In the last few posts I have explained how regression equations based upon 2005 through 2009 MLS data can be used to judge how well teams are performing in the 2010 season. Those posts culminated in this table, which I will update on a weekly basis with commentary throughout the season. I have done some further studies, mainly of the goal differential and points required to finish in the eighth spot in the table and make the 2010 playoffs.
Statistical Background
There are three ways to answer the question of what goal differential or points are required to finish 8th in the table. They revolve around the three statistical concepts below.
- Linear regression: A best fit line that represents the mean value of y for a given value of x.
- Confidence interval (CI): A range of values based upon the statistical spread within the data set and regression, often representing a range of values that are the likely distribution of the mean.
- Prediction interval (PI): A prediction of the range of future, individual observations based upon the data observed to date that is used to construct the regression.
In general, one can think of the regression as the single point mean of y for a given x value, the CI is the likely range of means for that same x, and the PI is the range of individual observations one would expect to see for the specific value of x. Those ranges, the CI and PI, are represented by the narrow and wider dashed lines, respectively, in the Figure 1.
Figure 1: Relationship of finish position and points from this earlier post.
Knowing the ranges of values expected for any value of x allows us to construct which values of x allow us greater certainty in finishing in the 8th position. In the case of my regression data, I have used the common 95th percentile distribution for constructing CI's and PI's. That means that for any value of x that I study, I will account for 95% of the expected values in the CI and PI when setting the bounds of any test.
Because the CI and PI involve distributions and not nominal values, the x-value that ensures the 8th place finishing position is below the range of potential finishing positions will be higher than that predicted by the regression equation.
Applying statistics to the League Table
In the case of my 2010 league table, I have constructed the following rules:
- Values are red if they fall lower than critical x-value from the regression. This means that they have less than a 50% chance of making the playoffs.
- Values are yellow if they fall between the critical x-value from the regression and the lower value of the 95% PI. This means that their chances of making the playoffs are between 50% and 95%.
- Values are green when they are greater than or equal to the lower end of the 95% PI. This means a team has less than a 5% chance of missing the playoffs.
Values for the goal differential and points that correspond to the CI and PI are shown in Figure 2 below. It shows that a team needs between a 3 and 16 goal differential and between 44 and 51 points to make sure they qualify for the playoffs.
Figure 2: x-values where lowest values in 95th% CI and PI are greater y-value that corresponds to 8th place in the table
The Modified League Table
Figure 3 shows the updated league table with the associated color codings. It gives a more complete picture of where teams are consistently performing at playoff form (LA, Columbus, and NY), the bulk in the middle that are showing mixed results, and those at the bottom who are already in danger of not making the playoffs. I have kept the average predicted finish from the three regressions shown in my earlier post, while collapsing the three constituent columns for those regressions to make the table easier to read.
Figure 3: League table as of May 3, 2010.
I will continue to update this table on a weekly basis, along with each of the columns and their colors. Hopefully it will shed some light on shifts in table position that will occur on a weekly basis - whether its goals or points.
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