ELynah Forum

Well the Northwestern-Cornell basketball game will be played in an arena close enough to my apartment that I could almost spit on it from my window, so it was going be a tough decision between doing that and going to watch the hockey game at Joe's.  But then a friend suggested that I could spit on the basketball arena earlier in the day, freeing me up to go watch hockey in the evening. Why choose, when I can do both?

I already live in Chicago, so I have no travel excuse for not going; but I already know I have to be home on Sunday morning for brunch. Am I crazy enough to drive back and forth to Green Bay twice in two days?

Here's hoping we get to find out.

Some of us are going to attempt to watch the games this weekend at Joe's on Weed here in Chicago.  They've got enough TV's and satellites that we should be able to get the games there.  Unlike some other cities, this isn't really organized in any significant way, so just come one down and find us by the TV that's not showing crapketball.

Thanks to everyone for your help; it appears they delivered the tickets to the doorway vestibule there without a signature required, so we seem to have averted any serious disasters arising from Elliot's boneheadedness.

See you all in Minnesota

There's a giant hole in my heart where my hockey tickets for Minnesota are supposed to be. The short version of the story is that I believe the tickets we ordered were sent to the address of my friend who is out-of-town until Saturday morning.

Can anyone who did receive overnighted tickets enlighten me as to what overnight service was used?  My fleeting hope is that perhaps I could get the courier to re-direct what will probably be the third and final delivery attempt tomorrow to my home instead of his, but I suspect I'd need to find them in the next few hours to even try. Unfortunately the cornell ticket office is closed today so I can't get any info from them.

Thanks!

[Q]Beeeej Wrote:
Hm... Beeeej->Stu->Ralph->Scrog->Newman isn't correct?  Who am I missing?  :-{)}

Beeeej[/q]

Beeeej->Stu->Ralph->Scrog->Yeast->Newman

[Q]Beeeej Wrote:
 He's my little brother's little brother's little brother's little brother, if I remember correctly...
Beeeej[/q]
I think you're short one, but you get the gist. And for those of you who don't know me: I was  a Tuba player back in the day, and now my schooling at Northwestern keeps me far from Lynah most of the time, but I've been to the Michigan games in recent years.

[Q]billhoward Wrote:

 Second try: KRACH can account for about one-fifth of the explanation of how any game turns out (for games among ~closely matched teams)? As opposed to (my bad interpolation) it can account for the outcome of one of every five games among closely matched teams - you're saying the second interpretation isn't the same as the first? [/q]

An approximate although not exact analogy can be explained thinking about a dice game: two sides each roll five dice, and the side with the highest total wins.

However, suppose instead of totally random dice, the each die represents a factor: inherent skill of the team, officiating calls in your favor, luck, etc. Some of the dice are rolled fresh every game, such as luck or officiating, but some are based on skill, coaching, or other persistent factors and are not actually rolled for each game, but predetermined beforehand. However, no one actually can see the individual dice when you play; the totals are counted by a referee, and a winner (or tie) is determined without anyone knowing the exact counts.  Now, two sides match up, and they only roll four dice each, and get points from the fifth (skill) die at the assigned levels, with the winner having the highest total from all five dice. With KRACH ratings, we can sort of reverse engineer the results and estimate the inherent skill die for each side after a reasonable number of games, since it is theoretically the same every time. (This is why KRACH is unavailable early in the season.) We still know nothing about the other dice in each game.

When two sides have very similar (identical) KRACH ratings, then their assigned dice values are the same, and the game is a toss-up - we cannot predict the outcome any better than if we didn't have KRACH ratings. When two sides have vastly different ratings, such that one side has a six and the other side has a one, then we know that the side with the six will win more often. When we say that KRACH explains 19% (about 20%, or 1/5) of the variance, it's similar to saying KRACH tells us the outcome of one of five dice.

When the KRACH table is calculated, the values in it are a mathematical best guess at what the number is: equivalent to saying "the skill die for Cornell is most likely a five" but we cannot tell with great precision. What to my knowledge was never done before yesterday was any analysis of how good a guess that table was. What the new table I made adds is upper and lower bounds for that guess, the rough equivalent of saying "the skill die for Cornell is most likely a five, but it could be as high as six or as low as four." Since we can never see the individual components, we can't really tell with great precision what the KRACH values are.

I hope this helps. And I'd also point out that Newman being my real name and an affinity for Seinfeld characters are not mutually exclusive events.

[Q]billhoward Wrote:
But the margin of error is always going to be there and it's as least as great as the sum of all posts and crossbars hit plus Zambonis making double passes across the ice and leaving it wet, and players coming out of the penalty box just as the puck is passed to center ice. [/q]

Indeed; I neglected to mention in my post (although it's in the excel file if you downloaded it) that the process for finding the variance also tells you, on average, what percentage of the variance in game outcomes is explained by KRACH. In other words, how much of a factor the different KRACH ratings are in determining who wins. This year it's 19% across the whole season so far, and a lot of that is from the expected blowouts. So in games between closer matched teams, the outcome is well over 80% based on the other factors (Zambonis and crossbars and baaaa-ad calls, oh my!)

[Q]elliotb Wrote:

 [Q2]Newman Wrote:

> Teams 1-15 in KRACH have a statistical claim on being #1 at a 95% confidence level (which, ignoring the fact the KRACH has little to do with tournament selection, makes having a 16 team tournament rather an auspicious size).[/Q]
Are you making that claim based on the fact that the confidence intervals for the top 15 teams all cover CC's actual KRACH rating? Or is it based on the analysis in your spreadsheet where everything was done relative to CC? (Perhaps those are equivalent, although I suspect not.)

- Elliot[/q]

I'm making that claim based on CI's covering CC's rating when CC is the reference team. Comparing two teams ratings when one team isn't the reference school is more complex, and requires incorporating the covariance.

Also, really this is just showing the chance of a team having a KRACH higher than Colorado College. To be #1 they'd have to have a rating higher than all teams, which is a much more complex calculation I'm not going to try to do.

[Q]KeithK Wrote:

 [Q2]So, setting the number of teams participating in the NCAA tournament at 16 is good practice, since it's close to the number of top teams whose KRACH ratings are not significantly different.[/Q]
I wouldn't be too quick to draw that conclusion.  This year's results may show a match between number of tournament bids and statistically comparable #1 teams according to KRACH.  But it remains to be seen whether this pattern holds for earlier years.  [/q]

What I meant was that it coincidentally happened to be a close match this year, not that 16 is a good number all around every year.  Also, no teams from AH or CHA are in that group, so at least one of those 15 teams will be shut out of the NCAAs.

The english version:

KRACH is not a holy grail of rankings. It may be slightly better than pairwise, but a season of about 30-35 games per team doesn't conclusively prove who's the best, or what the overall rankings should be. So when there are apparent "major" aberrations, such as Dartmouth ranked 9th in Pairwise but 24th in KRACH, these differences are not really that huge, since the differences between adjacent teams in the rankings are miniscule. In fact, any objective ranking criteria that would get the teams in an order that is anything close to reasonable would probably be mathematically unrejectable.

Put another way, the existence of KRACH does not prove Pairwise as invalid. There is a season length, probably on the order of hundreds of games per team with plenty of inter-conference play, when KRACH would invalidate Pairwise, but the actual college hockey season doesn't meet that criteria.

This is, in part, why we have playoffs. Every college hockey team but one (the cellar-dweller of Hockey East) makes their conference playoffs, and thus has one last opportunity to win out and be national champions. Therefore, people should treat the playoffs as we did when we got Mankato in the first round two years ago; the system may hand you tough opponents or easy ones, but you've got to be able to beat any team on any day.

The mathematical version:
(If you don't know what the standard error of the estimate is, you won't lose anything by stopping now)
I've figured out the necessary transformations to get a covariance matrix for KRACH. Since KRACH is a purely ratios system, it isn't possible to get a variance/covariance matrix for the whole league at once. This is solved for the scores themselves by arbitrarily setting the average to 100, but it doesn't work so neatly for variance, so I set Cornell's variance to zero and let all other teams vary compared to us. I've posted an excel workbook at http://pubweb.northwestern.edu/~jpn714/KRACH_Analysis_022805.xls that has most of the data, including covariance tables and hessians.

To sum up the results of the analysis, all at a 95% confidence level:

> Teams 1-15 in KRACH have a statistical claim on being #1 at a 95% confidence level (which, ignoring the fact the KRACH has little to do with tournament selection, makes having a 16 team tournament rather an auspicious size).
> Cornell has a statistically significant better rating than all teams from #30 SLU downwards at a 95% confidence level
> The standard error of the estimates are relatively consistent across teams, although it's smaller for WCHA teams, probably due to longer schedules and more in-conference play. WCHA covariances with other teams are also notably smaller.
> Generally the 95% confidence interval will allow teams to move up or down roughly 15 ranks on the list, or a little more towards the middle of the pack, although this isn't constant across teams.
> It looks like the most important games for a team are the ones played against others close to them in rank, whether in conference or not.

 KRACH with 95% Confidence Intervals

Rank   Team         KRACH   Lower    Upper
1   ColoradoCollege     954.75   218.30   4175.70
2   Denver              794.61   188.06   3357.42
3   Minnesota           601.30   147.85   2445.53
4   Wisconsin           557.28   132.53   2343.28
5   BostonCollege       520.47   135.92   1992.94
6   Michigan            519.19   129.15   2087.12
7   Cornell         479.60   n/a   n/a
8   NewHampshire        417.49   109.83   1587.00
9   BostonUniversity    397.14   105.27   1498.20
10   NorthDakota         370.03   91.44   1497.41
11   OhioState           361.70   89.71   1458.27
12   MassLowell          315.65   82.24   1211.51
13   Maine               284.07   78.32   1030.37
14   Harvard             279.72   78.48   996.93
15   NorthernMichigan    276.75   71.24   1075.04
16   MinnesotaDuluth     227.32   57.31   901.66
17   Northeastern        206.09   55.53   764.77
18   AlaskaAnchorage     204.62   48.98   854.76
19   Colgate             204.19   58.17   716.74
20   MichiganState       203.56   55.36   748.57
21   MinnesotaState      201.81   49.60   821.04
22   Vermont             186.25   54.23   639.72
23   StCloudState        184.26   46.08   736.74
24   Dartmouth           183.34   51.41   653.83
25   NebraskaOmaha       164.22   42.21   638.91
26   BowlingGreen        159.41   41.18   617.09
27   MichiganTech        144.17   34.60   600.75
28   Miami               136.63   35.54   525.18
29   AlaskaFairbanks     126.27   31.57   505.11
30   StLawrence          114.36   33.29   392.78
31   Brown               107.66   29.86   388.17
32   BemidjiState        92.67   22.91   374.87
33   WesternMichigan     92.28   23.47   362.91
34   FerrisState         82.78   21.24   322.61
35   Massachusetts       81.78   20.80   321.50
36   LakeSuperior        81.40   21.30   311.13
37   AlabamaHuntsville   78.64   18.85   328.06
38   Providence          69.97   18.05   271.27
39   Clarkson            51.62   14.43   184.71
40   NotreDame           50.32   12.33   205.38
41   Union               48.10   13.20   175.24
42   Rensselaer          44.57   12.38   160.40
43   Niagara             43.21   10.89   171.40
44   Merrimack           41.68   10.56   164.50
45   WayneState          38.77   9.96   150.96
46   Princeton           34.47   8.93   133.06
47   AirForce            21.23   4.99   90.36
48   Yale                18.40   4.23   80.05
49   Quinnipiac          16.78   3.77   74.81
50   HolyCross           15.78   3.66   68.10
51   Canisius            14.76   3.35   65.06
52   Mercyhurst          12.45   2.79   55.60
53   RobertMorris        12.34   2.83   53.76
54   SacredHeart         10.36   2.28   47.14
55   Connecticut         8.04   1.85   35.00
56   Bentley             4.94   1.06   22.93
57   Army                3.96   0.81   19.33
58   AmericanIntl        2.58   0.51   13.10

The covariance in KRACH is an interesting problem. I think it's pretty clear that goal scoring shouldn't be a factor; hockey is about win or lose, not racking up the goals - otherwise Cornell would be ranked pretty darn low. The Bradley-Terry underlying KRACH is a logit model, so it shouldn't be difficult to dig into this and find how to do it.  It might also be interesting to expand KRACH to a random parameters model, on the premise that teams can sometimes play well or poorly. I'm not sure there's enough information in a season of wins and losses to calculate good parameters, but it's certainly worth some investigation. (As if I need any more projects.)

As for the "[Pairwise] For Dummies" concept that started this thread, I'd offer this basically non-math way to think about the pairwise rankings: The difference between your team (not just Cornell, but basically every team) ranking as number 4, 14, or 24 in the pairwise is based on the quality and skill of your team. The difference between 4 and 5 (a la, "earning a one seed"), or between 14 and 15 (a la, "making the tournament") is based more on the vagaries of the records of other teams, especially teams with records near .500, even teams which your team did not play this year. This is a principle reason why people complain about this system.

In theory, if it was one set-in-stone set of values, it could be calculated after a finite and probably small number of years of observing the selections. But there's no reason to believe that it's a fixed set of numbers from year to year. Since it's secret, it can be changed on a whim.

[Q]RichH Wrote:

as did the whole "no shirts in the 3rd" thing.[/q]

The shirtless third period definitely dates at least to the 1998-99 season, when I was last in Ithaca with season tickets, and it was going on amongst a group of guys in the row behind me.