Methodology

In the interest of transparency, I have attempted to describe the RAMS process in detail here. I use an Excel spreadsheet to perform all steps listed below. Viewers not interested in the nuts and bolts of the process can find a simplified explanation of the rankings in the About tab.

Point Rating (PR)

A team’s Point Rating is simply the average Point Rating of its opponents, plus or minus its Average Scoring Margin.

STEPS:

  1. Create a Team Equation for each team. If Team A were to have played teams B, C, and D, and outscored them by a total of 25 points, the Team A equation would be

        \[{\mathsf{(A~-~B)~+~(A~-~C)~+~(A~-~D)~=~25}}\]

    or in simplified terms;

        \[{\mathsf{3A~-~1B~-~1C~-~1D~=~25}}\]

  2. If all teams are connected directly or indirectly through games played, eliminate one Team Equation and replace it with an equation that sets the sum of all member teams to zero. It doesn’t matter which Team Equation you replace, since using all of them is redundant.
  3. Conversely, if there are groups of teams that have not played anyone outside their group, eliminate one Team Equation from each group and replace it with an equation that sets the sum of the PR values for the teams in the group to a value equal to the sum of the PR values for those teams in the prior season.
  4. Solve for all variables, using matrix algebra.
  5. If every game was won by the team with the higher PR, a perfect correlation exists between the Point Rating and winning. In that case, the Point Rating is considered the final ranking and there is no need to calculate Winning Propensity.
Winning Propensity (WP)

Winning Propensity is a numerical value that represents a team’s likelihood of defeating an opponent. A team with a WP of 300 would be considered a 3:1 favorite over a team with a WP of 100. A team’s WP is based on its record and the WP values of its opponents, with the following twist – each team is credited with an artificial tie against a hypothetical Peer whose WP corresponds to that team’s Point Rating.

TERMS AND FORMULAS:

Probability of Winning (POW) – the probability of a team winning against a specific opponent. A team with a WP of 300 would have a 75% Probability of Winning against a team with a WP of 100.

    \[{\mathsf{POW~=~\frac{WP}{WP~+~WP_{Opp}}}}\]

Correlation Factor (CF) – a numeric value that represents the correlation between Point Rating and Winning Propensity. RAMS presumes that a one-point different in PR corresponds to a certain percent difference in WP. Adding 1 to that percent gives you the Correlation Factor. A CF of 1.20 would indicate that a one-point difference in PR equates to a 20% difference in WP. RAMS recalculates the Correlation Factor every rating cycle as part of the process in order to determine WP values for hypothetical Peers.

Estimated Winning Propensity (EWP) – the estimated winning propensity of a team based on its Point Rating and the Correlation Factor. A team’s EWP is used as the Winning Propensity for its hypothetical Peer.

    \[{\mathsf{EWP~=~CF^{PR}~x~100}\]

If the CF is 1.20, a team with a PR of 1 would have an EWP of 120 and a team with a PR of 2 would have an EWP of 144. A team with a PR of zero would always have an EWP of 100, regardless of the Correlation Factor. A negative PR always generates an EWP between zero and 100.

Upset – a game in which the winning team has a equal or lower PR than the losing team.

Favorite – the team with the higher Point Rating.

Underdog – the team with the lower Point Rating.

League Game – a game in which both teams are members of the organization being ranked.

Probability of Upset (POU) – the probability that an Underdog defeats a Favorite.

    \[{\mathsf{POU~=~\frac{EWP_{Und}}{EWP_{Und}~+~EWP_{Fav}}}}\]

STEPS:

  1. Determine the Estimated Winning Propensity for each team
    • Assign a tentative initial value to the Correlation Factor as a starting point (use previous CF if available).
    • Compute each team’s EWP based on the CF and the team’s PR.
    • Compute the Probability of Upset (POU) for every league game, based on EWPs.
    • Compare the sum of POUs to the total number of Upsets in league games over the course of the season.
    • Increase the CF if the sum of POUs exceeds the number of Upsets; reduce the CF if the number of Upsets exceeds the sum of POUs.
    • Repeat the previous four steps, computing EWPs and adjusting the CF until the sum of POUs equals the total number of Upsets. Once these totals are equal, we’ve established the Estimated Winning Propensity for each team and can proceed with the next step.
  2. Credit each team with an artificial tie (½ win and ½ loss) against its Peer. Each team’s Peer is assigned a Winning Propensity equal to that team’s Estimated Winning Propensity calculated in the previous step. The artificial tie serves two purposes. One, it adjusts a team’s Winning Propensity up or down based on its Point Rating. And two, it eliminates perfect records, which the Winning Propensity model cannot handle.
  3. Determine the Winning Propensity for each team.
    • Initially set each team’s Winning Propensity to 100 as a starting point.
    • Using WP values, compute the Probability of Winning (POW) for each game (including ties vs. Peers).
    • For each team, compare the sum of its POWs to its number of Wins (including ½ win vs. Peer).
    • Reduce a team’s WP if the sum of its POWs exceeds its number of Wins; increase a team’s WP if its number of Wins exceeds the sum of its POWs.
    • Repeat the previous three steps, computing POWs and adjusting WPs until the sum of each team’s POWs equals its total number of Wins. Once these totals are equal, we’ve established the Winning Propensity for each team.

As you may have figured out, the trial and error approach in calculating WP would be a monumental task if done by hand. Fortunately Excel can handle iterative calculations. I have Excel simultaneously adjust each team’s WP by 80% of the difference between its number of wins and the sum of its POWs, and then use the new WP value to recalculate its POW. After a couple of minutes and a thousand or so iterations every team has a Winning Propensity consistent with its record.

Playoff Consistent Ranking

The PC Ranking mirrors the WP ranking initially, but makes adjustments so that the winner in every playoff game is ranked ahead of the team it defeated. In the case of college sports, it also favors teams that qualify for the playoffs over those that don’t, where appropriate. There is no qualification process for Indiana High School sports, since all teams participate in the tournament.

Qualification Process (College Only)

TERMS:

Tournament – a term referring the preeminent post-season tournament for an association, subdivision, or class. Unless specifically stated, it does not pertain to conference tournaments or secondary tournaments such as the NIT.

Playoffs – RAMS uses the terms Playoffs and Tournament interchangeably

Conference Representative – in College Basketball, a team that automatically qualified for the tournament by winning its conference. In College Football, a team that qualified for the playoffs by being one of the top-ranked conference champions.

Tier 1 – a designation that includes all At-Large qualifiers, plus any Conference Representatives that 1) are seeded above at least one At-Large Team, 2) have a higher Winning Propensity than at least one At-Large Team, or 3) defeated a Tier 1 team in the playoffs. Tier 1 teams will finish ahead of all other teams in the final PC rankings.

Tier 2 – a designation that includes any team that did not meet Tier 1 standards, but won a tournament game against a Tier 2 or Tier 3 opponent. Tier 2 is typically comprised of play-in game winners. Tier 2 teams may be interspersed with Tier 3 teams and non-qualifiers in the final PC rankings.

Tier 3 – a designation for all Playoff teams that did not meet the standards for Tier 1 or Tier 2. Tier 3 teams may be interspersed with Tier 2 teams and non-qualifiers in the final PC rankings.

STEPS:

  1. Adjust the rankings of Tier 1 teams just enough to place them above non-qualifiers. If the top-ranked team to miss the playoffs is ranked 40th in the WP rankings, all Tier 1 teams ranked below 40th are given a tentative ranking of 39.999. Ties and decimal numbers will be resolved in a later step when renumbering takes place.
  2. Adjust the rankings for conferences in which the Tier 3 Conference Representative is ranked behind non-qualifiers. The process is explained through the following scenarios.
    • Scenario 1 – Suppose #150 Princeton won the Ivy League conference tourney, but was ranked behind non-qualifiers Harvard (#100) and Yale (#140). We take the rankings of the conference champion (Princeton) and the top-ranked non-qualifier (Harvard) and find the average (125). We then tentatively assign Princeton a ranking slightly above the average (124.999) and give Harvard a ranking slightly below the average (125.001). Yale’s ranking remains at 140.
    • Scenario 2 – Princeton again won the conference tourney, and non-qualifier Harvard was again ranked #100, but Yale was ranked #120. If we were to use the same process we used in Scenario 1, Yale would wind up ranked above both Harvard and Princeton. To avoid such an impropriety, we average the conference champion’s ranking with the rankings of the top two non-qualifiers and get 123.333. We assign Princeton a tentative ranking of 123.332 (slightly above the average) and give Harvard and Yale a tentative ranking of 123.334 (slightly below the average). Other scenarios might require including an even larger number of non-qualifiers in the re-ranking in order to ensure that the conference champion winds up in the highest spot.
  3. Adjust the ranking of any Tier 2 team that defeated a higher ranked opponent in the playoffs. If the 80th ranked team defeated the 75th ranked team, the winning team would be given a tentative ranking of 74.999.
  4. Adjust the rankings of any conference in which the Tier 2 Conference Representative is ranked behind non-qualifiers. Follow the process described in step 2.
  5. Sort the teams by the revised rank and Winning Propensity and RENUMBER.
  6. Proceed to Upset Reconciliation.
Upset Reconciliation (College and High School)

TERMS:

Upset – a tournament game in which a lower ranked team defeats a higher ranked team.

Cinderella Team – a team that upset two or more opponents in the tournament.

Cinderella String – a group of upsets won by the same Cinderella Team.

Upset Chain – a group of linked tournament upsets that does not include any wins by a Cinderella team. An example would be a scenario in which A upsets B, B upsets C, C upsets D, etc.

Simple Upset – an upset that is neither a part of a Cinderella String or an Upset Chain.

Upset Network – a group of linked tournament upsets that include a combination of Cinderella Strings, Upset Chains, and Simple Upsets.

STEPS:

  1. Resolve Simple Upsets by raising the rank of the winning team and lowering the rank of the losing team until the winning team has a slightly higher ranking. If #20 Colorado defeated #10 Alabama, Colorado’s ranking would be raised to #14.999 and Alabama’s ranking would be lowered to #15.001. Decimal numbers and ties will be eliminated in a subsequent step during renumbering.
  2. Resolve Upset Chains by finding the average ranking of the teams involved, and giving those teams tentative rankings on either side of the average. If #41 Fordham upset #34 Evansville, and Evansville upset #30 Dayton, the average rank would be 35, and the new rankings would be Fordham #34.999; Evansville #35; and Dayton #35.001.
  3. Resolve Cinderella Strings. Suppose #9 Iowa upset both #1 Georgia and #6 Houston. First, adjust the rankings of the three teams until Iowa (the Cinderella team) has the same ranking as one of its competitors. In order to keep the average ranking constant, raise Iowa’s ranking by the same amount as what we lower Georgia and Houston combined. After this step, Georgia tentatively drops to #2, Houston drops to #7, and Iowa rises to #7. Next, raise Iowa’s rank and lower Georgia’s until Iowa is ranked slightly higher. The revised rankings would be Iowa #4.499; Georgia #4.501; and Houston #7.
  4. Resolve Upset Networks. Suppose #8 Michigan beat #6 LSU in a Simple Upset. Suppose also the #25 Pitt upset #21 Oregon, and Oregon upset #14 Navy in an Upset Chain. And finally, in a Cinderella String, #14 Navy upset both #4 Kentucky and #8 Michigan.
    • Find the average ranking of all six teams – 13 in this case.
    • Resolve the Simple Upset, giving tentative rankings of #6.999 for Michigan and #7.001 for LSU.
    • Resolve the Upset Chain, giving tentative rankings of Pitt #19.999; Oregon #20; and Navy #20.001.
    • Resolve the Cinderella String, giving tentative rankings of Navy #8.999; Kentucky #9.001; and Michigan #10.
    • Note that Michigan and Navy each have conflicting ratings. Resolve this issue be adding 3.001 to each of the Simple Upset rankings and subtracting 11.002 from each of the Upset Chain rankings. This results in tentative rankings of Pitt #8.997; Oregon #8.998; Navy #8.999; Kentucky #9.001; Michigan #10; and LSU #10.002.
    • Note that the average ranking has shifted from #13 to #9.333. In order to revert back to the original average of 13, add 3.667 to each ranking, giving Pitt #12.664; Oregon #12.665; Navy #12.666; Kentucky #12.668; Michigan #13.667; and LSU #13.669.
  5. One extra twist – any re-ranking could generate additional upsets. After each of the preceding steps, check for new upsets and recalculate if any exist. Consider Step 1 in which Alabama’s ranking was lowered from #10 to #15.001. If Alabama had defeated #12 Baylor in the tournament, that game would now be classified as an upset, and what had been a Simple Upset would be reclassified as an Upset Chain, with the revised ranking of Colorado #13.999; Alabama #14; and Baylor #14.001.
  6. After all upsets are resolved, sort teams by their latest ranking and Winning Propensity; then RENUMBER.