Better Than Expected

More fun with math.

So, it started with a simple note that the @umichfootball Twitter account put out yesterday morning that Michigan is 40-13-4 all-time against Iowa.  I was stunned that Iowa and Michigan had only played 57 times all-time despite being conference mates for over 100 seasons, but more stunned to realize that Iowa has only beaten Michigan thirteen total times.  Ever.  But then I wondered if that was in line with what you would expect, so I started on the project with this question in mind:

Taking into account only B1G conference games, which team has most outperformed its overall conference winning percentage against a single opponent?  Which team has most underperformed its overall conference winning percentage?

Using the always awesome Stassen database, I culled all of the records of B1G games between 1896-2011.  I included Nebraska's eight games in the conference winning percentage calculations, but due to the ridiculously small sample size, they have been excluded from this conversation.  I did, however, include the University of Chicago in both calculations, because it's interesting.  I did not include Michigan's 1907-1916 conference interregnum, no Ohio State games prior to 1912, no Michigan State games prior to 1953, and no Penn State games prior to 1993. I then calculated the team's all-time winning percentage, calculating ties as half a win and half a loss.

So, with all of that done, I then calculated each school's winning percentage against each opponent.  I compared that to their overall winning percentage and got the following result:

The B1G team that has most outperformed its overall conference winning percentage against a single opponent is...complicated because of variable sample sizes.
The top five results are as follows:

• Penn State vs. Indiana-36.18 percentage points better (Penn State has never lost to Indiana in B1G play.)
• Chicago vs. Indiana-28.15 percentage points better (Indiana was glad to see the Maroons depart.)
• Michigan State vs. Indiana-21.98 percentage points better (Oh, and yes, that's Michigan State's protected cross-division game.)
• Minnesota vs. Chicago-21.95 percentage points better (back when Minnesota and Chicago were good at football!)
• Northwestern vs. Indiana-21.07 percentage points better (remember, these two teams are #1 and #2 in all-time losses among FBS schools.)

So, over the longest term, it's Northwestern over Indiana.  But really, it's everybody over Indiana!  (Also, for all of Iowa's collective grousing about their issues against Northwestern, they are in the top ten of this list.)

The B1G team has most underperformed its overall conference winning percentage against a single opponent is Minnesota vs. Ohio State, 33.49 percentage points worse than their overall conference winning percentage.

Now, since this is a Michigan blog, what about the Wolverines?  Well, let's have a look, shall we?

Michigan's all-time B1G winning percentage is 71.33%, which is just a shade behind Ohio State's 71.66% for first place all-time in the conference (this does not take into account Ohio's vacated games in 2010, if you want to feel better about yourself, since Michigan moves back into first place when it does..)

When we look at the chart, what we see is that Michigan doesn't really have a high positive variance against anyone team, which is to say that they have handled their business against pretty much everyone historically:

So much opposing red.

What we also can see is that while Michigan has "struggled" against Penn State, Ohio State, and Michigan State, relative to its overall conference success, every B1G team has done worse against Michigan than their overall conference winning percentage.  In a fit of ironic pique, Northwestern actually has the lowest variance because they are so bad overall in conference that even performing poorly against Michigan doesn't hurt them that much because they don't  have as far to fall.  Sparty's relatively strong performance against Michigan puts them in second, and when Ohio State is the greatest variance game for Michigan, Ohio's variance is more negative than Michigan's when compared side by side.  Also, Indiana football is still terrible.

The ten most positive and most negative variance comparisons, by the way:

Penn State's still relatively small sample size helps them (and hurts them a little too.)

If you'd like to see all of the data (to play with it), please drop me a line, and I will pass it along.

CCHA Tournament Odds

Now that the brackets have all been set for the second round of the CCHA tournament, I started thinking about what Michigan's chances are of advancing to the Joe. Western stole a game off of us, so they aren't to be taken too lightly, but they aren't a good hockey team either.

One of the neat things about the KRACH ratings system (as previously discussed) is that you're supposed to be able to compare teams' KRACHs to give you an expected winning percentage. Using a little (fairly simple) probability theory¹, it's then possible to predict the chances each CCHA team has of winning its series and predicting the chances of a sweep or whether Game 3 is on the horizon.

Chart? Chart.

	Notre Dame vs. UNO	Michigan vs. Western	Miami vs. Northern	Alaska vs. Ohio State
Team 1 KRACH	540.3	391.3	238.9	137.0
Team 2 KRACH	115.2	89.9	136.9	216.9
T1 sweep	67.94%	66.13%	40.41%	14.99%
T2 sweep	3.09%	3.49%	13.27%	37.56%
P(Game 3)	28.97%	30.38%	46.32%	47.45%
T1 series win	91.82%	90.83%	69.86%	33.35%
T2 series win	8.18%	9.17%	30.14%	66.65%

Well then. Notre Dame and Michigan are both-near locks to advance. Miami is fairly solid, though KRACH doesn't take into account how hot Northern has been lately, so that's probably a bit high.

Likewise, it's really down on Alaska, and it's kind of unfair, though not without its reasons. Alaska had a brutal and short list of non-conference opponents, losing to anyone with half a pulse. They did get a tie against a very good Northeastern team, but that was a. the season opener and b. in Alaska. Non-conference games are disproportionately valuable in any college hockey ranking system, so Alaska's KRACH got hammered by losing to Maine and getting swept by Anchorage. This is reflected in the fact that OSU is a 2:1 favorite to advance.

Realistically, I'd put it at more like a 55% chance that the Buckeyes make it through. The first game in the series is often a very tough one for a team that's just arrived in Alaska. Couple that with a well-rested Chad Johnson in goal and I like the Nanooks' chances in Game 1. If they take that game, suddenly the numbers say they're a 5:3 favorite to advance.

Projecting a little further out, the question then turns to what are the chances of each of these teams of hoisting the Mason Cup? Well, we'll assume that each team will have to play the toughest possible schedule (by KRACH) to claim the championship. For example, Michigan has to beat Western, then Miami, then Notre Dame. Northern would have to beat Miami, then Notre Dame, then Michigan on its theoretical path. Let's go to the chart:

Probability of winning...
	This Series	Semi- final	Final
Notre Dame	91.82%	65.52%	38.00%
Michigan	90.83%	56.40%	23.69%
Miami	69.86%	36.61%	11.23%
Alaska	33.35%	12.16%	6.41%
Ohio State	66.65%	31.71%	9.08%
Northern	30.14%	6.09%	1.58%
Western	9.17%	1.31%	0.36%
UNO	8.18%	1.86%	0.61%

Way to go, Captain Obvious. Yeah, I know. It's shocking to learn that Northern, Western, and Nebraska-Omaha are ridiculous longshots to win it all, and that their best shot at the finals is to buy tickets at the group rate. It is interesting, however, to see that Miami's such a longshot, and that we're slightly better than a 50/50 bet to make the finals. We'll see how this all plays out on the ice.

1 – If you really want the math, leave a comment².
2 – If you really want the math, you almost certainly know how to do it already. And are a dork.

More Pairwise Math

At the moment, Saturday's loss to OSU looks to be harmless. Michigan is still tied for #2 in the Pairwise. Sure, Notre Dame replaced Vermont as the team we're tied with, and since they have the tiebreakers on us, that means we're on the outside looking in at the Grand Rapids regional. But they flipped their comparison with the Catamounts and there was nothing we could do about that. The real damage is in the Michigan vs. Miami Pairwise comparison, which is now like a Sword of Damocles hanging over the Wolverines.

This is what the Michigan/Miami comparison looks like today:

					Michigan vs. Miami
			0.5706	1	RPI	0	0.5477
8	5	0	0.6154	0	Win % vs. TUC	0	0.6250	4	2	2
17	6	0	0.7391	0	Win % vs. COp	1	0.7750	14	3	3
2	2	0		2	Head-to-Head	2		2	2	0
				3	Total	3

Look at the "Win % vs. TUC" category again. Miami has a better record, but since they haven't reached the 10-win threshold (when the head-to-head series is taken out) it isn't counted. This weekend, they play OSU. A team under consideration. If they sweep, they win the category and the comparison, even if we sweep Ferris State. This is how the table will look then:

					Michigan vs. Miami
			0.5706	1	RPI	0	0.5477
8	5	0	0.6154	0	Win % vs. TUC	1	0.7000	6	2	2
19	6	0	0.7600	0	Win % vs. COp	1	0.7955	16	3	3
2	2	0		2	Head-to-Head	2		2	2	0
				3	Total	4

This would've been different had the Wolverines swept Ohio State. They still would've had a chance to control their own destiny. Assuming a Michigan sweep over Ferris and a Miami sweep over OSU, Michigan still would lose the TUC comparison (by .0077), but would've held onto the Common Opponents category by the barest of margins: 0.8000 to 0.7955. It would've been almost insignificant, but just enough, giving Michigan the 4-3 overall edge.

What has to happen now for Michigan to maintain the edge over Miami is 1.) Sweep Ferris and 2.) Ohio State has to take one game over Miami in regulation/standard OT. If that comes to pass, Michigan takes a very strong 5-2 lead over Miami in the eyes of the Pairwise.

All of this can still be rendered academic by the CCHA conference tournament. Even if Miami sweeps OSU and hangs onto the TUC and COp categories, they'll be hard-pressed to catch us in RPI. If Michigan holds an RPI advantage and defeats Miami in the semi-finals, that's another win added to the head-to-head comparison, which would put Michigan back into a tie, which would be broken by RPI in Michigan's favor.

In conclusion: This sucks, but the suckage can be rendered moot by winning. Go Blue.

The Mathematics of Beating OSU

Another week, another titanic shift in the Pairwise rankings. As far as we're concerned, the big news is Michigan jumping up to #2 after Notre Dame's split with Ohio State. The Wolverines are tied with Northeastern for that spot, but own the comparison with the Huskies, and thus the tiebreaker. It really is a knife's edge, though. Take a look at the full chart at CHN: Michigan, Northeaster, Vermont, and Notre Dame all have 21 or 22 comparison wins over the field. Lose a game, and suddenly you're the #5 overall seed. And sometimes you don't even have to lose to have a comparison flip on you. Remember the TUC cliff?

As a refresher, the Pairwise Rankings consider four factors when comparing two teams: head-to-head record, RPI, record against common opponents, record against teams under consideration (TUCs), but you have to have played 10 TUCs for that comparison to be used.

For instance, let's consider Michigan and Miami. Head-to-head record is 2-2, so two points to each team for that. Michigan's RPI is better, so they get a point. Miami's record against common opponents is better: point, Redhawks. Miami has only played 8 TUCs, so that comparison is not used. Michigan and Miami are tied 3-3 here, and we use RPI to break the tie. Michigan takes the comparison. HOWEVA. Miami has played 8 TUCs, which are defined as the top 25 teams by RPI. They play Ohio State twice to close out the season. As OSU is currently a TUC, that would allow them to meet the 10-game requirement. If both teams win out, Miami's record against common opponents is still better and now their record against TUCs is better, flipping the comparison over to them. Maddening, I know, but that's the way the Pairwise works.

What that means is that this weekend's series against the Buckeyes is much bigger than even a regular UM/OSU brawl: It's fighting for position at the top of the Pairwise, fighting for a spot in the Grand Rapids regional against the soft underbelly of Atlantic Hockey or College Hockey America. Win, get a Miami loss, and life is looking a lot easier. If not, drop below Notre Dame and you have to fight harder to not get sent to Minneapolis.

So, what are our chances? KRACH is here to answer that. KRACH is meant to use ratios between teams' KRACH ratings to give a predicted winning percentage. For example, the expected winning percentage of a team with a KRACH of 150 playing a team with a KRACH of 50 is .750.

Michigan's expected win pct. vs. OSU = Michigan KRACH / (OSU KRACH + Michigan KRACH)

Michigan currently sports a shiny KRACH rating of 402, while OSU is up to a rather healthy 197.8. That gives Michigan an expected winning percentage of .6702. The probability of two independent events both occurring is the product of the probability of either one occurring separately (i.e., P(A and B) = P(A)*P(B)). Michigan has a 67.02% chance of winning on either night against OSU, so the probability of a sweep is 44.92%, going by KRACH. Not great odds, but I'll take 'em. OSU only has a 10.88% chance of a sweep, leaving a 44.2% chance that we split.*

Michigan's been playing decent hockey since the Bowling Green loss, while OSU has been doing weird things, like only getting a pair of ties out of four opportunities against Western and State (though they got a win againt Western in the shootout) while beating Notre Dame and almost sweeping them but for some late-period heroics by the Irish.

Meanwhile, root for Northern to upset Miami down in Oxford. We'll take any help we can get. And really pull for the Bucks when Miami comes calling.

*Odds of Michigan winning out for the rest of the season? A moderately robust 30.1%.

UPDATE: Excellent catch/clarification by Brian, reproduced here verbatim: "Miami's TUC record is actually 6-4-2, but head to head games are excluded from that category in each individual comparison, leaving Miami at 4-2-2 for the purposes of its Michigan comparison." So for the purposes of everyone not in the CCHA, Miami is a TUC.

On Retired Numbers

This Joe Posnanski post got me thinking about retired numbers in other sports and the categories they fit into. College is a strange case, because I think you only have three or four years to evaluate a guy's contribution. I didn't even realize that we had any retired numbers for football until a few years ago.

No.	Player
11	Francis Wistert, Albert Wistert, Alvin Wistert
47	Bennie Oosterbaan
48	Gerald Ford
87	Ron Kramer
98	Tom Harmon

Of our guys, I think Oosterbaan, Harmon, and Kramer are easy to identify under the Greatness category. Oosterbaan was a three-time All-American in a time when I don't think freshmen were allowed on the varsity squad. Kramer was a multi-sport beast whose number was retired after his senior season. Harmon was Harmon; our first Heisman winner and the #1 pick of the NFL draft. The Wisterts were collectively 9-time letter-winners and 4-time All-Americans. None probably get it on their own, so you might argue an emotional component there. I don't know where you file Gerald Ford. His number wasn't officially retired until 1994 and he was never an All-American and his number was in wide circulation until then. It's mostly a recognition for his post-Michigan achievements, which isn't its own category in Posnanski's system. In the pros, the situation rarely arises where a guy has three or four good years, then leaves the sport entirely to go to law school and later becomes a prominent politician.

We do have the interesting situation where no one who played after 1956 has a retired number, and I like it that way. I like what the #1 jersey has come to mean, the recent flap notwithstanding. I like that #7 has some meaning of its own. I want the next guy who puts on #20 to understand what he's taking over. Michigan is a team with a history that stretches back to 1879, and the numbers in circulation make it a living history. The numbers mean different things to different people (#3 is always Marlin Jackson to me), but they're all Michigan.

Down The Stretch They Come: Hockey Projections

First thing: More to come tomorrow on the ND series. Craig and I were in attendance at both games, so we've got a few words to say about that bit of awesome.

I've been looking at Michigan's schedule for the rest of the regular season in the CCHA and wondering what we can expect in the second half of conference play. The Wolverines have a tough road ahead, with series against Michigan State and Miami in the next few weeks after having just despatched Notre Dame. They've been on a torrid pace, but with the high-quality opponents left on their schedule, there's no way they can keep it up, right? So I've tried to use a little math to project how teams in the conference will fall at the end of the regular season.

I broke it down, game by game, and I decided that the difference in current winning percentage is the best barometer of the chance teams have of winning a game. Two teams with .500 records should have the same chance of winning a game, but a team with a .700 record should have a better chance than one stuck at .300.

Projected Points = 2*(Number of games)*((Team winning % - Opponent winning %)*0.5 + 0.5)

2 = 2*(2*(0.600-0.600)*0.5+0.5)
3 = 2*(2*(0.750-0.250)*0.5+0.5)
4 = 2*(2*(1.000-0.000)*0.5+0.5)

In a few weeks, Michigan will be playing Miami. Right now, the stats for that series project like this:

2*(2 games)*((0.9375 UM winning % - 0.8333 Miami winning %)*0.5+0.5) = 2.2038 pts projected for Michigan

It basically projects a split, with Michigan having a slightly better chance than Miami of getting a sweep or an additional point. Before last weekend's sweep, it was predicting about 2.4 pts for Michigan off Notre Dame, which seems fair enough, considering the number of chances each team had in the third period of Friday's game before Caporusso buried the game winner with 20.3 seconds left on the clock.

This is how the numbers fall for the entire CCHA for the rest of the year. I've included the pace that each team is on (which is solely based on its winning percentage) for comparison.

Team	GP	Winning %	Points	Projected	(Pace)
Alaska	18	0.3611	13	21.98	(20.2)
Bowling Green	16	0.5625	18	30.80	(31.5)
Ferris State	16	0.4688	15	26.12	(26.3)
Lake Superior	16	0.1563	5	13.88	(8.8)
Miami	18	0.8333	30	44.11	(46.7)
Michigan	16	0.9375	30	46.31	(52.5)
Michigan State	16	0.8125	26	40.90	(45.5)
Nebraska-Omaha	20	0.4750	19	26.83	(26.6)
Northern Michigan	18	0.3889	14	22.69	(21.8)
Notre Dame	18	0.6389	23	35.92	(35.8)
Ohio State	16	0.2188	7	15.52	(12.3)
Western Michigan	16	0.1250	4	10.96	(7.0)

As you can see, the Wolverines are on a pace for a 52.5-pt season, but the projection knocks them down to only 46.3 pts, which would have them going about 8-4 down the stretch. Considering that the team is 22-2-0 right now, at first glance this seems excessive. But consider that they're playing the two next best teams in the league a total of six times during that stretch and it's a little less unlikely. Each game we play is like a loaded coin flip; eventually it'll come up tails, even if we play bad teams. That's what happened in part with OSU earlier in the season. We ran up against a hot goalie when we weren't playing our best and we couldn't get a bounce to bail us out.

Here's a breakdown by series of the remainder of our schedule:

Opponent	Projected
(Current)	30.00
MSU	2.25
Northern	3.10
Miami	2.21
LSSU	3.56
MSU	2.25
Ferris	2.94
Total	46.31

Ferris, Northern, and Lake State are mediocre-to-bad teams, but they get six chances to run into us on a bad night. Still, I think the numbers are conservative; 9-3 is what I'd call a successful conclusion to the season. I think we split with Miami, and State wins at least one game off of us, probably two. And if the Spartans only win one, there's a good chance someone else steals a game. A 48-point regular season would be a magnificent achievement and position us well for the CCHA playoffs and the NCAA tournament. A 50-point season would be simply astonishing, and I can't even believe anything north of that is possible. Down this stretch run, I just want to see our guys keep playing sound hockey, take care of business against teams they shouldn't lose to, and make it to the tournament without injury. Let's see how this plays out.

21 Jump Street

This post is Mathman approved!

As some of you may know, this past weekend, I pulled off the rare football trifecta, watching a high school game on Friday night at my alma mater, Saturday was the Michigan/Notre Dame game and Sunday, I went to Ford Field for the Lions/Vikings game. Three games, three wins. A pretty neat circumstance. As we were walking out of the game, I snarkily noted "Lions 2-0, Michigan State 3-0, Michigan 1-2. We are in the end times gentlemen."

And that is when my friend Mike stated the following premise (essentially):

"Each year, there's a pile of 20 wins and Michigan, Michigan State, and the Lions divvy them up between them."

So I did the math from 1970 (the year of the NFL-AFL merger and Bo's second year in Ann Arbor)

(and for the purposes of this, I did not count ties as half wins. They were ignored.)

The results after the jump...

The bold number is the highest win total for the year.

Graphically:

So what did we learn? Well, actually it's that there's a pile of 21 or so wins available and the teams seem to divvy them up amongst themselves. So Mike was very close to being right without any significant research, just off the top of his head.

Outliers: 1982: the NFL Strike Year, which cost the Lions seven chances to win. Presuming that they went 4-5 in the 9 games, an extrapolated 7-9 would bring that up to 17, which would be almost right there. Still a bad year.

1984: But if you're going to have a set of collective bad years in Detroit, that would be the year to do it.

1991: Mike's initial premise buster as stated to me isn't actually that far off because of how bad Michigan State was.

1997: Michigan goes undefeated and wins the (Mythical) National Championship, Lions make playoffs, Michigan State not awful under Nick Saban.

1999: Michigan goes to the Orange Bowl, Lions amazingly make playoffs post-Sanders retirement, and Saban gets his season to get him lured to LSU.

And, as Geoff pointed out to me in writing this up, Michigan is responsible for roughly 9 of those wins a year, leaving MSU and the Lions to fight over the other 12 between them.

Conclusions: Well, sir, the data support no conclusions as yet. The absence of activity in the Pacific suggests this could be just an exercise. But when you look at how often Michigan has carried the banner, it is amazing considering that the Lions have three more chances a year for most of those years, if not four. Let's hope that this year is the good kind of 1999 style outlier and everyone ends up happy (maybe flip the result of the Michigan/Michigan State game that year, just to be safe).