Conclusions

Finally, we can test the validity of the model. Recall, we are performing an equivalence test. For evidence of equivalence it is necessary to reject each test, otherwise we cannot reject our null hypothesis of non-equivalence. The decision rule is:

Analysis: Overall, only one team rejected the null hypothesis from each test, and that team is Grinnell. Grinnell is known for taking as many shots a game as possible, and a significant amount of those shots are three pointers. From the table you can see that around 37% of their shots are three pointers. Because of this, my original thought was that Grinnell would be the most inconsistent with the model, but their shot ratio of 1.32 is the closest to the optimal ratio of 1.5. Not surprisingly then, the Grinnell team became the only validating example. While the Grinnell opponents failed to reject the null hypothesis, their shot selection provides some valuable insight. Intuition tells us a team’s proportion of shots should increase with shooting percentage. In this case the shot % ratio for Grinnell opponents is 2.06, where almost 95% of the shots taken are field goals. This likely validates the corner solution where a team chooses to take only field goals. The shot % ratio is high enough that the reward does not outweigh the risk of a three point shot. This is likely the result of Grinnell’s refusal to play good defense and teams take advantage of open shots near the basket.

Data

TOST

Two one-sided t-tests can be conducted to test equivalence in this case.

If both tests are rejected jointly in favor of the alternatives, then their is statistical evidence of equivalence. However, if only one test fails to reject the null hypothesis, then no conclusions can be drawn. Invoking the central limit theorem, computations for the test are as follows:

where, θi is estimated for two point and three point shots as

Equivalence Tests

Ordinarily, a traditional test concerning differences in proportions could be used to test the significance of our results. Such a test could be outlined as follows:

However, we want to show that two and three point shooting proportions are equivalent, and this hypothesis test cannot accurately make such a claim. There are many ways that poor data and sampling can result in not rejecting the null hypothesis, which on any level would not support our assertion. Because failing to reject the above test does not imply equivalence, a different test must be developed.
Equivalence testing is used and developed for testing equivalence of generic drugs with their name brand counterpart. The FDA needs to be sure that a generic drug behaves similarly to the original. So, an equivalence region is defined and a test, H0:Groups are Different, versus the alternative, Ha:Groups are similar is conducted. This case is no different. I’ll define an equivalence region in which to perform the following test:

Wood Floor Solutions Analyzed

What information can we glean from these solutions? Basically, we have derived conditions that determine an optimal shooting mixture for given shooting probabilities. Therefore, if the probability of making a two point shot is more than 1.5 times as likely as the probability of making a three point shot, a team should only shoot two point field goals. If the probability of hitting a two point shot is less than 1.5 times as likely as the probability of making a three point shot, then the team should rely entirely on the three point shot. And, if the ratio of the two probabilities is 1.5, any mixture of two and three point shots will be optimal.

The results are interesting if not realistic. No teams choose to shoot exclusively one shot during a game. The solution in this case stems from maximizing over a linear objective function. If the objective function were concave, a more generalized mix of shots could be determined. However, a linear objective function seems appropriate given the scoring function. It is also possible that team shooting percentages lie with in an acceptable statistical interval of the conditions. If this were true, then teams would be behaving optimally as predicted by the model. At any rate, the solutions are still interesting. We have essentially developed conditions which can involve a team relying on a single shot. At least those conditions are defined and describe a situation where it would be optimal for a team to take all three point shots.

It’s time to test the model. Next post will develop and perform a statistical test using some college basketball data. Things should start to get interesting.

Wood Floor Portfolio Solutions

Wood Floor Portfolio

We’re looking to analyze the efficiency of shot ratios for a basketball team. Suppose the coach wants to maximize expected points scored for his team. Let x and y be random variables defined as the number of two and three point shots made; while n₁ and n₂ are the total two and three point shot attempts, respectively. Assume x and y are distributed binomially,

where θ₁ and θ₂ represent the probability of making a two or three point shot. Suppose the total number of shots per game is fixed at some level z, which becomes the constraint faced by the system. Now, the optimal mix of shots for a team can be determined by solving the following static optimization problem:

Essentially, we are choosing the optimal number of two and three point shots, n₁ and n₂, which maximize expected points.

Shot Management

There are many decisions which require choosing an optimal mix of several goods. Throw in some uncertainty and you have a classic portfolio problem. The portfolio problem answers this question:
What is the optimal mixture of securities (stocks and bonds) which optimizes a portfolio’s return, given an investor’s risk preference?

Recently, I’ve been wondering if the same techniques can be used to analyze basketball shot selection. Basketball teams choose to take a two or three point shot. The likelihood of making either shot should determine the optimal ratio of two to three point shots. In the next few days I’ll provide a simple model and derive conditions for optimal shot ratios.