Point Rating (PR) represents the strength of a team in terms of points. A team’s PR is determined by its average scoring margin and the average PR of its opponents. A team with a PR of 10 would be considered a 2-point favorite over a team with a PR of 8. A PR of zero is considered average.
Winning Propensity (WP) represents the strength of a team in terms of odds. A team’s WP is determined primarily by its record and by the WP values of its opponents, but is “fine-tuned” based on its Point Rating. A team with a WP of 300 would be considered a 3 to 1 favorite over a team with a WP of 100. A WP of 100 is considered average.
Correlation Factor (CF) represents the correlation between Point Rating and Winning Propensity. RAMS assumes that a 1-point increase in in Point Rating corresponds to a certain percent increase in Winning Propensity. Adding 1 to that percent gives you the Correlation Factor. A CF of 1.20 would indicate that a 1-point increase in PR corresponds to a 20% increase in WP.
Estimated Winning Propensity (EWP) is an estimate of a team’s Winning Propensity, based solely on its Point Rating and the Correlation Factor.
![\[{\mathsf{EWP~=~CF^{PR}~x~100}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-e3c6f94a8a1e1ccb116b325878c81447_l3.png "Rendered by QuickLaTeX.com")
Likelihood of Result (LOR). The Result of a game is simply “which team won”. The Likelihood of a Result is determined by the EWP values of the Winning team and the Losing team.
![\[{\mathsf{LOR~=~\frac{EWP_{W}}{EWP_{W}~+~EWP_{L}}}}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-d936d1fdf7466bcf67b5193cd31fe437_l3.png "Rendered by QuickLaTeX.com")
Let’s say Army has an EWP of 300 and Navy has an EWP of 100. If Army defeated Navy, the LOR for that game would be .75, but if Navy won the LOR would be .25
Likelihood of ALL Results (LAR). The Likelihood of All Results is the likelihood that EVERY game turned out the way it did. To calculate the Likelihood of multiple Results occurring, you can simply multiply the Likelihoods together. If the Likelihood of one result occurring is 90% and the Likelihood that a second result occurs is 50%, the Likelihood of BOTH results occurring is 45% (.9 times .5).
To find the Likelihood of All Results, you could theoretically multiply the LOR for every game together. However, if you have more than a handful of games, the LAR would be too miniscule to easily calculate. Instead of multiplying the LOR’s together, you can find the logarithm of each LOR and add the logarithms together to achieve the same thing. Using the example above, if the likelihood of two results are .9 and .5, their respective logs would be -.04576 and -.030103 respectively. Adding the two logs gives you -034679, which is the log of .45.
Logistic Regression. The goal of Logistic Regression is to find the parameters that maximize the Likelihood of All Results. In our example, the only parameter we’re looking for is the Correlation Factor. Using the Excel function Solver, we can assign a temporary value to the Correlation Factor, and let Solver adjust the CF value until it generates the maximum Likelihood of all results.
Consider a two-team league consisting of Army and Navy. Army is 4-1 against Navy, and has outscored Navy by an average of two points per game. Let’s give the Correlation Factor a temporary value of 2 and see what the resulting LAR is. A CF of 2 means that a one-point favorite in terms of points would be a 2-to-1 favorite in terms of odds.
Since RAMS conventions specify that the average Point Rating is zero, Army would have a PR of 1 and Navy would have a PR of -1. If the CF is 2, we can calculate the EWP for Army and Navy as follows:
![\[{\mathsf{Army~EWP:~=~2^{1}~x~100~=~200}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-96f164bad4db81bc9d013bd7081ae077_l3.png "Rendered by QuickLaTeX.com")
![\[{\mathsf{Navy~EWP:~=~2^{-1}~x~100~=~50}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-fd5fdacd6e09351d350af1a45063ee09_l3.png "Rendered by QuickLaTeX.com")
We then can determine the likelihood of each team winning:
![\[{\mathsf{Army~Likelihood:~\frac{200}{200~+~50}~=~.8}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-1cad6c04f0bf30f28f7b883ed2952bca_l3.png "Rendered by QuickLaTeX.com")
![\[{\mathsf{Navy~Likelihood:~\frac{50}{50~+~200}~=~.2}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-6741b50b360d896d00f14edc7526c3b9_l3.png "Rendered by QuickLaTeX.com")
There were four Army wins and one Navy win, so the Likelihood of All Results would be
![\[\mathsf{LAR~=~.8~x~.8~x~.8~x~.8~x~.2~=.08192}\]](https://ramsrating.com/wp-content/ql-cache/quicklatex.com-d4c52debb00fb70850cc7e77e98372bb_l3.png "Rendered by QuickLaTeX.com")
If we assume a CF of 2, the likelihood of all the Army-Navy games turning out the way they did would be 8.192% based on the above calculations. We can determine whether another CF value would produce a higher likelihood by directing the Excel Solver function to find the CF value that maximizes the LAR. As it turns out, .08192 is the maximum LAR possible, so we conclude that 2 is the appropriate value for the Configuration Factor.
In the real world, we would typically be looking at hundreds of games, and the likelihood that EVERY game turned out the way it did would be miniscule. So to avoid having Solver try to maximize a miniscule LAR, we would have it maximize the LOGARITHM of the LAR instead. The CF value that generates the maximum LAR also generates the maximum logarithm of the LAR, so both methods are equally valid.