Recently the USA Today ran a story on NBA team payrolls. The story seemed to conclude that NBA teams that have the highest paid players do the best and thus teams that have lower paid players do not perform so well. They make their point by using player salaries from the LA Lakers and the Boston Celtics as examples. This is a variant on the argument that teams with high payroll will perform better than teams with lower payroll, and I have to disagree that NBA (or for that matter NHL, MLB or NFL) teams that have high payrolls result in higher winning percentages; nor am I the first to say this.
In essence, the main premise of Michael Lewis' book, Moneyball was to examine how the Oakland A's did so well with one of the lowest payrolls in Major League Baseball. Additionally, as we state in The Wages of Wins, team payroll does not explain a high degree of team performance. How do we back up this statement statistically? We analyized team performance and relative team payroll data (to account for increasing overall payrolls over multiple seasons), and calculated the coefficient of determination, also called r-squared or R2. We use R2 since we are interested in the proportion of variance that is in common between NBA team payroll and NBA team performance. Since R2 is between zero and one, the number is the percentage of the variance that is in common between NBA team payroll and NBA team performance. What we find is that the proportion of variance that is in common between NBA team performance and NBA team payroll is rather small.
Some have argued - incorrectly - that we use the wrong statistical measure. They say the true measure is the correlation coefficient - also called r. Why is this incorrect? As I explained in this post on The Wages of Wins Journal, the correlation coefficient does not measure how much of the variaition between NBA team payroll and NBA team performance is in common, but rather whether NBA payroll and NBA performance change together or change oppositely.
Sometimes correlations can lead us astray. For example my blog about there being is a high positive correlation between vocublary and corporate success. If we use correlation as our guide to the importance that one variable has on another, we would conclude that studying the dictionary (or watching The Daily Show) will allow us to climb higher on the corportate ladder. While I do not have the data, my guess is that the R2 is rather low, since the amount of variation that is common between these two variables is most likely tiny. These cases where you get very high correlations (positive or negative) are referred to as spurious correlation.
So with the stats stuff briefly discussed, let me show you why I disagree with the USA Today's inferences about NBA payroll and team performance. If we calculate the coefficient of determination (R2) for NBA team payroll - using the USA Today's NBA salary database and the NBA's final season team performance the R2 is 0.041. What this means is that the proportion of variance that is common between NBA team payroll and NBA team performance is 4.1%. Just to be clear, the correlation coefficient is 0.202.
Not only that, but I also tested to see if the correlations between this past years NBA team payroll and team performance were related, and using the test statistic: ((n-2)*R2)/(1-R2)) for 1 degree of freedom and 30 degrees of freedom, found that the calculated test statistic was less than found at the 5% probability level in the F Distribution, so we would accept the null hypothesis, which is that the correlations between the two variables (NBA payroll and NBA performance) are unrelated. So not only the proportion of variance that is common between the two tiny, but here I am able to show that the correlation coefficient between the two populations (NBA payroll and NBA performance) for the 2008-2009 season is statistically zero.
Now since I am only looking at the 2008-09 NBA season, I did not calculate relative payroll as we did in The Wages of Wins. If I were to calculate relative payroll - like we did in The Wages of Wins - we will get the same answer since relative payroll is a monotonic transformation of total payroll.
Earlier this year, an unnamed NHL executive and I looked at NHL payroll (using their data) and NHL team performance, and we found in essence the exact same result - which was a surprise to him, but not to me.
Bottom line: team payrolls are poor gauges in measuring team performance.
Click here for more information on correlation.