KRACH-Modified Pairwise Comparisons and KPWR for D1 College Hockey (2001-2002)

© 1999-2002, Joe Schlobotnik (archives)

URL for this frameset: http://www.slack.net/~whelan/tbrw/tbrw.cgi?2002/kpwr.shtml

Game results taken from US College Hockey Online's Division I composite schedule

See also

• A comparison of college hockey ranking systems
• Standard Pairwise Comparisons and PWR
• The NCAA Selection Procedure
• You Are The Committee, an interactive run through the selection process (also available in a versatile self-service hypothetical interface)
• A Proposed Modification to the NCAA Selection Criteria

Current KRACH-Modified Pairwise Comparisons (including games of 2002 March 17)

Explanation of the Table

The table above lists all of the Teams Under Consideration (TUCs) for the NCAA tournament. This includes all tournament-eligible Division 1 teams with overall winning percentages of .500 or above, plus any additional teams receiving automatic bids to the NCAAs by virtue of winning their conference tournaments. (For a listing of tournament-eligible teams and conferences receiving automatic bids, see the NCAA selection procedure page.) Each team has been compared to each other team on the basis of proposed modified versions of the NCAA selection criteria.

The standard criteria are described on our PWR page; the modifications, which are also described in a proposal written in the summer of 2000, aim to improved the assessment of strength of schedule while maintaining as much of the spirit of the standard criteria as possible.

The modified criteria are:

KRACH or RRWP

Our proposed replacement for the Ratings Percentage Index (RPI; q.v.). As described in detail on our KRACH page, the KRACH rating accomplishes what the RPI is supposed to: representing a team's strength on the basis of its won-lost-tied record and the strengths of its opponents. The comparison table lists, rather than the KRACH rating itself, the Round-Robin Winning Percentage to which it corresponds.

TUC

Performance vs other Teams Under Consideration. Head-to-head games are explicitly excluded from this criterion. This is evaluated on the basis of a criterion KRACH defined below.

L16

Performance in the Last 16 games. Note that head-to-head games are not excluded this criterion, which is evaluated on the basis of a criterion KRACH defined below.

COp

Performace vs Common Opponents. This is evaluated on the basis of a criterion KRACH defined below.

H2H

Head-to-head results. Each win in head-to-head competition carries the same weight as each of the other criteria. (This criterion is unmodified.)

The TUC, L16 and COp criteria, which in the standard selection criteria are judged on the basis of winning percentage without any adjustment for strength of schedule, are handled in the modified system with a Criterion KRACH. In each case, when comparing two teams, we look at the set of games relevant to criterion for each team, and the winning percentage they amassed in those games. We then look for the KRACH rating which each team would have needed to get an expected winning percentage equal to this actual winning percentage, holding all of the other teams' KRACH ratings constant. This is the Criterion KRACH If two teams have played the same opponents the same number of times in games counting towards a criterion, the team with the higher winning percentage will have the higher Criterion KRACH.

Aside from the changes described above to improve and expand inclusion of strength of schedule considerations, the KRACH-modified PairWise Comarison system is identical to the existing one. In particular:

A team gets one point towards the comparison for each of the first four criteria it wins, plus one point for each head-to-head victory. Whichever team has more points according to this method wins the criterion. In case of a tie, the team with the higher RRWP wins the criterion.

In each team's row, in the "Comparisons Won" part of the grid, are listed the abbreviations of all the teams with which they win comparisons. Each of these cells is a link to a mini-table (which will appear in a pop-up window under most browser setups) detailing the results of the five criteria. The KRACH row of the mini-table contains the overall record and KRACH for each team, the TUC, L16, and COp rows contain the record and Criterion KRACH in the games relevant to each criterion, and the H2H row contains the head-to-head record of each team against the other.

The KPWR column in the main table gives the total number of comparisons won by each team. The teams are ordered according the their KPWR; if two or more teams are tied in the KPWR, the tie is broken if possible according to the number of comparisons each wins against the other tied teams; if this fails to resolve the tie (which can be thought of a ro-sham-bo situation: Rock crushes Scissors, Scissors cut Paper, Paper covers Rock), the RRWP is used to break the tie.

The Nitty-Gritty

The mathematical definition of the Criterion KRACH is as follows. When comparing team i to team j in a particular criterion, if W_ik is the number of times team i has beaten team k in games relevant to the criterion, M_ik is the number of times they've played, W_i=∑_kW_ik is the total number of wins for team i in games contributing to the criterion and M_i=∑_jM_ij is the total number of games they've played that count towards the criterion, and similarly for team j, and K_k is the ordinary KRACH of any team k, then the Criterion KRACHes X_i and X_j are defined by

W_i = ∑_k≠j M_ik*X_i/(X_i+K_k) + M_ij*X_i/(X_i+X_j)
W_j = ∑_k≠j M_jk*X_j/(X_j+K_k) + M_ji*X_j/(X_j+X_i)

The use of X_j rather than K_j in the definition of X_i is a subtle correction which can only come into play when comparing two teams who have played each other in the last 16 games (since that is the only criterion from which head-to-head games are not excluded). Basically, it's designed to ensure that if two teams play identical schedules aside from head-to-head games, the team with the higher overall KRACH is not penalized for playing a "weaker" schedule including the team with the lower KRACH. (This is almost never going to make a difference; the KPWR was calculated for the past few seasons with and without this correction, and no comparisons were changed from one case to the other.)

The definition of Criterion KRACH can be written in the "winning ratio times strength of schedule" format as

X_i = [W_i/(M_i-W_i)] * [∑_k≠jf_ik*K_k + f_ij*X_j]
X_j = [W_j/(M_j-W_j)] * [∑_k≠if_jk*K_k + f_ji*X_i]

where the weighting factor is

f_ik = [M_ik/(X_i+K_k)] / [∑_l≠jM_il/(X_i+K_l) + M_ij/(X_i+X_j)] for k ≠ j
f_ij = [M_ij/(X_i+X_j)] / [∑_l≠jM_il/(X_i+K_l) + M_ij/(X_i+X_j)]
f_jk = [M_jk/(X_j+K_k)] / [∑_l≠iM_jl/(X_j+K_l) + M_ji/(X_j+X_i)] for k ≠ j
f_jj = [M_ji/(X_j+X_i)] / [∑_l≠iM_jl/(X_j+K_l) + M_ji/(X_j+X_i)]

Note that once again the weighting factor is normalized to one, i.e., ∑_kf_ik=1=∑_jf_jk, so in each case a team's Criterion KRACH can be thought of as the product of the PF/PA ratio for the criterion times a weighted average of their opponents' KRACH ratings.

Breakdown of Criteria

The following table lists, for each Team Under Consideration, the three selection criteria (complete with Criterion KRACH rating) which are more or less the same in each comparison: KRACH/RRWP, record vs TUCs, and record in the last 16 games. Each team's name in the table is a link to a rundown of the games contributing to these three criteria.

Note a team's record and Criterion KRACH Rating in the "vs TUCs" column is that for games against all TUCs; since head-to-head games are left out of this criterion, the record and rating used in an actual comparison will be different if the two teams have played each other. Note also that the Criterion KRACH given for the last 16 games is that which would apply in comparisons with no head-to-head games in the last 16; the rating used in a given pairwise comparison will be slightly different if the teams being compared played each other in the last 16 games.