KRACH Ratings for D1 College Hockey (2013-2014)

© 1999-2013, Joe Schlobotnik (archives)

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

Today's KRACH (including games of 2014 March 22)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Minnesota 1 493.1 .8057 2T 25-6-6 3.111 4 158.5
Union 2 483.0 .8027 1 28-6-4 3.750 28 128.8
Boston Coll 3 481.0 .8020 2T 26-7-4 3.111 6 154.6
Mass-Lowell 4 310.7 .7302 6T 25-10-4 2.250 13 138.1
Ferris State 5 273.9 .7071 4 28-10-3 2.565 47 106.8
Wisconsin 6 272.0 .7059 5 24-10-2 2.273 35 119.7
Quinnipiac 7 271.1 .7053 6T 24-9-6 2.250 33 120.5
St Cloud 8 261.3 .6983 9 21-10-5 1.880 11 139.0
Notre Dame 9 255.1 .6938 12 23-14-2 1.600 3 159.4
Providence 10 237.9 .6804 10 21-10-6 1.846 27 128.9
MSU-Mankato 11 219.1 .6642 8 26-13-1 1.963 44 111.6
Colgate 12 217.8 .6630 15 20-13-5 1.452 8 150.1
Vermont 13 212.4 .6580 18 20-14-3 1.387 7 153.1
North Dakota 14 210.7 .6564 11 23-13-3 1.690 29 124.7
Cornell 15 206.9 .6528 13 17-10-5 1.560 21 132.6
New Hampshire 16 192.6 .6384 25 22-18-1 1.216 5 158.4
Michigan 17 181.8 .6265 19 18-13-4 1.333 14 136.3
Northeastern 18 170.8 .6137 20T 19-14-4 1.312 25 130.1
Yale 19 167.1 .6091 16 17-11-5 1.444 40 115.7
Minn-Duluth 20 161.8 .6023 33 16-16-4 1.000 2 161.8
Western Mich 21 152.5 .5900 29 19-16-5 1.162 23 131.2
Ohio State 22 146.5 .5816 24 18-14-5 1.242 37 117.9
Denver U 23 144.7 .5788 22 20-15-6 1.278 43 113.2
Maine 24 137.5 .5681 32 16-15-4 1.059 26 129.9
Bowling Green 25 135.3 .5646 28 18-15-6 1.167 39 116.0
Clarkson 26 131.0 .5578 26 21-17-4 1.211 45 108.3
NE-Omaha 27 127.4 .5519 34T 17-18-2 .9474 17 134.5
AK-Fairbanks 28 126.3 .5499 27 18-15-4 1.176 46 107.3
AK-Anchorage 29 126.2 .5499 30 18-16-4 1.111 42 113.6
RPI 30 117.1 .5338 34T 15-16-6 .9474 30 123.6
Miami 31 114.5 .5290 41 15-20-3 .7674 9 149.2
St Lawrence 32 109.7 .5198 38 15-19-4 .8095 15 135.5
Lake Superior 33 102.0 .5042 36 16-19-1 .8462 34 120.5
Michigan Tech 34 92.68 .4838 39T 14-19-7 .7778 36 119.2
Brown 35 90.55 .4788 43 11-17-3 .6757 19 134.0
Northern Mich 36 85.73 .4672 42 15-21-2 .7273 38 117.9
Harvard 37 83.16 .4607 47 10-17-4 .6316 22 131.7
Mich State 38 77.44 .4457 44 11-18-7 .6744 41 114.8
Dartmouth 39 73.31 .4342 49 10-20-4 .5455 18 134.4
Boston Univ 40 72.04 .4306 50 10-21-4 .5217 12 138.1
Bemidji State 41 66.56 .4142 48 10-21-7 .5510 32 120.8
Mercyhurst 42 63.64 .4050 14 21-13-7 1.485 48 42.86
Mass-Amherst 43 60.58 .3950 52 8-22-4 .4167 10 145.4
CO College 44 60.06 .3932 55 7-24-6 .3704 1 162.2
Air Force 45 55.52 .3775 17 21-14-4 1.438 49 38.63
Merrimack 46 54.64 .3743 54 8-22-3 .4043 16 135.2
Bentley 47 44.66 .3352 20T 19-14-4 1.312 55 34.02
Connecticut 48 42.63 .3264 23 18-14-4 1.250 54 34.10
Penn State 49 41.15 .3199 56 8-26-2 .3333 31 123.4
Robert Morris 50 36.06 .2959 31 19-17-5 1.103 56 32.71
Canisius 51 31.46 .2722 37 17-21-3 .8222 50 38.26
Princeton 52 30.67 .2680 57 6-26 .2308 20 132.9
Niagara 53 29.46 .2612 39T 15-20-5 .7778 51 37.87
Holy Cross 54 23.00 .2222 45 14-22-3 .6596 53 34.87
RIT 55 22.99 .2222 46 12-20-5 .6444 52 35.68
Sacred Heart 56 16.21 .1738 51 12-24 .5000 58 32.42
American Intl 57 13.43 .1510 53 10-25-1 .4118 57 32.62
AL-Huntsville 58 9.180 .1116 59 2-35-1 .0704 24 130.4
Army 59 6.694 .0851 58 6-28 .2143 59 31.24

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: 2020 February 1

Joe Schlobotnik / joe@amurgsval.org

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