NCAA Lacrosse Bradley-Terry

[CowbellGuy]

[quote KeithK]Of course, if the draft isn't done yet it's a different story...[/quote]
IF? We're talking about Hayes here.

[jtwcornell91]

[quote jtwcornell91]One of the nice things about this method is that it also comes with a probability distribution for the ratings, and so you can estimate the uncertainties in each of them.  I've got the raw numbers for those, but I don't have time to put them into a nice form tonight.  But I have some ideas for cool graphs...[/quote]

Okay, so as promised.  What you end up predicting is a probability for each possible set of values for all the ratings of the 56 teams, meaning that it's a function of 56 variables.  But you can look at the probability distribution for one team's rating by "marginalizing" over all the other ratings; here are the marginalized probability distributions for the top six teams, each scaled so that the maximum is one:

If we want to normalize the distributions, so that the total probability in each case adds up to 1, we have to scale the broader curves down relative to the more sharply-peaked ones:

[ugarte]

Well, that explains it.

[Jacob '06]

Is the lower probability center for Cornell due to their weaker SOS, or less exposure to OOC teams? (By lower I meant the center of our distribution is only at ~.5 and we have a wider distribution)

[jtwcornell91]

[quote Jacob '06]Is the lower probability center for Cornell due to their weaker SOS, or less exposure to OOC teams? (By lower I meant the center of our distribution is only at ~.5 and we have a wider distribution)[/quote]

It's just that in the second plot everything's scaled so that the area under all of the curves is the same.  Since ours is broader, the peak is lower.  The broader distribution means our rating is less precisely determined.  That may be because of the slightly weaker schedule, but it may also just be that our rating is farther from 100, where everybody's prior was peaked at the start of the season.  Note that Duke's distribution is the second-broadest.

[jtwcornell91]

Yeesh, let me finish...

[ebilmes]

JTW for President! B-]

[DeltaOne81]

[quote ebilmes]JTW for President! B-][/quote]

... of the NCAA lacrosse committee

[jtwcornell91]

Okay, so actually marginalizing over everything is not quite the right thing to do.  I.e., if you want to ask what we know about the strengths of Cornell and Duke, you have to think not just about the overall probability of Cornell having a certain rating and of Duke having a certain rating, but rather the probabilities of combinations of their ratings.  (If we'd used the Jeeeeffreys prior, this would be even more extreme, since we'd actually know nothing about the product of two teams' rating, since the overall scale would not be defined.)  Take, for example, Cornell and Duke; our knowledge of their ratings is correlated; if you marginalize over all the other teams' ratings and plot the probability distribution of combinations of Cornell's and Duke's ratings, it looks like this:

What's really interesting is the ratio of these two teams' ratings, which tells you, e.g., the expected odds that one will win a game against the other.  The probability distribution of that (marginalizing over everything else) looks like this:

About three times as much of the area under that curve lies to the right of one as lies to the left, which says there's a 25.5% chance that Duke is actually stronger than Cornell and just got unluckier in their game results.  But it's 74.5% likely that Cornell is indeed stronger than Duke, but not necessarily by the factor of two that the most likely ratings indicate.  It could be more or less.

[jtwcornell91]

If we do the same thing with Virginia, the contour plot looks like this:

And the ratio of the ratings is distributed as follows.

There's only a 9.3% chance that UVa is actually stronger than Cornell.

[jtwcornell91]

Finally, let's do the same thing with Hopkins (although they're actually only the sixth strongest team according to this evaluation).


There's only a 3.8% chance that Johns Hopkins is actually better than Cornell.

[Jacob '06]

Please forward to the person in charge of the selections. Of course that person will probably have no clue what you are talking about.

[Jim Hyla]

[quote Jacob '06]Please forward to the person in charge of the selections. Of course that person will probably have no clue what you are talking about. [/quote]

And we do?::wow::

[Swampy]

[quote jtwcornell91][quote Jacob '06]Is the lower probability center for Cornell due to their weaker SOS, or less exposure to OOC teams? (By lower I meant the center of our distribution is only at ~.5 and we have a wider distribution)[/quote]

It's just that in the second plot everything's scaled so that the area under all of the curves is the same.  Since ours is broader, the peak is lower.  The broader distribution means our rating is less precisely determined.  That may be because of the slightly weaker schedule, but it may also just be that our rating is farther from 100, where everybody's prior was peaked at the start of the season.  Note that Duke's distribution is the second-broadest.[/quote]

Wouldn't the spread reflect the total number of games played? The standard deviation is inversely proportional to the sample size, and the curves have a more than passing resemblance to a normal curve.

[French Rage]

What's a nubian?