Robust sports ratings based on least absolute errors

Rating sports teams based on scores of previous games can be done by estimating the parameters of the standard linear model. Previously proposed estimates have been based on a modified version of least squares in which games with large victory margins were downweighted due to the sensitivity of least squares to outlying observations. This paper considers the least absolute value or L-1 estimate as an alternative method for rating teams. An advantage of L-1 is that it is intrinsically less sensitive to outlying observations, and hence automatically accounts for extreme outcomes. For the ratings version of the linear model it is shown that L-1 is the median of a team's normalized scores and least squares is the average. For teaching purposes the rating problem provides a nonstandard instance of the linear model that students find interesting. Students can tune the estimators to give recent games more weight, and they can compare L-1 to other regression quantiles. Because the estimators can be expressed in terms of the familiar average and median statistics, the ratings context allows insight into how linear model estimators work. Comparison of least squares and L-1 is presented for the 1993 NFL season.