KRACH Ratings for D1 College Hockey (2004-2005)

© 1999-2004, Joe Schlobotnik (archives)

URL for this frameset: http://slack.net/~whelan/tbrw/tbrw.cgi?2005/krach.shtml

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

Today's KRACH (including games of 2005 March 19)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
CO College 1 1031 .8677 3 29-8-3 3.211 4 321.1
Denver U 2 975.7 .8619 5 28-9-2 2.900 3 336.5
Boston Coll 3 642.5 .8117 4 25-6-7 3.000 13 214.2
Minnesota 4 620.4 .8070 11 26-14-1 1.828 2 339.5
Michigan 5 619.7 .8069 2 30-7-3 3.706 18 167.2
Cornell 6 592.6 .8008 1 26-4-3 5.000 35 118.5
Wisconsin 7 486.9 .7730 13T 23-13-4 1.667 8 292.1
North Dakota 8 469.1 .7676 20 22-14-5 1.485 5 315.9
New Hampshire 9 428.9 .7541 8 25-10-5 2.200 15 195.0
Boston Univ 10 384.4 .7371 13T 23-13-4 1.667 11 230.7
Ohio State 11 347.5 .7209 6 27-10-4 2.417 26 143.8
Harvard 12 314.7 .7047 9 21-9-3 2.143 22 146.9
Maine 13 302.8 .6982 18 20-12-7 1.516 14 199.7
Northern Mich 14 266.3 .6765 12 22-11-7 1.759 21 151.4
Mass-Lowell 15 250.6 .6660 16 20-12-4 1.571 20 159.5
Minn-Duluth 16 238.5 .6574 30T 15-17-6 .9000 10 265.0
Colgate 17 222.5 .6452 7 25-10-3 2.304 37 96.57
Mich State 18 220.1 .6433 24 20-17-4 1.158 16 190.1
MSU-Mankato 19 218.4 .6419 37 13-19-6 .7273 6 300.3
AK-Anchorage 20 203.1 .6290 38T 12-19-6 .6818 7 297.9
Northeastern 21 185.4 .6126 32T 15-18-5 .8537 12 217.2
St Cloud 22 182.6 .6099 41 14-23-3 .6327 9 288.7
Vermont 23 177.2 .6045 21 21-14-4 1.438 32 123.3
AK-Fairbanks 24 172.3 .5994 28 17-16-4 1.056 19 163.3
NE-Omaha 25 152.9 .5776 23 19-16-4 1.167 31 131.1
Dartmouth 26 151.5 .5760 19 20-13-2 1.500 36 101.0
Michigan Tech 27 135.9 .5560 54 8-25-4 .3704 1 366.8
Bowling Green 28 133.7 .5531 29 16-16-4 1.000 30 133.7
Miami 29 124.0 .5393 32T 15-18-5 .8537 24 145.3
St Lawrence 30 110.8 .5186 30T 17-19-2 .9000 33 123.1
Bemidji State 31 106.4 .5112 10 23-12-1 1.880 44 56.57
Brown 32 104.9 .5087 25 16-14-3 1.129 38 92.90
Western Mich 33 93.23 .4873 38T 14-21-2 .6818 29 136.7
Ferris State 34 87.30 .4754 43 13-22-4 .6250 27 139.7
Mass-Amherst 35 84.29 .4691 47 13-23-2 .5833 25 144.5
Providence 36 83.30 .4670 44T 12-21-4 .6087 28 136.9
AL-Huntsville 37 73.31 .4442 13T 18-10-4 1.667 46 43.99
Lake Superior 38 71.92 .4408 49 9-22-7 .4902 23 146.7
Clarkson 39 52.19 .3854 46 13-23-3 .5918 40 88.18
Notre Dame 40 45.23 .3616 56 5-27-6 .2667 17 169.6
Union 41 44.04 .3572 44T 13-22-2 .6087 41 72.35
Merrimack 42 40.56 .3439 55 8-26-2 .3333 34 121.7
RPI 43 40.20 .3425 40 14-22-2 .6522 43 61.64
Niagara 44 39.22 .3386 35 15-19-2 .8000 45 49.02
Wayne State 45 36.75 .3283 34 14-17-4 .8421 47 43.63
Princeton 46 31.04 .3024 51 8-20-3 .4419 42 70.25
Air Force 47 21.04 .2470 36 14-19-3 .7561 49 27.82
Yale 48 20.82 .2456 58 5-25-2 .2308 39 90.24
Quinnipiac 49 17.37 .2220 17 21-13-3 1.552 55 11.19
Mercyhurst 50 16.77 .2176 22 18-15-4 1.176 50 14.25
Holy Cross 51 15.19 .2055 26 16-14-6 1.118 52 13.59
Canisius 52 13.98 .1957 27 16-15-4 1.059 54 13.21
Robert Morris 53 12.97 .1870 52 8-21-4 .4348 48 29.82
Sacred Heart 54 8.478 .1426 42 13-21-1 .6279 53 13.50
Connecticut 55 6.972 .1248 48 11-23-3 .5102 51 13.67
Bentley 56 5.220 .1012 50 8-20-6 .4783 56 10.91
Army 57 3.699 .0774 53 7-21-3 .3778 58 9.792
American Intl 58 2.368 .0527 57 4-23-4 .2400 57 9.867

Explanation of the Table

KRACH
Ken's Rating for American College Hockey is an application of the Bradley-Terry method to college hockey; a team's rating is meant to indicate its relative strength on a multiplicative scale, so that the ratio of two teams' ratings gives the expected odds of each of them winning a game between them. The ratings are chosen so that the expected winning percentage for each team based on its schedule is equal to its actual winning percentage. Equivalently, the KRACH rating can be found by multiplying a team's PF/PA (q.v.) by its Strength of Schedule (SOS; q.v.). The Round-Robin Winning Percentage (RRWP) is the winning percentage a team would be expected to accumulate if they played each other team an equal number of times.
Record
A multiplicative analogue to the winning percentage is Points For divided by Points Against (PF/PA). Here PF consists of two points for each win and one for each tie, while PA consists of two points for each loss and one for each tie.
Sched Strength
The effective measure of Strength Of Schedule (SOS) from a KRACH point of view is a weighted average of the KRACH ratings of its opponents, where the relative weighting factor is the number of games against each opponent divided by the sum of the original team's rating and the opponent's rating. (Each team's name in the table above is a link to a rundown of their opponents with their KRACH ratings, which determine each opponent's contribution to the strength of schedle.)

The Nitty-Gritty

To spell out the definition of the KRACH explicitly, if Vij is the number of times team i has beaten team j (with ties as always counting as half a win and half a loss), Nij=Vij+Vji is the number of times they've played, Vi=∑jVij is the total number of wins for team i, and Ni=∑jNij is the total number of games they've played, the team i's KRACH Ki is defined indirectly by

Vi = ∑j Nij*Ki/(Ki+Kj)

An equivalent definition, less fundamental but more useful for understanding KRACH as a combination of game results and strength of schedule, is

Ki = [Vi/(Ni-Vi)] * [∑jfij*Kj]

where the weighting factor is

fij = [Nij/(Ki+Kj)] / [∑kNik/(Ki+Kk)]

Note that fij is defined so that ∑jfij=1, which means that, for example, if all of a team's opponents have the same KRACH rating, their strength of schedule will equal that rating.

Finally, the definition of the KRACH given so far allows us to multiply everyone's rating by the same number without changing anything. This ambiguity is resolved by defining a rating of 100 to correspond to a RRWP of .500, i.e., a hypothetical team which would be expected to win exactly half their games if they played all 60 Division 1 schools the same number of times.

KRACH vs RPI

KRACH has been put forth as a replacement for the Ratings Percentage Index because it does what RPI in intended to do, namely judge a team's results taking into account the strength of their opposition. It does this without some of the shortcomings exhibited by RPI, such as a team's rating going down when they defeat a bad team, or a semi-isolated group of teams accumulating inflated winning percentages and showing up on other teams' schedules as stronger than they really are. The two properties which make KRACH a more robust rating system are recursion (the strength of schedule measure used in calculating a team's KRACH rating comes from the the KRACH ratings of that team's opponents) and multiplication (record and strength of schedule are multiplied rather than added).

Recursion

The strength-of-schedule contribution to RPI is made up of 2 parts opponents' winning percentage and 1 part opponents' opponents' winning percentage. This means what while a team's RPI is only 1 parts winning percentage and 3 parts strength of schedule, i.e., strength of schedule is not taken at face value when evaluating a team overall, it is taken more or less at face value when evaluating the strength of a team as an opponent. (You can see how big an impact this has by looking at the "RPIStr" column on our RPI page.) So in the case of the early days of the MAAC, RPI was judging the value of a MAAC team's wins against other MAAC teams on the basis of those teams' records, mostly against other MAAC teams. Information on how the conference as a whole stacked up, based on the few non-conference games, was swamped by the impact of games between MAAC teams. Recently, with the MAAC involved in more interconference games, the average winning percentage of MAAC teams has gone down and thus the strength of schedule of the top MAAC teams is bringing down their RPI substantially. However, when teams from other conferences play those top MAAC teams, the MAAC opponents look strong to RPI because of their high winning percentages. (In response to this problem, the NCAA has changed the relative weightings of the components of the RPI from 35% winning percentage/50% opponents' winning percentage/15% opponents' opponents' winning percentage back to the original 25%/50%/25% weighting. However, this intensifies RPI's other drawback of allowing the strength of an opponent to overwhelm the actual outcome of the game.)

KRACH, on the other hand, defines the strength of schedule using the KRACH ratings themselves. This recursive property allows games further down the chain of opponents' opponents' opponents etc to have some impact on the ratings. Games among the teams in a conference are very good for giving information about the relative strengths of those teams, but KRACH manages to use even a few non-conference games to set the relative strength of that group to the rest of the NCAA. And if a team from a weak conference is judged to have a low KRACH desipte amassing a good record against bad competition, they are considered a weak opponent for strength-of-schedule purposes, since the KRACH itself is used for that as well.

Multiplication

One might consider bringing the power of recursion to RPI by defining an "RRPI" which was made up of 25% of a team's winning percentage and 75% of the average RRPI of their opponents. (This sort of modification is how the RHEAL rankings are defined.) However, this would not change the fact that the rating is additive. So, for example, a team with a .500 winning percentage would have an RPI between .125 and .875, no matter what their strength of schedule was. Similarly, a team playing against an extremely weak or strong schedule only has .250 of leeway based on their actual results.

With KRACH, on the other hand, one is multiplying two numbers (PF/PA) and SOS which could be anywhere from zero to infinity, and so no matter how low your SOS rating is, you could in principle have a high KRACH by having a high enough ratio of wins to losses.

The Nittier-Grittier

How To Calculate the KRACH Ratings

This definition defines the KRACH indirectly, so it can be used to check that a given set of ratings is correct, but to actually calculate them, one needs to do something like rewrite the definition in the form

Ki = Vi / [∑jNij/(Ki+Kj)]

This still defines the KRACH ratings recursively, i.e., in terms of themselves, but this equation can be solved by a method known as iteration, where you put in any guess for the KRACH ratings on the right hand side, see what comes out on the left hand side, then put those numbers back in on the right hand side and try again. When you've gotten close to the correct set of ratings, the numbers coming out on the left-hand side will be indistinguishable from the numbers going in on the right-hand side.

The other (equivalent) definition is already written as a recursive expression for the KRACH ratings, and it can be iterated in the same way to get the same results.

How to Verify the KRACH Ratings

It should be pointed out that if someone hands you a set of KRACH ratings and you only want to check that they are correct, it's much easier. You just calculate the expected number of wins for each team according to

Vi = ∑j Nij*Ki/(Ki+Kj)

And check that you come up with the actual number of wins. (Once again, a tie counts as half a win and half a loss.)

Dealing With Perfection

As described in Ken Butler's explanation of the KRACH, the methods described so far break down if a team has won all of their games. This is because their actual winning percentage is 1.000, and it's only possible for that to be their expected winning percentage if their rating is infinitely compared to those of their opponents. Now, if it's only one team, we could just set their KRACH to infinity (or zero in the case of a team which has lost all of their games), but there are more complicated scenarios in which, for example, two teams have only lost to each other, and so their KRACH ratings need to be infinite compared to everybody else's and finite compared to each other. The good news is that this sort of situation almost never exists at the end of the season; the only case in recent memory was Fairfield's first Division I season, when they went 0-23 against tournament-eligible competition.

An older version of KRACH got around this by adding a "fictitious team" against which each team was assumed to have played and tied one game, which was enough to make everyone's KRACH finite. However, this had the disadvantage that it could still effect the ratings even when it was no longer needed to avoid infinities.

The current version of KRACH does not include this "fictitious team", but rather checks to see if any ratios of ratings will end up needing to be infinite to produce the correct expected winning percentages. The key turns out to be related to the old game of trying to prove that the last-place team is better than the first-place team because they beat someone who beat someone who beat someone who beat the champions. If you can take any two teams and make a chain of wins or ties from one to the other, then all of the KRACH ratings will be finite.

If that's not the case, you need to work out the relationships teams have to each other. If you can make a chain of wins and ties from team A to team B but not the other way around, team A's rating will need to be infinite compared to team B's, and for shorthand we say A>B (and B<A). If you can make a chain of wins and ties from team A to team B and also from team B to team A, the ratio of their ratings will be a unique finite number and we say A~B. If you can't make a chain of wins and ties connecting team A and team B in either direction, the ratio of their ratings could be anything you like and you'd still get a set of ratings which satisfied the definition of the KRACH, so we say A%B (since the ratio of their ratings can be thought of as the undetermined zero divided by zero). Because of the nature of these relationships, we can split all the teams into groups so that every team in a group has the ~ relationship with every other team in the group, but not with any team outside of its groups. Furthermore if we look at two different groups, each team in the first group will have the same relationship (>, <, or %) with each team in the second group. We can then define finite KRACH ratings based only on games played between members of the same group, and use those as usual to define the expected head-to-head winning percentages for teams within the same group. For teams in different groups, we don't use the KRACH ratings, but rather the relationships between teams. If A>B, then A has an expected winning percentage of 1.000 in games against B and B has an expected winning percentage of .000 in games against A. In the case where A%B there's no basis for comparison, so we arbitrarily assign an expected head-to-head winning percentage of .500 to each team.

In the case where everyone is in the same group (again, usually true by the middle of the season) we can define a single KRACH rating with no hassle. If they're not, we need the ratings plus the group structure to describe things fully. However, the Round-Robin Winning Percentage (RRWP) can still be defined in this case and used to rank the teams, which is another reason why it's a convenient figure to work with.

See also


Last Modified: 2012 March 26

Joe Schlobotnik / joe@amurgsval.org

HTML 4.0 compliant CSS2 compliant