With this blog post, we are finishing our Offensive Contribution and Impact Trilogy. In the previous two posts we presented Involvement Rate (INV Rate) and Direct Offensive Contribution (DOC). INV Rate is an advanced indicator for measuring player’s offensive involvement and DOC is an advance indicator for presenting direct offensive impact of players, measured in the number of points those players either score or directly create with passing, setting screens or offensive rebounding. Moreover, we believe that there is another offensive game indicator required. Players’ usage tells one story, effectiveness another, but there is also efficiency which should not be ignored. That is why this time we are presenting the third indicator, Direct Offensive Efficiency (DOE).

In mathematics and science, the definition of efficiency is “*a measure of the extent to which input is well used for an intended task or function (output)*”, and in business, the definition of efficiency is “*the comparison of what is actually produced or performed with what can be achieved with the same consumption of resources*”. However in NBA basketball, efficiency is considered *“a composite basketball statistics derived from basic individual stats with simple addition”*, and as such confused with effectiveness. That is why we need to consider both input – an adjusted version of Involvement Rate, and output – Direct Offensive Contributions. But this is not a simple task since in practice we can use most output statistics, recorded and reported by NBA stats collectors, but there are certain basic input statistics that are not recorded and reported. Let us illustrate this issue with the next table. For better understanding of certain abbreviations and how statistics are derived, please see our previous blog post.

When we talk about efficiency, we have to take into account both successful, output generating events, and unsuccessful, PTS not generating events. The problem arises when there are only successful events reported by official stats collectors or automatic trackers. In the table above, the unsuccessful events not recorded are potential secondary assists and potential screen assists. If we didn’t take into account other potential screen assists, then centers and power forwards who set most screens would have significantly higher Direct Offensive Contribution values than perimeter players. That is why those potential events should be estimated. The simplest (and probably the most accurate) way is to use the same coefficient as for assists and potential assists – to get one assist a player should on average create 1.96 potential assists. In the end, the league average for assists efficiency is consequently the same as for 2^{nd} assists efficiency and slightly higher than for screen assists efficiency (since we found no evidence of higher percentages of screen assisted 3-pointers).

The Direct Offensive Efficiency (DOE) formula in the end looks like this:

__DOE__ = DOC / Adjusted INV Rate = (PTS + ASTPTS + 2ndAST * 2.36 + SAST * 2.25 + OREB * 1.26 + FTAST * 1.79) / (FGA + FTA * 0.44 + PAST + 2ndAST * 1.96 + SAST * 1.96 + OREB + FTAST + TOV)

Direct Offensive Efficiency is, put very simplistically, an average number of points generated per offensive event that a player is directly involved in. The more he is involved on the offensive end (INV), the more points he should create (DOC) to have the same Direct Offensive Efficiency (DOE) score. The mean for all 355 qualified players in 2016/2017 is 1.074, median 1.072, with the highest value 1.243 (Kevin Durant) and the lowest value 0.900 (Mario Hezonja), while the variable is perfectly normally distributed, which doesn’t really happen too often.

Now we will show Direct Offensive Efficiency statistics for the same 10 players as in the previous two blog posts (INV Rate, DOC).

The chart shows that the order of players is very different from those for INV Rate or DOC indicators. The most offensively efficient player is Draymond Green and the least efficient out of selected players is Dwyane Wade, however the whole group is pretty average, and nobody really stands out (Green 84^{th} percentile, Wade 19^{th} percentile).

The scatterplots show the relationship between Direct Offensive Efficiency and the other two offensive indicators. The multivariate relationship between all three indicators tells a more complete story of players’ offensive games. For example, Russell Westbrook is a highly engaged offensive player, probably one of the most if not the most involved in the history of the game, who contributes a lot on the offensive end, despite of the fact that his offensive efficiency is quite average. The most average offensive player out of all ten selected players is Serge Ibaka with all three values close to the league averages for qualified players. Dwyane Wade is on the other hand quite involved on the offensive end, but his efficiency is well below average, therefore his offensive contribution is lower than for some other players who are less engaged offensively.

The developed offensive indicators presented in the last three blog posts could definitely be improved, but we would need additional advance statistics such as potential screen assists or direct effectiveness of secondary assists, which for now have to be estimated. But until then we can use our already tested offensive indicators, we can carry out additional analysis, including all players for the last four seasons and post-seasons in the analysis, we can investigate additional relationships between those three indicators and already established advanced NBA statistics and go even deeper into offensive games of players and teams. And that is exactly what we intent to do in the upcoming weeks and months.