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

© 1999-2002, Joe Schlobotnik (archives)

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

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

See also

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

Bradley-Terry Modified Pairwise Comparisons
Rk Team KPWR RRWP Comparisons Won
1 Denver U  (W) 26 .9080 Mn SC NH CC Mi MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
2 Minnesota  (W) 25 .8873 SC NH CC Mi MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
3 SCloud  (W) 23 .8540     CC Mi MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
4 New Hampshire (H) 23 .8419   SC   Mi MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
5 CCollege  (W) 23 .8339     NH Mi MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
6 Michigan  (C) 21 .7795         MS BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
7 Mich State  (C) 20 .7945           BU Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
8 Boston Univ  (H) 19 .7785             Me Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
9 Maine  (H) 18 .7578               Ak NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
10 AK-Fairbanks  (C) 17 .7339                 NM ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
11 Northern Mich (C) 16 .7335                   ML OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
12 Mass-Lowell  (H) 15 .6952                     OS NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
13 Ohio State  (C) 14 .6694                       NO Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
14 NE-Omaha  (C) 13 .6764                         Cr WM NE BC WS RP Ck Ha Da Mh Qn SH HC
15 Cornell  (E) 12 .6712                           WM NE BC WS RP Ck Ha Da Mh Qn SH HC
16 Western Mich  (C) 11 .6647                             NE BC WS RP Ck Ha Da Mh Qn SH HC
17 NorthEastern  (H) 10 .6312                               BC WS RP Ck Ha Da Mh Qn SH HC
18 Boston Coll  (H) 9 .6440                                 WS RP Ck Ha Da Mh Qn SH HC
19 Wayne State  (A) 8 .5225                                   RP Ck Ha Da Mh Qn SH HC
20 RPI  (E) 7 .5211                                     Ck Ha Da Mh Qn SH HC
21 Clarkson  (E) 6 .5039                                       Ha Da Mh Qn SH HC
22 Harvard  (E) 5 .5032                                         Da Mh Qn SH HC
23 Dartmouth  (E) 4 .4569                                           Mh Qn SH HC
24 Mercyhurst  (M) 3 .2724                                             Qn SH HC
25 Quinnipiac  (M) 2 .1922                                               SH HC
26 Sacred Heart  (M) 1 .1682                                                 HC
27 Holy Cross  (M) 0 .1604                                                  

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 Wik is the number of times team i has beaten team k in games relevant to the criterion, Mik is the number of times they've played, Wi=∑kWik is the total number of wins for team i in games contributing to the criterion and Mi=∑jMij is the total number of games they've played that count towards the criterion, and similarly for team j, and Kk is the ordinary KRACH of any team k, then the Criterion KRACHes Xi and Xj are defined by

Wi = ∑k≠j Mik*Xi/(Xi+Kk) + Mij*Xi/(Xi+Xj)
Wj = ∑k≠j Mjk*Xj/(Xj+Kk) + Mji*Xj/(Xj+Xi)

The use of Xj rather than Kj in the definition of Xi 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

Xi = [Wi/(Mi-Wi)] * [∑k≠jfik*Kk + fij*Xj]
Xj = [Wj/(Mj-Wj)] * [∑k≠ifjk*Kk + fji*Xi]

where the weighting factor is

fik = [Mik/(Xi+Kk)] / [∑l≠jMil/(Xi+Kl) + Mij/(Xi+Xj)] for k ≠ j
fij = [Mij/(Xi+Xj)] / [∑l≠jMil/(Xi+Kl) + Mij/(Xi+Xj)]
fjk = [Mjk/(Xj+Kk)] / [∑l≠iMjl/(Xj+Kl) + Mji/(Xj+Xi)] for k ≠ j
fjj = [Mji/(Xj+Xi)] / [∑l≠iMjl/(Xj+Kl) + Mji/(Xj+Xi)]

Note that once again the weighting factor is normalized to one, i.e., ∑kfik=1=∑jfjk, 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.

Team Comps Won KRACH vs TUCs Last 16
Rk KPWR Rk W-L-T Rating RRWP Rk W-L-T Rating Rk W-L-T Rating
Denver U 1 26 1 32-7-1 1872 .9080 2 10-6 1476 4 10-5-1 1110
Minnesota 2 25 2 29-8-4 1462 .8873 1 7-5-2 1561 1 12-4 2159
St Cloud 3 23 3 29-10-2 1043 .8540 5 6-7-1 860.2 8 9-6-1 626.1
New Hampshire 4 23 4 29-6-3 933.3 .8419 4 15-5-3 898.7 3 13-2-1 1437
CO College 5 23 5 26-12-3 870.4 .8339 3 8-7 1224 2 11-4-1 1483
Michigan 6 21 7 26-10-5 565.6 .7795 6 13-6-4 702.5 5 13-3 862.9
Mich State 7 20 6 27-8-5 633.0 .7945 7 10-5-5 561.3 9 10-3-3 614.0
Boston Univ 8 19 8 25-9-3 561.8 .7785 9 13-9-2 444.3 7 11-4-1 679.7
Maine 9 18 9 23-10-7 484.5 .7578 8 13-8-6 476.8 6 10-3-3 836.8
AK-Fairbanks 10 17 10 22-12-3 411.4 .7339 12 9-8-2 404.7 10 10-4-2 607.7
Northern Mich 11 16 11 26-12-2 410.5 .7335 10 9-7-1 438.3 12 11-5 375.6
Mass-Lowell 12 15 14 22-13-3 320.1 .6952 14 10-12 302.0 19 6-8-2 173.8
Ohio State 13 14 18 20-16-4 272.5 .6694 11 11-10-2 420.2 14 6-8-2 270.9
NE-Omaha 14 13 16 21-16-4 284.5 .6764 16 6-11-3 245.9 15 8-7-1 233.8
Cornell 15 12 17 24-7-2 275.6 .6712 18 7-7 147.7 11 13-2-1 388.6
Western Mich 16 11 19 19-15-4 264.9 .6647 15 10-12-2 276.9 13 8-7-1 347.6
Northeastern 17 10 22 19-17-3 216.3 .6312 13 11-11-2 338.2 17 8-8 220.5
Boston Coll 18 9 20 18-18-2 233.6 .6440 17 6-14-1 174.9 20 6-10 145.6
Wayne State 19 8 29 17-11-4 114.2 .5225 22 0-4 0 16 12-3-1 230.7
RPI 20 7 30 20-13-4 113.3 .5211 20 7-7-1 120.1 18 11-4-1 201.5
Clarkson 21 6 31 17-15-6 102.3 .5039 23 6-10-2 95.85 21 8-6-2 86.95
Harvard 22 5 32 15-14-4 101.9 .5032 19 5-8-3 122.3 23 7-8-1 74.85
Dartmouth 23 4 36 14-13-5 77.20 .4569 21 4-5-3 116.8 22 6-6-4 75.14
Mercyhurst 24 3 48 24-8-3 21.50 .2724 25 4-6-3 11.08 24 12-3-1 16.62
Quinnipiac 25 2 50 20-12-5 10.12 .1922 24 4-5-4 19.46 25 10-2-4 14.94
Sacred Heart 26 1 51 16-14-4 7.739 .1682 27 2-8-3 5.792 26 9-5-2 9.840
Holy Cross 27 0 52 15-12-5 7.053 .1604 26 2-5-3 8.629 27 8-5-3 8.670

Last Modified: 2012 March 25

Joe Schlobotnik / joe@amurgsval.org

HTML 4.0 compliant CSS2 compliant