Status will be -1 for them and their equation can be rewritten as
Change in Start = 11.276 - 0.155 x HAM in Previous Week
Their start, therefore, will tend to increase by about 11.3 points relative to the previous week (because they're away this week) less 1 point for every 6.5 points lower their HAM was in the previous week.
Away teams, therefore, are penalised more heavily for larger HAMs than are home teams.
This I offer as one source of potential bias, similar to the bias that was found in the original article I read.
Proving the Bias
As a simple way of quantifying any bias I've fitted what's called a binary logit to estimate the following model:
Probability of Winning on Line Betting = f(Result on Line Betting in Previous Week, Start Received, Home Team Status)
This model will detect any bias in line betting results that's due to an over-reaction to the previous week's line betting results, a tendency for teams receiving particular sized starts to win or lose too often, or to a team's home team status.
The result is as follows:
logit(Probability of Winning on Line Betting) = -0.0269 + 0.054 x Previous Line Result + 0.001 x Start Received + 0.124 x Home Team Status
The only coefficient that's statistically significant in that equation is the one on Home Team Status and it's significant at the 5% level. This coefficient is positive, which implies that home teams win on line betting more often than they should.
Using this equation we can quantify how much more often. An away team, we find, has about a 46% probability of winning on line betting, a team playing at a neutral venue has about a 49% probability, and a team playing at home has about a 52% probability.
That is undoubtedly a bias, but I have two pieces of bad news about it. Firstly, it's not large enough to overcome the vig on line betting at $1.90 and secondly, it disappeared in 2010.
Do Margins Behave Like Starts?
We now know something about how the points start given by the TAB Sportsbet bookie responds to a team's change in estimated quality and to a change in a team's home status. Do the actual game margins respond similarly?
One way to find this out is to use exactly the same equation as we used above, replacing Change in Start with Change in Margin and defining the change in a team's margin as its margin of victory this week less its margin of victory last week (where victory margins are negative for losses).
If we do that and run the new regression model, we get the following:
For a team (designated to be) playing at home in the current week:
Change in Margin = 4.058 - 0.865 x HAM in Previous Week + 8.801 x Change in Home Status
For a team (designated to be) playing away in the current week:
Change in Margin = -4.571 - 0.865 x HAM in Previous Week + 8.801 x Change in Home Status
These equations explain an impressive 38.7% of the variability in the change in margin. We can simplify them, as we did for the regression equations for Change in Start, by using the fact that the draw tends to alternate team's home and away status from one week to the next.
So, for home teams:
Change in Margin = 12.859 - 0.865 x HAM in Previous Week
While, for away teams:
Change in Margin = -13.372 - 0.865 x HAM in Previous Week
At first blush it seems a little surprising that a team's HAM in the previous week is negatively correlated with its change in margin. Why should that be the case?
It's a phenomenon that we've discussed before: regression to the mean. What these equations are saying are that teams that perform better than expected in one week - where expectation is measured relative to the line betting handicap - are likely to win by slightly less than they did in the previous week or lose by slightly more.
What's particularly interesting is that home teams and away teams show mean regression to the same degree. The TAB Sportsbet bookie, however, hasn't always behaved as if this was the case.
Another Approach to the Source of the Bias
Bringing the Change in Start and Change in Margin equations together provides another window into the home team bias.
The simplified equations for Change in Start were:
Home Teams: Change in Start = -10.694 - 0.072 x HAM in Previous Week
Away Teams: Change in Start = 11.276 - 0.155 x HAM in Previous Week
So, for teams whose previous HAM was around zero (which is what the average HAM should be), the typical change in start will be around 11 points - a reduction for home teams, and an increase for away teams.
The simplified equations for Change in Margin were:
Home Teams: Change in Margin = 12.859 - 0.865 x HAM in Previous Week
Away Teams: Change in Margin = -13.372 - 0.865 x HAM in Previous Week
So, for teams whose previous HAM was around zero, the typical change in margin will be around 13 points - an increase for home teams, and a decrease for away teams.
Overall the 11-point v 13-point disparity favours home teams since they enjoy the larger margin increase relative to the smaller decrease in start, and it disfavours away teams since they suffer a larger margin decrease relative to the smaller increase in start.
Historically, home teams win on line betting more often than away teams. That means home teams tend to receive too much start and away teams too little.
I've offered two possible reasons for this:
- Away teams suffer larger reductions in their handicaps for a given previous weeks' HAM
- For teams with near-zero previous week HAMs, starts only adjust by about 11 points when a team's home status changes but margins change by about 13 points. This favours home teams because the increase in their expected margin exceeds the expected decrease in their start, and works against away teams for the opposite reason.
If you've made it this far, my sincere thanks. I reckon your brain's earned a spell; mine certainly has.
As so many traders discovered to their individual and often, regrettably, our collective cost over the past few years, betting against longshots, deliberately or implicitly, can be a very lucrative gig until an event you thought was a once-in-a-virtually-never affair crops up a couple of times in a week. And then a few more times again after that.
To put a footballing context on the topic, let's imagine that a friend puts the following proposition bet to you: if none of the first 100 home-and-away games next season includes one with a handicap-adjusted margin (HAM) for the home team of -150 or less he'll pay you $100; if there is one or more games with a HAM of -150 or less, however, you pay him $10,000.
For clarity, by "handicap-adjusted margin" I mean the number that you get if you subtract the away team's score from the home team's score and then add the home team's handicap. So, for example, if the home team was a 10.5 point favourite but lost 100-75, then the handicap adjusted margin would be 75-100-10.5, or -35.5 points.
A First Assessment
At first blush, does the bet seem fair?
We might start by relying on the availability heuristic and ask ourselves how often we can recall a game that might have produced a HAM of -150 or less. To make that a tad more tangible, how often can you recall a team losing by more than 150 points when it was roughly an equal favourite or by, say, 175 points when it was a 25-point underdog?
Almost never, I'd venture. So, offering 100/1 odds about this outcome occurring once or more in 100 games probably seems attractive.
Ahem ... the data?
Maybe you're a little more empirical than that and you'd like to know something about the history of HAMs. Well, since 2006, which is a period covering just under 1,000 games and that spans the entire extent - the whole hog, if you will - of my HAM data, there's never been a HAM under -150.
One game produced a -143.5 HAM; the next lowest after that was -113.5. Clearly then, the HAM of -143.5 was an outlier, and we'd need to see another couple of scoring shots on top of that effort in order to crack the -150 mark. That seems unlikely.
In short, we've never witnessed a HAM of -150 or less in about 1,000 games. On that basis, the bet's still looking good.
But didn't you once tell me that HAMs were Normal?
Before we commit ourselves to the bet, let's consider what else we know about HAMs.
Previously, I've claimed that HAMs seemed to follow a normal distribution and, in fact, the HAM data comfortably passes the Kolmogorov-Smirnov test of Normality (one of the few statistical tests I can think of that shares at least part of its name with the founder of a distillery).
Now technically the HAM data's passing this test means only that we can't reject the null hypothesis that it follows a Normal distribution, not that we can positively assert that it does. But given the ubiquity of the Normal distribution, that's enough prima facie evidence to proceed down this path of enquiry.
To do that we need to calculate a couple of summary statistics for the HAM data. Firstly, we need to calculate the mean, which is +2.32 points, and then we need to calculate the standard deviation, which is 36.97 points. A HAM of -150 therefore represents an event approximately 4.12 standard deviations from the mean.
If HAMs are Normal, that's certainly a once-in-a-very-long-time event. Specifically, it's an event we should expect to see only about every 52,788 games, which, to put it in some context, is almost exactly 300 times the length of the 2010 home-and-away season.
With a numerical estimate of the likelihood of seeing one such game we can proceed to calculate the likelihood of seeing one or more such game within the span of 100 games. The calculation is 1-(1-1/52,788)^100 or 0.19%, which is about 525/1 odds. At those odds you should expect to pay out that $10,000 about 1 time in 526, and collect that $100 on the 525 other occasions, which gives you an expected profit of $80.81 every time you take the bet.
That still looks like a good deal.
Does my tail look fat in this?
This latest estimate carries all the trappings of statistically soundness, but it does hinge on the faith we're putting in that 1 in 52,788 estimate, which, in turn hinges on our faith that HAMs are Normal. In the current instance this faith needs to hold not just in the range of HAMs that we see for most games - somewhere in the -30 to +30 range - but way out in the arctic regions of the distribution rarely seen by man, the part of the distribution that is technically called the 'tails'.
There are a variety of phenomena that can be perfectly adequately modelled by a Normal distribution for most of their range - financial returns are a good example - but that exhibit what are called 'fat tails', which means that extreme values occur more often than we would expect if the phenomenon faithfully followed a Normal distribution across its entire range of potential values. For most purposes 'fat tails' are statistically vestigial in their effect - they're an irrelevance. But when you're worried about extreme events, as we are in our proposition bet, they matter a great deal.
A class of distributions that don't get a lot of press - probably because the branding committee that named them clearly had no idea - but that are ideal for modelling data that might have fat tails are the Stable Distributions. They include the Normal Distribution as a special case - Normal by name, but abnormal within its family.
If we fit (using Maximum Likelihood Estimation if you're curious) a Stable Distribution to the HAM data we find that the best fit corresponds to a distribution that's almost Normal, but isn't quite. The apparently small difference in the distributional assumption - so small that I abandoned any hope of illustrating the difference with a chart - makes a huge difference in our estimate of the probability of losing the bet. Using the best fitted Stable Distribution, we'd now expect to see a HAM of -150 or lower about 1 game in every 1,578 which makes the likelihood of paying out that $10,000 about 7%.
Suddenly, our seemingly attractive wager has a -$607 expectation.
Since we almost saw - if that makes any sense - a HAM of -150 in our sample of under 1,000 games, there's some intuitive appeal in an estimate that's only a bit smaller than 1 in 1,000 and not a lot smaller, which we obtained when we used the Normal approximation.
Is there any practically robust way to decide whether HAMs truly follow a Normal distribution or a Stable Distribution? Given the sample that we have, not in the part of the distribution that matters to us in this instance: the tails. We'd need a sample many times larger than the one we have in order to estimate the true probability to an acceptably high level of certainty, and by then would we still trust what we'd learned from games that were decades, possibly centuries old?
Is There a Lesson in There Somewhere?
The issue here, and what inspired me to write this blog, is the oft-neglected truism - an observation that I've read and heard Nassim Taleb of "Black Swan" fame make on a number of occasions - that rare events are, well, rare, and so estimating their likelihood is inherently difficult and, if you've a significant interest in the outcome, financially or otherwise dangerous.
For many very rare events we simply don't have sufficiently large or lengthy datasets on which to base robust probability estimates for those events. Even where we do have large datasets we still need to justify a belief that the past can serve as a reasonable indicator of the future.
What if, for example, the Gold Coast team prove to be particularly awful next year and get thumped regularly and mercilessly by teams of the Cats' and the Pies' pedigrees? How good would you feel than about betting against a -150 HAM?
So when some group or other tells you that a potential catastrophe is a 1-in-100,000 year event, ask them what empirical basis they have for claiming this. And don't bet too much on the fact that they're right.
Okay, this is probably going to be a long blog so you might want to make yourself comfortable.
For some time now I've been wondering about the statistical properties of the Handicap-Adjusted Margin (HAM). Does it, for example, follow a normal distribution with zero mean?
Well firstly we need to deal with the definition of the term HAM, for which there is - at least - two logical definitions.
The first definition, which is the one I usually use, is calculated from the Home Team perspective and is Home Team Score - Away Team Score + Home Team's Handicap (where the Handicap is negative if the Home Team is giving start and positive otherwise). Let's call this Home HAM.
As an example, if the Home Team wins 112 to 80 and was giving 20.5 points start, then Home HAM is 112-80-20.5 = +11.5 points, meaning that the Home Team won by 11.5 points on handicap.
The other approach defines HAM in terms of the Favourite Team and is Favourite Team Score - Underdog Team Score + Favourite Team's Handicap (where the Handicap is always negative as, by definition the Favourite Team is giving start). Let's call this Favourite HAM.
So, if the Favourite Team wins 82 to 75 and was giving 15.5 points start, then Favourite HAM is 82-75-15.5 = -7.5 points, meaning that the Favourite Team lost by 7.5 points on handicap.
Home HAM will be the same as Favourite HAM if the Home Team is Favourite. Otherwise Home HAM and Favourite HAM will have opposite signs.
There is one other definitional detail we need to deal with and that is which handicap to use. Each week a number of betting shops publish line markets and they often differ in the starts and the prices offered for each team. For this blog I'm going to use TAB Sportsbet's handicap markets.
TAB Sportsbet Handicap markets work by offering even money odds (less the vigorish) on both teams, with one team receiving start and the other offering that same start. The only exception to this is when the teams are fairly evenly matched in which case the start is fixed at 6.5 points and the prices varied away from even money as required. So, for example, we might see Essendon +6.5 points against Carlton but priced at $1.70 reflecting the fact that 6.5 points makes Essendon in the bookie's opinion more likely to win on handicap than to lose. Games such as this are problematic for the current analysis because the 'true' handicap is not 6.5 points but is instead something less than 6.5 points. Including these games would bias the analysis - and adjusting the start is too complex - so we'll exclude them.
So, the question now becomes is HAM Home, defined as above and using the TAB Sportsbet handicap and excluding games with 6.5 points start or fewer, normally distributed with zero mean? Similarly, is HAM Favourite so distributed?
We should expect HAM Home and HAM Favourite to have zero means because, if they don't it suggests that the Sportsbet bookie has a bias towards or against Home teams of Favourites. And, as we know, in gambling, bias is often financially exploitable.
There's no particular reason to believe that HAM Home and HAM Favourite should follow a normal distribution, however, apart from the startling ubiquity of that distribution across a range of phenomena.
Consider first the issue of zero means.
The following table provides information about Home HAMs for seasons 2006 to 2008 combined, for season 2009, and for seasons 2006 to 2009. I've isolated this season because, as we'll see, it's been a slightly unusual season for handicap betting.
Each row of this table aggregates the results for different ranges of Home Team handicaps. The first row looks at those games where the Home Team was offering start of 30.5 points or more. In these games, of which there were 53 across seasons 2006 to 2008, the average Home HAM was 1.1 and the standard deviation of the Home HAMs was 39.7. In season 2009 there have been 17 such games for which the average Home HAM has been 14.7 and the standard deviation of the Home HAMs has been 29.1.
The asterisk next to the 14.7 average denotes that this average is statistically significantly different from zero at the 10% level (using a two-tailed test). Looking at other rows you'll see there are a handful more asterisks, most notably two against the 12.5 to 17.5 points row for season 2009 denoting that the average Home HAM of 32.0 is significant at the 5% level (though it is based on only 8 games).
At the foot of the table you can see that the overall average Home HAM across seasons 2006 to 2008 was, as we expected approximately zero. Casting an eye down the column of standard deviations for these same seasons suggests that these are broadly independent of the Home Team handicap, though there is some weak evidence that larger absolute starts are associated with slightly larger standard deviations.
For season 2009, the story's a little different. The overall average is +8.4 points which, the asterisks tell us, is statistically significantly different from zero at the 5% level. The standard deviations are much smaller and, if anything, larger absolute margins seem to be associated with smaller standard deviations.
Combining all the seasons, the aberrations of 2009 are mostly washed out and we find an average Home HAM of just +1.6 points.
Next, consider Favourite HAMs, the data for which appears below:
The first thing to note about this table is the fact that none of the Favourite HAMs are significantly different from zero.
Overall, across seasons 2006 to 2008 the average Favourite HAM is just 0.1 point; in 2009 it's just -3.7 points.
In general there appears to be no systematic relationship between the start given by favourites and the standard deviation of the resulting Favourite HAMs.
- Across seasons 2006 to 2009, Home HAMs and Favourite HAMs average around zero, as we hoped
- With a few notable exceptions, mainly for Home HAMs in 2009, the average is also around zero if we condition on either the handicap given by the Home Team (looking at Home HAMs) or that given by the Favourite Team (looking at Favourite HAMs).
Okay then, are Home HAMs and Favourite HAMs normally distributed?
Here's a histogram of Home HAMs:
And here's a histogram of Favourite HAMs:
There's nothing in either of those that argues strongly for the negative.
More formally, Shapiro-Wilks tests fail to reject the null hypothesis that both distributions are Normal.
Using this fact, I've drawn up a couple of tables that compare the observed frequency of various results with what we'd expect if the generating distributions were Normal.
Here's the one for Home HAMs:
There is a slight over-prediction of negative Home HAMs and a corresponding under-prediction of positive Home HAMs but, overall, the fit is good and the appropriate Chi-Squared test of Goodness of Fit is passed.
And, lastly, here's the one for Home Favourites:
In this case the fit is even better.
We conclude then that it seems reasonable to treat Home HAMs as being normally distributed with zero mean and a standard deviation of 37.7 points and to treat Favourite HAMs as being normally distributed with zero mean and, curiously, the same standard deviation. I should point out for any lurking pedant that I realise neither Home HAMs nor Favourite HAMs can strictly follow a normal distribution since Home HAMs and Favourite HAMs take on only discrete values. The issue really is: practically, how good is the approximation?
This conclusion of normality has important implications for detecting possible imbalances between the line and head-to-head markets for the same game. But, for now, enough.