The term "strength of schedule" has no meaning without the context in which it is being used, so by itself it doesn't mean anything. And even for a well-defined measurement there are many different ways to define "schedule strength".

At the end of the 2004 season, for instance, we find quotations of Miami's SOS as first (by RPI standards) and 45^th (by ISR standards). The PASOS version had Miami's schedule 10^th-toughest. So which is "right"?

None of them are - like the rating systems themselves, there's no right or wrong, there's only different.

The first thing to keep in mind is that the term "SOS" has a different meaning if it is a part of the rating system used to derive it - one could be talking, for instance, about the SOS component of the RPI for a team or could be characterizing a team's schedule be the values of its opponents' RPIs. In the case of the RPI - which is defined to be 75 percent SOS - we assume it means the SOS component of the RPI. But it would be equally valid to define an SOS that was the average of the opponents' RPIs, and those are decidedly not the same thing.

The Iterative Strength Ratings do define "SOS" as the average of opponents' ISRs. Although the ISR depends as much or more on SOS to determine a team's rank than the RPI, there isn't a specific number that is a part of the final ISR value that can be pointed to as the "SOS part of it". The "RPI SOS" and the "ISR SOS" are calculated differently, and there's no way to calculate one the same as the other except by relative ranking, which only shows that they are different.

It's not apples and oranges, it's "fruits and vegetables".

The PASOS algorithm actually produces three different values that in one way or another can be called "strength of schedule." It is like the ISR in that the result of the SOS affect on the final PASOS value can't be explicitly written out, but the definitions of the components are very simple.

The RPI's SOS component is a combination of Opponents' Winning Percentage (OWP) and Opponents' Opponents' Winning Percentage (OOWP). Besides the problems caused by the RPI definitions of OWP and OOWP, the RPI SOS definition gives higher weight to an opponent who's opponents have done well, even if the opponent lost every game to them. So if team A beat team B who's lost to team Z, if team Z has a good record team A benefits without ever having played team Z.

The first, and most basic, component of the PASOS is a true OWP. This is just the number of games won divided by the number of games played by a team's opponents, not including games against the team. This is the basic "SOS", but it is a crude measurement, since there is no way to tell whether the opponents' wins were obtained against weak or strong opponents.

The PASOS avoids this problem by not using a team's OWP to calculate the team's own rating. Instead, the team's OWP is a value used to determine the teams' opponents' SOS. If team B beats team A, then team B's adjusted SOS is increased by team A's OWP, but the increase is to team B's OOWP. Team A gets no credit in its OOWP category, since losing to team B doesn't mean that A is better or worse than other teams that lost to team B, and certainly is not better than teams that beat team B.

So aSOS is the second SOS component in the PASOS algorithm, but that still doesn't say much about the team. It is a combination of the team's winning percentage and the team's opponents' OWP. Within the PASOS context, the aSOS is the value of a win against the team. When team B beats A it adds to its OOWP and OOOWP all of the OWP and OOWP that A was entitled to, along with its WP and A's reduction in WP.

The third "SOS" in the PASOS context is the only one that can be used to compare schedules. The dSOS is what a team's PASOS would be if the team had won every game. In other words, it is the PASOS without the team's winning percentage. It does include opponents' winning percentage, opponents' opponents winning percentage, and even a trace of opponents' opponents' opponents winning percentage.

The "Performance Against" algorithm is actually very general. All it does is sort out head-to-head-to-head wins. If you replaced OWP with the RPI's or ISR's definition of SOS, you'd still have a "Performance Against SOS". The only difference between one of those and the PASOS is that you couldn't interpret the final result as an adjusted Winning Percentage. Each would be an essentially new rating system to complement the one the one that provided the "SOS". The PASOS definitions for aSOS and dSOS would be just as valid: the aSOS would still be the value of a win over the team and the dSOS would characterize a team's schedule without regard to how SOS was originally defined.

One could also use the algorithm to define a "Performance Against ISR", "Performance Against RPI", or even a "Performance Against PASOS". When one does this, the "SOS" part of the PASOS is a meta-SOS that refines the definition of SOS in the rating method that is the one team performance is being compared against, and is an easier way to perform the analysis than many that are currently used, which is the subject of the next installment.