Welcome back. This is the seventh part in a series on answering the question "Which teams are deserving of a playoff invitation?" In it I'll outline a model I'll refer to as "Extended Standings" and over the course of the series I'll provide the exact details so anyone interested can independently verify the results.
Index: Part I | Part II | Part III | Part IV | Part V | Part VI
To this point I've provided everything needed to calculate the Extended Standings except how to factor in each team's classification, so I'll quickly wrap up the mechanics of the Extended Standings here and then provide an Excel spreadsheet so that you can study, dissect, calculate, etc., the Extended Standings all on your own. I'll provide an updated version of this spreadsheet each week in my Extended Standings column.
First let's challenge the inclusion of each team's classification by considering again what is the fundamental data required to perform meaningful comparisons between teams. In Part I of the series I asserted that given an almanac or newspaper, sports fans intuitively know the essential data required to compare teams – the opponents and the scores.
In an ideal world, every team would have played every other team at least once if not multiple times so that all direct comparisons could be made and no other information would need to be included. If we constructed a graph showing a network of teams and games with this characteristic, it would be considered "dense".
This is typically the case for leagues such as Major League Baseball, but the logistics of football pushes any sizeable league toward a sparse network of games. For example, there are 417 teams in the GHSA so there are 417 x 416 = 173,472 total possible matchups. However only about 2,100 games, or 1.2% of the total possible, are actually played during the regular season.
Additionally, for the 2015 season, there are 48 regions in the GHSA, or an average of 8.7 teams per region, so an average team will play roughly seven region opponents out of their ten regular season games. And although not the case for this specific season, in the past there have been regions with 11 members, meaning that each of those 11 teams only played against other region opponents, isolating them from the remainder of the network and rendering direct or indirect comparisons with teams outside the region impossible without considering some other information.
We can visualize the network for the GHSA's 2015 regular season by referring to the graph below. Each node in the graph represents a team and each connection between the nodes represents a game. The teams are color coded by the region and a quick glance highlights the provincial scheduling tendencies driven by the region structure. Although outside the scope of this post, the study of this graph would fall under social network analysis, where we could explore many measures of overall network density and the value of each node and connection, Additionally, social network analysis would allow is to identify various "cliques", "clans", "clubs", "clusters", "components", etc., depending on the specific density of connections between different groups of nodes.
Credit: Loren Maxwell
Credit: Loren Maxwell
So back to challenging the inclusion of each team's classification, the question becomes does that information augment this sparse network and help us identify which teams are deserving of a playoff invitation? The answer is yes, especially for the more isolated regions.
David Mease, an associate professor at San Jose State University, considered a similar problem for Division I college football when using the B-T model without margin of victory. Mease found that without using the margin of victory, the B-T Update Function (we explored in Part VI) pushed all undefeated team ratings upward to infinity. The end result is that an undefeated team from the Southeastern Conference and an undefeated team from the Sunbelt Conference appeared equal in the ratings, both at infinity. The Extended Standings resolves this particular problem through using the margin of victory to apportion a win between each team (our Rothman Grading as outlined in Part III and Part IV). The end result is that no team actually appears undefeated and so no team ratings are pushed upward to infinity.
However Mease elected to preserve the binary win-loss outcome and instead assigned every team a "virtual win" and a "virtual loss" against a "virtual team". Mease called his approach "penalized" since he considered each team to be penalized by the virtual loss to the virtual team so that no team is undefeated.
For the Extended Standing we'll use a variation of his idea by assigning each of our GHSA teams a virtual tie against a virtual team that represents its classification. So instead of one virtual team, we'll have one for each classification and instead of assigning one win and one loss, we'll assign a tie.
And since we're using the virtual team in a different context than Mease, we'll refer to each of these virtual teams as "anchors" instead of a penalizer.
So let's break this down.
In 2015 there are six classification, so we have six anchors, one for each classification. Each team, in addition to its actual games, is also considered to have played a single tie game against the classification anchor. So Colquitt County, Grayson, Roswell, Archer, etc., are considered to have played a tie with the Class AAAAAA anchor while Marion County, Prince Avenue Christian, Mount Paran Christian, Eagle's Landing Christian, etc. are considered to have played a tie with the Class A anchor.
As the ratings for all the teams in a specific classification are updated by considering their actual games, the virtual ties they played against the classification anchor updates the classification anchor's rating as well. In turn, the classification anchor pulls the ratings for teams in that classification up or down to stay relative to the other teams in the same classification, aiding us in rating those teams in relatively or completely isolated regions.
So let's look at everything together by exploring the spreadsheet that I personally use to calculate the Extended Standings. I've included the 2014 Extended Standings spreadsheet as a zipped file and will make the 2015 spreadsheet available each week in my Extended Standings articles.
In the spreadsheet, you'll notice four tabs, "Team Ratings", "Anchor Ratings, "Games", and "Anchors".
The main tab we'll work from is the Team Ratings. Along with some other information at the top, the Team Ratings tab has two buttons, "Reset" and "Calculate", each of which triggers a macro. Note that you may have to enable macros for the spreadsheet to work.
Pressing the Reset button sets all of our team and anchor ratings to 1 and zeros out each team's xWins and xLosses. Hitting the Calculate button updates the ratings by copying from the Updated column (column P) to the Rating column (column Q) and then normalizes the ratings by copying from the Normalized column (column R) to the Rating column (column Q). Additionally the spreadsheet sorts the teams by ratings as it updates. This process repeats until the difference in the Sum of the Negative Log-Likelihood from one iteration to another (shown in cell E5) is less than 0.00000001. The number of iterations is tracked in cell E1.
After the solution converges, the spreadsheet then calculates the xWins and xLosses for each team.
Well, that's it for today's post, so feel free to test the spreadsheet, follow the formulas through, and investigate the VBA modules. Change scores or delete or add games to run what if scenarios or plug in other leagues to see how the Extended Standings would work for them as well.
Next time I'll look a little closer at the 2014 season by exploring how we can most effectively seed teams for a playoff using the Extended Standings.
Until then, I look forward to your comments!
