Performance against Strength of Schedule FAQ

© Copyright 2004, Paul Kislanko

What is the Peformance Against Strength of Schedule?

The PASOS is a ranking of teams based upon how well they've done and how well the teams they've beaten have done.

Isn't that what the RPI is for?

Not exactly. The way the RPI is calculated allows a team's own record to be incorporated into its own OOWP, so playing weak teams can help your RPI if those play a lot of strong teams. Your higher WP and OOWP is half of the formula, and you can get that to be higher than the other half (OWP) by careful choices of opponents. Also, because seventy five percent of the RPI is strength of schedule, a winless team can have a high RPI.

How does the PASOS avoid that?

The PASOS doesn't calculate OWP and OOWP and then figure a rating. Instead, it is an iterative approach built from OWP as a pure percentage -

(Opponents' Wins - Opponents' Wins over this team)
divided by
(Opponents' games - Opponents' games vs this team)
The resulting OWP is not used to calculate a team's PASOS. It is just input for calculating its opponents adjusted Strength of Schedule.

What's the adjusted Strength of Schedule?

A team's aSOS is calculated by adding its opponents' OWP for each win over the opponent and dividing by games played. So if a team wins 2 of 3 against an opponent, the opponent's OWP is counted twice. If the team is swept by its opponent, that opponent's OWP isn't counted at all for this team.

Like the OWP, the aSOS for a team is not used to calculate its PASOS - it is used its calculate its opponents' PASOS, and its opponents' aSOS values are used to calculate the team's PASOS.

And the PASOS?

It's just the sum of the opponent's aSOS for each win divided by the number of games played. The opponents' aSOS combine OWP and OOWP, and the PASOS adds the team's WP to the mix.

Can you give an example?

The easiest way to see how it works is as an array of head-to-head results. Suppose 7 teams have played 27 games so far, with these results (Wins by team in row over team in column header):

	A	B	C	D	E	F	G	Wins
A		2	3	1				6
B	1		2	2	2			7
C	0	1				1	1	3
D	2	1					2	5
E		1						1
F			2					2
G			2	1				3
Losses	3	5	9	4	2	1	3	27

1. Calculate the OWPs. For example, team A played B, C, and D, who are 15-18, but 3-6 vs A (just the reverse of A's record) so A's OWP is based on their 12-12 record against other teams = .500, B has played A, C, D, and E, who are 10-11 against teams other than B.

   	A	B	C	D	E	F	G
   OW-OL	12-12	10-11	9-9	12-6	5-4	2-7	7-11
   OWP	0.500	0.476	0.500	0.667	0.556	0.222	0.389

2. Calculate the teams' aSOS. For this example we'll use team C. They've played A, B, F, and G.

   A B F G Totals Games ASOS OWP 0.500 0.476 0.222 0.389 C's wins 0 1 1 1 C's Sums 0.000 0.476 0.222 0.389 1.087 12 0.0906

   Do the same calculation for all teams and you get:

   	A	B	C	D	E	F	G
   aSOS	0.3466	0.3287	0.0906	0.2504	0.1587	0.3333	0.2778

3. Now replace OWP with aSOS and perform the same calculations:

   	A	B	C	D	E	F	G	Totals	Games	PASOS
   aSOS	0.3466	0.3287	0.0906	0.2504	0.1587	0.3333	0.2778
   A's Wns		2	3	1
   A's Sums		0.6574	0.2718	0.2540				1.1796	9	0.13107
   B's Wins	1		2	2	2
   B's Sums	0.3466		0.1812	0.5800	0.3174			1.3460	12	0.11217
   C's wins	0	1				1	1
   C's Sums	0.0000	0.3287				0.3333	0.2778	0.9398	12	0.07832
   D's Wins	2	1					2
   D's Sums	0.6932	0.3287					0.5556	1.5775	9	0.17528
   E's Wins		1
   E's Sums		0.3287						0.3287	3	0.10957
   F's Wins			2
   F's Sums			0.1812					0.1812	3	0.0604
   G;s Wins			2	1
   G's Sums			0.1812	0.2540				0.4316	6	0.07193

And that's it. The table of wins that has been input has been turned into an ordered list:

Team Rec PASOS 1 D    5-4       .17528 2 A    6-3       .13107 3 B    7-5       .11217 4 E    1-2       .10957 5 C    3-9       .07832 6 G    3-3       .07193 7 F    2-1       .06040

Why are the numbers so small?

Instead of an average of averages like the RPI, the PASOS is more nearly percentage of percentages: 60 percent of 60 percent is 36 percent. The values decrease with every iteration. This makes no difference in the relative rankings, it's just that while the RPI and its version of the SOS cluster around 0.5 (they must because they are averages of averages), the PASOS values do not.

You can translate the PASOS value into the same range as winning percentages by taking its cube root, since it's basically a percentage of a percentage of winning percentages. The team order stays the same but the values are in the same range as winning percentage:

Team Rec WP       aWP 1 D    5-4       .556       .5596 2 A    6-3       .667       .5080 3 B    7-5       .583       .4823 4 E    1-2       .333       .4785 5 C    3-9       .250       .4278 6 G    3-3       .500       .4159 7 F    2-1       .667       .3924

Why perform the iteration twice instead of stopping at aSOS or continuing on?

The number of iterations required depends upon the starting point. To go another iteration instead of beginning with using OWP to calculate opponents' aSOS to include OOWP for this team, you'd begin with OOWP, calculate opponents' aOWP (a combination of their OWP and OOWP), use that to calcualte aSOS (their WP, OWP and OOWP), and then the next iteration would be "Performance Against PASOS" and would be a combination of WP, OWP, OOWP, and OOOWP.

What's the dSOS?

The dSOS is what a team's PASOS value would be if it had won every game it played. The PASOS combines WP, OWP, and OOWP, while the dSOS combines just OWP and OOWP. So it is the "SOS" part of "performance against SOS".

A team's aSOS is the one to use for scheduling purposes since that is the value of a win over that team. The dSOS is a record of how the team's opponents have done.

Is this just for baseball?

No, the PASOS applies to any sport. The basic ideas were developed using the 2003-2004 basketball season to analyze the RPI in terms of number of opponents and opponents' opponents and the effect of common opponents. When applied to the 2003 football season (not including the bowl games) the list was very similar to the BCS rankings at that time:

PASOS	BCS
Oklahoma 12-1 Ohio State 10-2 Lousisana State 11-1 Southern California 11-1 Miami Florida 10-2 Michigan 10-2 Florida State 10-2 Kansas State 9-3 Texas 10-2 Miami-Ohio 12-1	Oklahoma 12-1 Lousisana State 11-1 Southern California 11-1 Michigan 10-2 Ohio State 10-2 Texas 10-2 Florida State 10-2 Tennessee 10-2 Miami Florida 10-2 Kansas State 9-3

Mimia-Ohio was 11^th in the BCS and Tennessee 12^th in the PASOS.

How valid the PASOS is for a given sport depends upon how well the teams are connected via Opponents' Opponents' opponents (implicit in the OWP that is calculated). For basketball, better results are obtained if the process begins with OOWP instead of OWP, so that the PASOS is a combination of WP, OWP, OOWP and OOOWP.

How does it work?

What the PASOS does is sort out the "A beat B beat C beat A" chains by assigning a specific value that determines the "quality" of a win. One characteristic of the PASOS is that the concept of a "quality win" has a precise definition. Unlike the RPI, there is no such thing as a "quality loss" - losing to a good team doesn't increase the rating and losing to a bad team doesn't hurt it (that's a "quality win" for the bad team, so it becomes less of a "bad win" for the teams that beat the "bad" team).