Statistical power rankings

[NoSix]

I have calculated power rankings for the 12 remaining CONCACAF teams in the 2010 World Cup qualifying tournament based on the results of all international matches over the past five years (source: www.eloratings.net).

The ranking scheme assumes goals are poisson-distributed, and takes into account home field advantage and strength of schedule differences. Match results are weighted in time so that more recent results weigh more heavily in the rankings.

The rankings are simple and easy to interpret. Teams are ranked based on the predicted points percentage (PPCT) each would win in a complete round-robin tournament vs. the other eleven teams in the region. Ties, if any, are broken based on the predicted goal differential per game (GD) over the same hypothetical round-robin tournament.


Rank Team PPCT GD
1 MEX 0.747 1.58
2 USA 0.738 1.41
3 HON 0.613 0.78
4 CRC 0.553 0.44
5 CAN 0.538 0.36
6 HAI 0.404 -0.26
7 GUA 0.380 -0.36
8 CUB 0.380 -0.47
9 JAM 0.377 -0.49
10 TRI 0.373 -0.48
11 SLV 0.264 -0.91
12 SUR 0.190 -1.60


MEX justs pips USA on goal difference as the current Kings of CONCACAF, by a margin corresponding to only sixth tenths of a point of the 66 available in the hypothetical round-robin tournament. Based on the power rankings, the teams favored to advance to the (Hexagonal) final round of qualifying are USA and GUA from group A, MEX and HON from group B, and CRC and HAI from Group C, leaving #5 CAN as the odds-on favorite to be cursing their luck of the draw come November.

Detailed group predictions to follow...

[NoSix]

Group A predictions:


RANK TEAM W L D P PPCT GD
1 USA 4.4 0.5 1.1 14.2 0.791 1.64
2 GUA 1.6 2.8 1.6 6.4 0.358 -0.45
3 CUB 1.7 3.0 1.4 6.4 0.353 -0.59
4 TRI 1.6 3.0 1.4 6.2 0.346 -0.60


USA are predicted to win the group comfortably, taking an expected 14 of 18 available points. The real intrigue is in the battle for the second finals slot, with all three remaining squads predicted to claim six points and GUA the slightest of favorites to claim second place on goal difference.

[scaryice]

What percentage chance does each team have of advancing?

[NoSix]

Group B predictions:


RANK TEAM W L D P PPCT GD
1 MEX 3.9 1.0 1.1 12.7 0.708 1.31
2 HON 2.7 2.1 1.3 9.3 0.517 0.28
3 CAN 2.1 2.6 1.3 7.6 0.423 -0.21
4 JAM 1.0 4.0 1.1 4.0 0.221 -1.38


MEX are the clear favorites to win group B with an expected 13 points, while JAM are anticipated to struggle. Second place in the group looks to come down to a dogfight between HON and CAN, with the Catrachos expected to prevail 9 points to 8.

[NoSix]

What percentage chance does each team have of advancing?

That's a good question. For World Cup first round groups with three matches, I wrote an excel spreadsheet enumerating the 3^6=729 possible outcomes to perform that calculation. For a six match round-robin, the number of possible outcomes grows to 3^12=531,441, beyond the capabilities of my spreadsheet, and I haven't had time to write a program.

[NoSix]

Group C predictions:


RANK TEAM W L D P PPCT GD
1 CRC 3.8 0.9 1.3 12.8 0.712 1.23
2 HAI 2.6 1.7 1.7 9.5 0.526 0.29
3 SLV 1.5 2.8 1.7 6.2 0.342 -0.47
4 SUR 1.0 3.6 1.4 4.5 0.250 -1.05


With an expected 13 points, CRC are the clear favorites in the most predictable and perhaps, therefore, least exciting group. Neither SLV or SUR appear likely to muster enough to challenge HAI, with an anticipated 9-10 pts, for the second spot in the group.

Come to think of it, based on past results SLV would not have appeared likely to muster enough to defeat PAN in a home-and-home series, either, and we all know how that turned out. ;)

[EvanJ]

That's a good question. For World Cup first round groups with three matches, I wrote an excel spreadsheet enumerating the 3^6=729 possible outcomes to perform that calculation. For a six match round-robin, the number of possible outcomes grows to 3^12=531,441, beyond the capabilities of my spreadsheet, and I haven't had time to write a program.
Can you calculate a standard deviation for every teams' expected point total? I don't know how to do it but I remember that if you have the mean and standard deviation for every teams' point total you can calculate the probability of each team finishing ahead of the other one for any pair of teams.

[tomwilhelm]

Hey NoSix... you and your power rankings... get out of my head!

:p

Group A predictions:


RANK TEAM W L D P PPCT GD
1 USA 4.4 0.5 1.1 14.2 0.791 1.64
2 GUA 1.6 2.8 1.6 6.4 0.358 -0.45
3 CUB 1.7 3.0 1.4 6.4 0.353 -0.59
4 TRI 1.6 3.0 1.4 6.2 0.346 -0.60


US 13
Guat 9
Cuba 6
TT 5
http://www.bigsoccer.com/forum/showpost.php?p=14981760&postcount=55

Group B predictions:


RANK TEAM W L D P PPCT GD
1 MEX 3.9 1.0 1.1 12.7 0.708 1.31
2 HON 2.7 2.1 1.3 9.3 0.517 0.28
3 CAN 2.1 2.6 1.3 7.6 0.423 -0.21
4 JAM 1.0 4.0 1.1 4.0 0.221 -1.38


Mexico 13
Honduras 9
Canada 6
Jamaica 4
http://www.bigsoccer.com/forum/showpost.php?p=14981790&postcount=29

Group C predictions:


RANK TEAM W L D P PPCT GD
1 CRC 3.8 0.9 1.3 12.8 0.712 1.23
2 HAI 2.6 1.7 1.7 9.5 0.526 0.29
3 SLV 1.5 2.8 1.7 6.2 0.342 -0.47
4 SUR 1.0 3.6 1.4 4.5 0.250 -1.05


Costa Rica 14
Haiti 9
El Salvador 6
Surinam 2
http://www.bigsoccer.com/forum/showpost.php?p=14982248&postcount=41

[NoSix]

Hey NoSix... you and your power rankings... get out of my head!

You unlock this door with the key of imagination.
Beyond it is another dimension...

[NoSix]

Can you calculate a standard deviation for every teams' expected point total? I don't know how to do it but I remember that if you have the mean and standard deviation for every teams' point total you can calculate the probability of each team finishing ahead of the other one for any pair of teams.

I think you may be thinking of a normal distribution, but the distribution of each team's likely points is not normally distributed.

To calculate the probability of each team advancing is a two part process: first you calculate the probability that each team will win a certain number of points 0, 1, 2, ... up to 18, in this case. I can do that and post the results here if people are interested. Then you need to know the conditional probability of advancing given that a team has a certain number of points. That's the part that I haven't written an algorithm for.

[tomwilhelm]

You unlock this door with the key of imagination.
Beyond it is another dimension...

You've just crossed over into... the NoSix Zone.

[EvanJ]

To calculate the probability of each team advancing is a two part process: first you calculate the probability that each team will win a certain number of points 0, 1, 2, ... up to 18, in this case. I can do that and post the results here if people are interested. Then you need to know the conditional probability of advancing given that a team has a certain number of points. That's the part that I haven't written an algorithm for.

I'm interested. Am I correct in saying that any team with 5 points or fewer mathematically can't finish in the top two and any team with 13 points or more must finish in the top two. For the conditional probability of advancing for a certain amount of points you could look at previous CONCACAF semifinal groups. A * means the team advanced to the Hexagonal:

Semifinals for World Cup 2006: 18*, 12*, 12*, 10*, 10* 8*, 7, 7, 6, 5, 4, 0
Semifinals for World Cup 2002: 15*, 15*, 13*, 12*, 11*, 10*, 10, 9, 5, 3, 1, 0
Semifinals for World Cup 1998: 16*, 13*, 13*, 12*, 12*, 10* 10, 8, 5, 3, 1, 0

Out of this sample of 36 teams, every team with at least 11 points has advanced and every team with at most 7 points has been eliminated.

[NoSix]

Am I correct in saying that any team with 5 points or fewer mathematically can't finish in the top two and any team with 13 points or more must finish in the top two.

Not sure off the top of my head about 13 points, but if you imagine that team A in a group wins all of its matches, and teams B, C, and D draw all of their matches not involving team A, then the final point totals would be 18, 4, 4, 4 and a team with 4 points would go through.

[NoSix]

Not sure off the top of my head about 13 points...

Agree that 13 points would guarantee advancement, with a 12, 12, 12, 0 split being the 3rd place finish with highest possible point total.

[tomwilhelm]

Agree that 13 points would guarantee advancement, with a 12, 12, 12, 0 split being the 3rd place finish with highest possible point total.

Did you just agree with yourself? :confused:;)

[EvanJ]

Semifinals for World Cup 2006: 18*, 12*, 12*, 10*, 10* 8*, 7, 7, 6, 5, 4, 0
Semifinals for World Cup 2002: 15*, 15*, 13*, 12*, 11*, 10*, 10, 9, 5, 3, 1, 0
Semifinals for World Cup 1998: 16*, 13*, 13*, 12*, 12*, 10* 10, 8, 5, 3, 1, 0

For each of the above three sets the number of draws equals 108 minus the combined points of the 12 teams. The semifinals had 9 draws (25% of all games), 5 draws (13.9% of all games), and 5 draws (13.9% of all games) for an overall draw percenatge of 17.6. The mean number of draws to the nearest whole number is 6, which is 2 per group. Therefore if we expect the upcoming semifinals to be similar to the previous three we would expect the teams in each group to combine for about 34 points. NoSix expects Group A to have 33.2 points, Group B to have 33.6 points, and Group C to have 33.0 points. The consensus of NoSix's predictions and past results yield an estimate of 33 to 34 points per group (2 or 3 draws out of 12 per group). NoSix's 18-4-4-4 example shows that a team must have at least thirteen and a third percent of the points in the group to have a chance at the top two. With a 9 group sample of semifinals from the past three cycles, one can look at what percentages of the total points in the group always, sometimes, and never resulted in advancement to the Hexagonal. I made a spreadsheet on my computer and may post more data about the semifinals of the past three cycles later.

[NoSix]

The probability of each team in group A winning a given number of points is summarized in the following table:


PTS CUB GUA TRI USA
0 0.0% 0.0% 0.0% 0.0%
1 0.2% 0.4% 0.2% 0.0%
2 1.1% 1.6% 1.2% 0.0%
3 3.0% 3.8% 3.4% 0.1%
4 5.2% 6.0% 5.7% 0.1%
5 8.2% 9.5% 8.8% 0.1%
6 12.7% 13.3% 13.2% 0.8%
7 13.9% 13.6% 14.2% 1.4%
8 13.7% 13.9% 13.7% 1.0%
9 15.1% 13.9% 14.6% 4.8%
10 10.2% 9.0% 9.8% 6.7%
11 7.2% 7.0% 6.7% 3.3%
12 6.0% 4.8% 5.4% 14.7%
13 1.3% 1.4% 1.1% 14.6%
14 1.9% 1.6% 1.6% 3.7%
15 0.1% 0.1% 0.0% 22.6%
16 0.2% 0.2% 0.2% 11.8%
17 0.0% 0.0% 0.0% 0.0%
18 0.0% 0.0% 0.0% 14.1%
100.0% 100.0% 100.0% 100.0%


Note that, in general, for each team the expected point value may differ from the most likely point value. In particular, while Guatemala's expected point value is slightly higher than Cuba's, comparing their distributions, it appears that Cuba might have a slightly higher probability of advancing.

The probability distribution table for group B:


PTS CAN HON JAM MEX
0 0.0% 0.0% 0.0% 0.0%
1 0.1% 0.1% 0.1% 0.0%
2 0.6% 0.3% 0.7% 0.0%
3 1.5% 0.6% 3.5% 0.1%
4 2.8% 1.4% 9.8% 0.3%
5 5.5% 3.1% 15.0% 0.5%
6 9.0% 4.9% 16.1% 1.1%
7 10.8% 7.1% 18.3% 2.7%
8 13.6% 11.3% 15.3% 3.3%
9 15.6% 12.4% 8.3% 5.3%
10 11.9% 12.6% 8.5% 10.3%
11 11.4% 15.4% 1.6% 9.0%
12 8.7% 10.3% 2.3% 11.7%
13 3.3% 8.2% 0.1% 17.9%
14 4.1% 7.8% 0.3% 8.5%
15 0.3% 1.2% 0.0% 11.6%
16 0.8% 2.9% 0.0% 11.9%
17 0.0% 0.0% 0.0% 0.0%
18 0.1% 0.3% 0.0% 5.8%
100.0% 100.0% 100.0% 100.0%


and group C:



PTS CRC HAI SLV SUR
0 0.0% 0.0% 0.0% 0.0%
1 0.0% 0.3% 0.6% 0.2%
2 0.0% 0.6% 2.3% 1.7%
3 0.2% 1.1% 5.0% 6.0%
4 0.5% 2.8% 7.4% 11.4%
5 0.5% 4.9% 11.1% 13.9%
6 1.6% 5.9% 14.7% 16.2%
7 3.4% 9.6% 14.1% 17.4%
8 3.2% 12.5% 13.5% 12.3%
9 6.7% 10.8% 12.4% 9.4%
10 11.4% 13.6% 7.6% 6.7%
11 7.4% 13.0% 5.6% 2.1%
12 13.5% 7.4% 3.5% 2.2%
13 17.3% 8.5% 1.0% 0.2%
14 5.9% 4.9% 1.0% 0.3%
15 13.0% 1.5% 0.1% 0.0%
16 9.9% 2.3% 0.1% 0.0%
17 0.0% 0.0% 0.0% 0.0%
18 5.5% 0.3% 0.0% 0.0%
100.0% 100.0% 100.0% 100.0%

[EduarDitto]

Did you just agree with yourself? :confused:;)

LOL

[NoSix]

Did you just agree with yourself? :confused:;)

No, I was agreeing with EvanJ's post above mine. I wanted to continue the thought from my previous post, but the time limit on editing the post had expired.

[EvanJ]

Thank you NoSix for the percentages. The next step might be to do cumulative percentages for every point total (meaning for X points the probability of getting X points or less).