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. 

It's a clever suggestion and got me to thinking about statistically modelling teams' expected scoring shot production separately from their expected conversion rate, then combining the two models to create game predictions - an extension, in retrospect, on this blog - but for now and for this blog I'll be concentrating simply on quantifying the variability in the two scoring metrics by team and by season and exploring the extent to which this variability might or might not have been foreseen by the TAB Bookmaker.

THE DATA

Because I want to incorporate TAB Bookmaker prices in the analysis I'll be restricting my attention to the period 2006 to 2013, including from those seasons all home-and-away and Finals contests.

Let's start by reviewing the Points per Scoring Shot averages by team and by season.

The table on the right provides the summary data, separating each team's performances when playing at home (top table) from those when playing away (bottom table).

(NB: If you're attempting to reconcile my analysis remember that I determine home team status in Finals on the basis of the team with the higher MARS Rating at the time.)

The first thing I'd note is that home teams and away teams have, in aggregate, scored about the same per shot in every season until the last two, when Away teams have been slightly more accurate than Home teams. Whilst the 2012 difference in points scored per scoring shot of just 0.07 points might seem small, it's equivalent to about a 1.6% point difference in scoring shot conversion at the levels we're considering. An increase of that magnitude has a material affect on a team's winning chances.

Adelaide and Port Adelaide have contributed most conspicuously to the phenomenon in those seasons, registering 0.14 and 0.17 more points per scoring shot away compared to at home in 2012, and 0.33 and 0.40 more points per scoring shot at home than away in 2013. Essendon, Sydney and West Coast are other teams who've being more accurate when playing away than when playing at home during these two seasons.

From season to season, teams have exhibited quite widely varying performances on the Points Scored per Scoring Shot metric. The Western Bulldogs, when playing at home, for example, scored only 3.22 points per scoring shot in 2012 after having scoring 3.95 per scoring shot in 2008 (and 3.92 in 2013, just a year after their nadir). Richmond have unfurled a similarly broad range of performances when playing away, ekeing out a paltry 3.34 points per scoring shot in 2010, after a whopping 4.06 points per shot haul in 2008. Generating 4.06 points per scoring shot requires that a team kick goals from about 61% of their scoring shots.

Taking all eight seasons as a whole, the Western Bulldogs (3.77) have generated more points per scoring shot than any other team when playing at home, and Gold Coast (3.45) have generated the fewest. Playing away, Geelong have the best record (3.73 points per scoring shot), though Port Adelaide, Sydney and the Western Bulldogs have almost identical averages. The Gold Coast also have the worst away record having registered just 3.46 points per scoring shot.

The all-time (well, if you'll let me call the eight seasons considered here as "all-time") single season records are as follows:

• Fewest points per scoring shot when playing at home: Western Bulldogs (2012; 3.22)
• Most points per scoring shot when playing at home: St Kilda (2010; 4.00)
• Fewest points per scoring shot when playing away: West Coast (2008; 3.24)
• Most points per scoring shot when playing away: Richmond (2008; 4.06)

Next,  reviewing Scoring Shots per Game (see table at left) we find that home teams, in aggregate, in every season have generated more scoring shots than away teams. The overall difference has been about 2.1 scoring shots per game though for a single season the difference has been as high as 2.9 scoring shots per game (2012) and as low as 1.5 (2008).

On a team-by-team basis, variability on this metric has also been substantial.

Playing at home, Hawthorn has gone from registering just 23.3 scoring shots per game in 2009 to accumulating 34.2 shots per game in 2012, and Port Adelaide has swung from a low of 20.2 in 2011 to a high of 31.1 in 2007.

Away from home Hawthorn also has the widest range of performances, with a low of 20.3 shots per game in 2006 and a high of 30.7 in 2012.

Only one team has produced more scoring shots per game playing away than when playing at home across the eight seasons combined: Carlton, who have recorded 26.6 scoring shots per game at home and 26.9 when playing away; yet more evidence for the virtual non-existence of a home ground advantage for Victorian teams.

The record for the largest difference between scoring shot production at home versus scoring shot production away belongs to the Brisbane Lions, who've totted up 26.7 scoring shots per game at the Gabba but only 22.2 when playing elsewhere, for a difference of 4.5 scoring shots per game. GWS (4.0), Sydney (3.6) and Fremantle (3.5) also have relatively large differences between their home and their away scoring shot production.

The all-time single season records for this metric are:

• Fewest scoring shots per game when playing at home: Melbourne (2013; 18.7)
• Most scoring shots per game when playing at home: Hawthorn (2012; 34.2)
• Fewest scoring shots per game when playing away: GWS (2012; 13.6)
• Most scoring shots per game when playing away: Geelong (2007; 33.6)

THE METHODOLOGY

One way of investigating the effects of variability in scoring shot production and conversion on the final result of a game is to decompose the game margin, taken as usual from the Home team's viewpoint, into an expected component, which we can derive from the TAB Bookmaker's pre-game prices, and an unexpected component. The latter can be further split into three pieces, as in the derivation in the box at right, which we can conceptualise as:

• a piece due to the teams' generating, in net terms, greater or fewer scoring shots than the Bookmaker (can be inferred to have) expected
• a piece due to the teams' generating, again in net terms, greater or fewer points per scoring shot than the Bookmaker expected
• a piece due to both of these factors

(For those of you familiar with rate-volume decompositions that are often used to split the change in interest income or interest expense on a deposit or loan portfolio in a financial institution, the approach used here is very similar.)

To make this approach operational, we need to make some assumptions that allow us to convert the Bookmaker's head-to-head prices into, firstly, an implicit game margin and then, secondly, into expected scoring shot production and conversion for the two teams.

The details of these assumptions are in the box above, but the essence is that we:

• convert the prices to a home team probability using the Risk-Equalising methodology
• convert the home team probability to a home team margin by assuming a Normal distribution for handicap-adjusted game margins with a mean of zero and a standard deviation chosen "optimally" (it maximises the correlation between actual and expected margins over the 2006 to 2013 period)
• assume the team predicted to win will score points equal to the all-team average across the period 2006 to 2013 (92.9 points) plus one-half of the predicted margin (and that the losing team will score this same amount less the expected game margin)
• assume that both teams will score the same number of points per scoring shot as the all-team average across the entire period 2006 to 2013 (3.65 points per shot)
• derive the expected number of scoring shots as the expected score divided by the expected points per scoring shot

It's possible, of course, that the Bookmaker might think otherwise, for example by adopting team-specific scoring shot conversion rates rather than using the same figure for all teams as I have him doing, but the analysis is already quite complex and would probably be not much altered by complicating it further. Not that I didn't think about it ...

THE RESULTS

After applying this methodology to all of the games from 2006 to 2013 let's look firstly at the results on a season-by-season basis.

From the Total Difference column we see that home teams have, in every season but 2013, won by larger margins (or lost by smaller margins) than the TAB Bookmaker expected, by average amounts ranging from about 1 to 3.5 points. This is a finding we've established before. Last season, however, home team margins were almost 2 points less than expected - a slightly disturbing development for me because MatterOfStats wagering depends partly on the bias in the TAB Bookmaker's assessment of home team abilities continuing.

The decomposition of the Total Difference shows that the variability due to unexpected net Scoring Shot generation has, in every season, had the same sign as the Total Difference, and that the variability due to unexpected net Scoring per Shot has also carried the same sign except in 2006, 2010 and 2012. Further, in every year except 2011 and 2013 the magnitude of the difference due to Scoring Shot variability has exceeded, in absolute terms, that due to Scoring per Shot variability. The contribution from the Both component has been relatively small in every season. (In rate-volume analysis this amount is often split 50:50 between the two sources.)

In summary then, the largest contribution to the error in Bookmaker Margin predictions across the eight years - to the extent that my methodology for inferring his assumptions about team scoring shot production and conversion is accurate - has generally been unexpected net differences between the Home and Away teams' Scoring Shot production, though in two of the last three seasons unexpected net differences between the teams' Points per Scoring Shot generation has played the more significant role.

We can also look at this same data on a team-by-team basis, subdividing the data on the basis of whether the team in question was playing at home or away. The results of doing this are recorded in the table at left.

There we see that St Kilda, Geelong, Sydney, Adelaide and the Roos have recorded the highest average net excess in game margins when playing at home, and Carlton and Richmond the highest average net deficit. Of the five teams who've generated excess margins (ie who've scored more than expected when playing at home), all but Sydney have done so mostly by generating additional net Scoring Shots. Similarly, Richmond's and, especially, Carlton's deficits have been mostly due to their generating fewer net Scoring Shots than expected.

GWS' and West Coast's decompositions when playing at home are especially interesting as, in both cases, their net Scoring Shot production has been less than expected whereas their net Points Scored per Shot has been greater than expected but insufficient to leave them with a net positive Total Difference. In other words, they've kicked straighter than expected but had fewer opportunities to capitalise on this virtue.

Finally, looking at the teams' records when playing away - and recognising that the margins here are still shown from the Home teams' viewpoint and so carry the opposite sign to what you might otherwise expect - we find GWS, Melbourne and Essendon with the largest net negative Total Differences, in GWS' and Melbourne's cases due mostly to unexpectedly small net Scoring Shot production. For GWS, in fact, over 22 points worth of unexpected game margins has been attributable to their generating net fewer Scoring Shots than expected.

Carlton and Richmond are, again, amongst the more notable teams, though here because of their association with net positive Total Differences from their point of view.

In all, five teams are associated with positive Total Differences when playing at home and when playing away (Adelaide, Collingwood, Geelong, Kangaroos (barely) and Sydney), and four teams are associated with negative Total Differences when playing at home and away (Essendon, GWS, Port Adelaide and West Coast).

Also, six teams have generated excess net Points Scoring per Shot whether playing at home or playing away (Fremantle, Geelong, Hawthorn, St Kilda, Sydney and the Western Bulldogs) and three have generated deficits (Essendon, Gold Coast, GWS and West Coast) at both venue types. Just two teams (Collingwood and Gold Coast) have generated more net Scoring Shots than expected whether playing at home or away, while four (Essendon, Fremantle, Port Adelaide and the Western Bulldogs) have generated net fewer than expected.

In reviewing all of these results in light of the earlier information about teams' scoring shot production and conversion, bear in mind that the numbers in this latter section are relative to the TAB Bookmaker's (inferred) expectations. So, for example, the fact that Gold Coast playing away have the second-poorest record in terms of scoring shot production (at 20.8 scoring shots per game) can be reconciled with the figure in the table above showing that they've generated about 3.3 points more per game as a result of unexpectedly high net scoring shot production by positing that, although their scoring shot production has been low in absolute terms when playing away, it's still been better than the TAB Bookmaker expected.