## Are the Wolves better than they were last year?

I am among those who have pointed out that the Wolves record is nearly identical to what it was on this date last year (13-41 vs. 13-40), reflecting a troubling lack of progress.  In response, Ailuridae has made the good point that the record does not tell the whole story, and the improvement in the team is reflected by the reduction in the Wolves' point differential.

In fact, the Wolves point differential is 2.8 ppg better than it was at this time last year, and 3.9 ppg better than it was relative to the entire season last year.  So it looks like there has been improvement.  However, a formal statistical test can be conducted to determine whether this improvement is statistically significant (i.e., not due to random chance).

The test is called ANOVA, or Analysis of Variance, which can be implemented using a "dummy", or indicator variable in an Ordinary Least Squares regression model.  This is done by collecting game-by-game point differentials and matching them to whether each one occurred this year (dummy variable = 1) or last year (dummy variable = 0).

When the model is estimated, the coefficient on this dummy variable simply returns the mean value of the change in the point differential across years (e.g., 3.9 in a model using all of the games from last year).  But it also estimates the standard error associated with the mean.  That is, think of each game as just drawing an outcome from a hat, where the values in the hat reflect the "true" quality of the team.  The more draws you make from the hat, the more the outcomes across those draws will reflect the true quality of the team.  The standard error allows us to properly account for the number of draws that we've taken.

The results:

Comparing games from this season to this time last season, the change in the point differential is 2.8 ppg.  However, the standard error of the difference is relatively high at 2.6.  Therefore, we cannot statistically distinguish between the point differentials in the two years.  In stats speak, we say that we cannot reject the null hypothesis that the coefficient is zero, or that there is no difference in the point differentials across seasons.

Using all of the data from last season produces a slightly more favorable result.  In this case, the coefficient is 3.9 ppg, with a standard error of 2.2.  The p-value associated with this estimate is 0.08, which means (roughly) that there's an 8 percent chance that we observe a difference across seasons where there actually is none.  People use different thresholds for statistical significance, but this is a borderline number.  Everyone would call a p-value of 0.01 statistically significant, the vast majority would call 0.05 statistically significant, and 0.10 is sometimes used, but is recognized as a more lax standard.

In short, the evidence that the Wolves have improved relative to last year is pretty shaky, even when we look at changes in point differentials across seasons.

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