NBA Fouls – Substitutions and Discussion

This is part 4 of my series on DeMarcus Cousins and how NBA players accrue personal fouls.
Part 3 can be found here.
Part 2 can be found here.
Part 1 can be found here.

I strongly recommend reading parts 2 and 3 before continuing as this series builds on the past.


A natural question that arises from our previous analysis is to question if anything can be done to prevent a player from “tilting.” We now show that making quick substitutions can change how a player accrues fouls and reduce his “tilt.” We define a quick substitution (QS) as a substitution that occurs within 30 seconds of a personal foul. While this definition may capture substitutions that are not a reaction to the player committing a foul, we believe it is adequate for the purposes of this paper. Fouls are then classified as happening before or after the QS. As a result, games without a QS will classify every foul as happening before a hypothetical QS, which may never be observed. Furthermore, for ease of analysis, we only consider the first time a player has a QS, despite the possibility of it happening more than once per game.

Table 4 gives the output for survival analysis that includes an indicator for being before or after a quick substitution in the conditional risk set model for DeMarcus Cousins, Al Horford, Robin Lopez, and all centers pooled. The coefficient on QS for all players examined is negative, indicating that a quick substitution is associated with a lower chance that a player will foul at any time after the substitution. However, for all players examined, it is not always a significant difference. Quick substitutions seem to be associated with a reduction in Al Horford’s foul tendencies, though not significantly and the effect size is smaller for Horford than for Cousins. The players still have significant positive coefficients for later fouls, indicating that while they may still “tilt,” the QS may mitigate some of it.

Table 4: Conditional risk set model for ordered events for Cousins, Horford, Lopez, and all centers pooled, including an indicator for whether a foul was after a quick substitution (QS). For Cousins, a QS is associated with a e^{-0.58}\approx56\% reduction in foul chance.

Focusing on Cousins, Table 5 displays the survival model output for all fouls before a QS and after a QS side-by-side to facilitate comparison. The analysis of fouls before a quick substitution shows a significant increase in the chance that he commits a foul once he has 3 or 4 fouls. However, after the substitution, the coefficients are smaller, indicating that he is no longer as “tilted.” We visualize this change in foul behavior in Figures 3a and 3b which show the survival curves before and after a quick substitution. Cousins’s foul tendencies prior to a QS (Figure 3a) are similar to those seen across a whole game (Figure 2a). However, after a QS (Figure 3b), there is much less of a stark contrast. He does appear to commit his 4th fouls faster than his 3rd, but not as significantly as before the QS.

Table 5: Conditional risk set model for ordered events for Cousins before and after a quick substitution (QS). All coefficients are decreased indicating that a QS could potential mitigate his aggressive fouling tendencies.
Figure 3: Survival curves for the first 4 fouls for Cousins, before and after a quick substitution (QS), when he commits a minimum of 5 fouls. Prior to the QS, Cousins has clear ordering, but it is disrupted after.

Al Horford, by contrast does not seem to be significantly affected by a QS, though throughout we have seen that Horford does not seem to “tilt” as much as other centers in general.  Figures 4a and 4b show Horford’s survival curves before and after a quick substitution. While there may be some distinction between the fouls before a QS, it is not as extreme as seen with Cousins, and there is certainly little order after a QS. Al Horford simply does not foul, “tilt,” or get affected by quick substitutions as much as other centers.

Figure 4: Survival curves for the first 4 fouls for Horford, before and after a quick substitution (QS), when he commits a minimum of 5 fouls. Horford shows no strong trends before or after a QS.

Discussion – Further Research

While we focused on only centers for this research, the methods used here can easily be used for all players in the NBA to identify players who “tilt.” In addition to looking at quick substitutions, it would be interesting to note other events which may reduce the effect of a “tilting” player, particularly other stoppages of play like timeouts or breaks in a period. We chose to look at substitutions shortly after a foul in the hopes of best capturing a direct coaching reaction to the foul. A timeout following shortly after a foul may also reflect a direct reaction to the foul and is a clear avenue for further analysis. Furthermore, while we only considered personal fouls in this study, it would be interesting to note how technical fouls play a role in “tilting” players. Technical fouls are especially interesting since they are rarely a part of strategy in the way a normal personal foul can be. Our overall aim is to examine players who are considered by many to be emotional, so how these players accumulate, or their teammates accumulate, technical fouls may have an impact on their foul rates and overall “tilt.” Additionally, we only adjusted for time and score, but there are many other factors that could be included such as the player being guarded (a player may be more likely to “tilt” against players who tend to play more aggressively or are known trash talkers) or if the rate at which the player of interest draws fouls (players may become more upset if they feel they are not receiving foul calls on their behalf). Moreover, while this paper was limited to a select few centers, the methods could easily be applied to all NBA players. Expanding the number of players analyzed would allow for greater understanding of how different players and positions accrue fouls.

Finally, we did not do any causal inference. Any effects we see are just associations. Proper causal inference analysis is a clear area for further research.


In this analysis, we used a survival model for fouls to show that fouling rates are not always independent of the number of fouls a player has accumulated. Emotional players, such as DeMarcus Cousins, often “tilt”, increasing the likelihood of committing another foul as they accrue more fouls. Our analysis also indicates that quickly substituting a player could influence an emotional player’s foul rate, reducing the likelihood of them picking up another foul.

We cannot say for certain the precise reason why a quick substitution has an effect. It could be that taking a player out of the game gives him time to calm down and become level-headed. However, it may also be related to the common strategy of attacking a player that is in “foul trouble”, often defined as approaching 3 fouls by halftime or 6 by the end of the game. Before the player is substituted, he may be in “foul trouble” causing the opposing team to attempt to draw a foul against him. After a QS and the player returns to the game, there is less incentive to attack since he is no longer in “foul trouble” due to the passage of game time. It may well be that a QS is simply a good indicator of keeping that player from being attacked. This hypothesis certainly merits further investigation.

While the scope of this paper is somewhat limited, we hope it will encourage others to explore the process by which players accrue fouls. We believe that further research in this area will reveal new insights into how players can remain effective throughout the game, especially if something as simple as a coach making a quick substitution can have such a significant impact. It may not be easy to stop “tilting” entirely, but there are ways to mitigate the effects.


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