# A Statistical X-Ray on Westbrook’s Triple-Double Performance from Wald and Wolfowitz

Everyone that follows the NBA even loosely has certainly heard about the monstrous performances that Russell Westbrook has put up this year. Every relevant website you visit talks about how “hot”he is – especially during the 7 game span of consecutive triple doubles that ended at the game with my Celtics.  This “hotness” in sports has been a theme for many years and people have been trying to understand whether such thing as the “hot hand” exists.  Psychologist Thomas Gilovich studied this problem and basically found that there is not such thing as “hot hand”.  The main argument is that if the hot hand existed this essentially would mean that a shot by a player would have a higher probability to be made if he had made the previous one.  However, his analysis did not find any significant evidence to support this hypothesis.  Gilovich was concerned with the performance within a game, but if we want to examine Westbrook’s performance with respect to his triple-double performance we should look across games, and every game will be one observation.  So let’s do that by using the Wald-Wolfowitz Runs Test (WWRT).

Let’s first give some background on WWRT.  WWRT is a statistical test that operates on a binary sequence.  The null hypothesis of the test is that the individual elements of the sequence are mutually independent, that is, the value of the $i^{th}$ element of the sequence does not depend on any other previous element.  Why this is relevant to examining Westbrook’s performance? Because if he indeed got “hot” then this means that the fact that he put up his fifth consecutive triple-double is dependent on the fact that he had a triple-double in the previous game as well – this is the definition of being hot, i.e., having an “unusual” streak that you do not expect if events are independent.  In our case, the performance of Westbrook will be captured by a binary vector, where the element $i$ will be 0 if he did not have a triple-double during his $i^{th}$ game of the season and 1 otherwise.  Under the null hypothesis Wald and Wolfowitz found that in a sequence of mutually independent elements with $S$ successes (i.e., triple-double performances) and $F$ failures ($S+F=N$), the expected number of runs (i.e.,  a maximal non-empty segment of the sequence consisting of adjacent equal elements) is:

$\mu = \dfrac{2\cdot F\cdot S}{N} + 1$

while the standard deviation of the number of runs is:

$\sigma = \sqrt(\dfrac{(\mu-1)(\mu-2)}{N-1})$

Using the above we can calculate the z-sc0re as:

$z = \dfrac{R-\mu}{\sigma}$

where $R$ is the number of actual runs in the sequence. If we are looking for a hot Westbrook the z score should be negative – i.e., the actual number of runs are less than the ones expected if triple-double performances were independent, and thus, longer runs. Of course, if you prefer to just not do the work you can just use the runs.test function of the R package randtests.  Following is the series of the first 25 games (including last nights at Portland) for Russell Westbrook.  The red circles correspond to performances with no triple-double, while the black circles are the triple-double performances.

The WWRT in Russell Westbrook’s case rejects the null hypothesis, i.e., that the performances are independent.  Even though a statistical test cannot accept the alternative it is telling that Westbrook might be a player that can get “hot” after a good performance.  Hence, we might see more streaks like that throughout the year.  I will repeat this exercise at the end of the season too.

Notes:

1. It is very possible that this is a false positive! This is a constant problem with hypothesis testing so always keep this in mind
2. If you noticed we examined a specific direction of the effect (i.e., z-score negative), that is, we performed a one-sided test.  In general, this is not recommended mainly because it is a way to p-hacking.  However, if you know a priori which direction of  the effect you are interested in (as in our case) then it is fine to use a one-sided test.
3. Someone might – rightly so – argue that you have 25 data points, what type of statistical testing can you do with 25 data points? This is a valid concern and this is why I will repeat the analysis later in the season.