FCS Pure Strength-Of-Schedule

Methodology

Simple Overview

We adapt Kenneth Massey’s core rating method but apply it only to FCS vs. FCS games. This derives team strengths from point differentials, accounts for home advantage, and calculates SOS as the average strength of FCS opponents (adjusted for venue). Data comes from NCAA.com scoreboards.

In basic terms, it works like this: We first calculate a “rating” for each FCS team by looking at who they played and the game outcomes (wins, losses, and score differences). Stronger teams get higher ratings if they beat good opponents or perform well against them. Then, a team’s SOS is simply the average rating of all the FCS opponents they’ve faced. To make it fair, we weight the games using Sagarin ratings (a well-known college football ranking system) so that matchups between top teams count more heavily in the calculations.

Finally, we shift the SOS numbers so they’re all positive (starting from 0 for the easiest schedule), and rank teams from toughest to easiest schedule. This helps show which teams have faced the hardest competition, regardless of their own win-loss record.

Technical Details

Our SOS model adapts the Massey Ratings approach, as described in his 1997 paper “Statistical Models Applied to the Rating of Sports Teams” which uses a weighted least squares regression to derive team ratings from game results. Here’s the step-by-step breakdown:

  1. Data Preparation: We start with the CSV dataset of FCS vs. FCS games (Data comes from NCAA.com scoreboards). Team names are normalized using a name mapping graph to handle discrepancies between the game data and external sources like Sagarin. This ensures consistency across 129 unique FCS teams. For each game, we record the home/away teams, scores, and compute the point differential (home score – away score).
  2. Weighting with Sagarin Ratings: Sagarin ratings for FCS teams (denoted as “AA” on sagarin.com) are fetched and mapped to our teams. These ratings provide a pre-existing strength measure. For each game, the weight is the average Sagarin rating of the two teams involved—higher-rated matchups (e.g., top FCS powers) get more influence in the model, emphasizing quality games over lopsided ones.
  3. Least Squares Setup: We model team ratings ( r_i ) for each team ( i ) using a system of linear equations. For every game between teams ( i ) (home) and ( j ) (away), we add an equation: ( r_i – r_j \approx d ), where ( d ) is the point differential (adjusted for home advantage if desired, though we kept it simple here). This forms a large matrix equation ( A \mathbf{r} = \mathbf{b} ), where ( A ) is the design matrix (rows for games, columns for teams), ( \mathbf{b} ) is the vector of differentials, and ( \mathbf{r} ) is the vector of unknown ratings. To normalize, we add a constraint that the average rating is zero (or fix one team’s rating). The system is solved via weighted least squares (using libraries like NumPy’s lstsq), with weights from Sagarin to minimize the squared error more for important games.
  4. Computing SOS: Once ratings are derived, a team’s SOS is the average of its opponents’ ratings. This reflects the cumulative strength of their schedule. We only consider FCS vs. FCS games to focus on divisional parity.
  5. Scaling and Ranking: Raw SOS can be negative (indicating below-average opponents), so we shift all values by adding the absolute minimum (here, 27.75) to make the scale non-negative while preserving differences. Teams are ranked descending by scaled SOS, with records (wins-losses in FCS games) added for context.

This method is robust because least squares handles overdetermined systems (more games than teams) and minimizes inconsistencies in transitive results (e.g., A beats B, B beats C, C beats A). It’s similar to Colley or other matrix-based ratings but weighted for quality. Limitations include sensitivity to early-season data and no adjustment for injuries or blowouts, but it provides a quantitative, data-driven SOS tailored to this dataset.

Citations: Massey’s methodology emphasizes past performance measurement via merit-based quantity (http://masseyratings.com/theory/massey.htm).
Sagarin College Football ratings (http://sagarin.com/sports/cfsend.htm)