By Elliot Chin
Do you ever turn on a game in the fourth quarter, only to see that you’ve caught the tail end of a barely competitive blowout? Alternatively, do you ever watch your team go blow-for-blow for three quarters (or one-and-a-half halves!) but fall apart at the end, making it feel like all the effort was for naught? The flow of a game is crucial to a fan’s experience. It’s even more crucial to a coach, whose decisions on when and who to sit and start can decide the outcome of the game.
This Spring, as I devoured every minute of March Madness I could in between midterms, I began wondering what part, temporally, of the game of basketball was most important. The easy answer, of course, is that every minute is equally important—a point scored right after the tipoff is worth the same amount as the free throw scored off “garbage time” fouls. But it certainly doesn’t feel that way to a fan.
Teams want to start off strong, and usually, a team’s starting five is their best five-player set. And while a strong start is great, the rubber-band effect means that a dominant—or even just strong—performance early in the game has diminishing returns. This is part of what makes basketball so exciting, with frequent swings and comebacks.
Simultaneously, the closing minutes of a game are crucial, and a point in a tied game will add significantly more win probability as the clock winds down than at the onset of the game. But if a lead is too insurmountable, both the winning and losing team will sit their stars, and the points they score don’t matter too much. Subjectively, fans may leave early, and objectively, statheads may filter those points out of their quantitative analysis.
To coalesce this question into one that is more easily answerable, we chose to evaluate college basketball on a minute-by-minute basis, seeking to quantify which minute matters the most. We first examine the correlation between scoring within a given minute and winning the game. Then, to control for game state and team strength and to add some degree of causal strength, we control for score differential in our analysis. We then compare these results with actual game state data to see which parts of the game, empirically, are most “important.”
WIN A MINUTE, WIN A GAME
Success in which minutes of a college basketball game correlates most with winning the game? Using all of 2022 college men’s basketball as our dataset, we approached this problem with two strategies.1 First, we found the correlation between points scored within a given minute and game outcome. Second, we fit a logistic regression for each game minute, predicting the outcome of the game given the points scored within a given minute.
As shown below, both models produce similar results. Game location was standardized as both graphs reflect the relationship between intra-minute score differential for the home team and the home team’s probability of winning the game.
Results are, across the board, positive, as one would expect. The last minute, when the foul game is on, shows a high correlation between scoring and winning. There is also a generally positive trend throughout the data. While scoring in the first minute is important—doing so may be correlated with other factors such as height and a won tip-off—the predictive value quickly drops. We can reasonably conclude that performance in the second half of the game is more predictive of a team’s probability of winning than performance in the first half of the game. This finding is much more clear if we bucket the game into 5-minute periods.
This statistically backs up an intuition that many fans likely have, but offers far from causal evidence. Scoring points earlier is clearly less predictive of success than scoring points later. There may be a causal reason for this: the rubber band effect, for example, would imply that earlier points are less important as they weaken the leading team or strengthen the trailing team. But many other factors are likely at play that lack a strong causal component. Being up within the tenth minute, for example, may be correlated with playing strongly throughout the first nine minutes, which could be the cause of winning. Alternatively, simply being a strong team would impact both scoring within a given minute and the probability of winning the entire game.
SEARCHING FOR CAUSATION
To attempt to control for team strength and game state, we can replicate our above experiments controlling for some key variables. While this by no means will remove all confounding factors, it will help control for two of the biggest ones. Our new equation will be a binomial regression between which team wins and the cumulative game spread up to but not including a given minute, points scored within that minute, and pre-game betting spread.
This graph actually looks quite similar to another graph: the relationship between cumulative score differential and who wins the game. This similarity makes sense: points scored help your team win whether they are scored in the current minute or in a past minute.
Comparing these two graphs demonstrates a clear exponentially increasing importance of points scored as a game goes on. Our results, here, are less interesting: the significance of points scored in different minutes mirrors the significance of score differential at different points in the game, meaning under this framework, minute importance is just correlated with time left in the game.
What this perhaps uninteresting correlation really implies is that what makes minutes matter is inherently their relationship to team strength and game state. If certain minutes matter more, they matter more because they imply that one team is stronger, or because they indicate that one team is winning on the scoreboard.
WHEN DO POINTS REALLY MATTER?
Our answer above, however, is not necessarily satisfying. If minutes matter more near the end of the game, after controlling for confounding variables, that impact is at least partially nullified by the fact that many games close to their conclusion are locked up with no chance of the losing team winning.
To understand this mathematically, we may graph a regression between score differential and who wins a game. As the game nears its end, the coefficient of the logistic regression may increase—indicating that a point is much more valuable—but many more games will be on the wings of the regression where the game is already decided. Excitement is concentrated around a few close games, with many games boring and already decided.
To adjust for this, we aim to answer a slightly different question: During which minute do points matter most? In the beginning of a game, baskets will always matter, but with so much of the game ahead their impact is minimized. Conversely, at the end of a game, baskets are often garbage time points, mattering a lot in only a small subset of games. To account for this, we find to what degree points matter in each minute by taking the slope of the logistic regression found in the previous section at the current game state. This differs from taking the coefficient of the regression, as we did previously, because the slope is dependent on whether the current game is close or not (ie. where the game rests on the x axis of the above graphs).
In the graph below, we classify each minute of each game into one of twenty quantiles based on the slope of the logistic regression created earlier at a given point within a given game. For interpretability, the first bucket (a Game Excitement Score of 1) contains minutes of games where scoring a basket changes the win probability by on average of 0-1.3%, while the last bucket (a Game Excitement Score of 20) contains games where scoring a bucket changes the win probability on average by 24.9-26.2%.
During the first few minutes of the game, scoring points is never of huge significance, but also rarely doesn’t matter at all. At the end of the game, as we expect, we see that while the majority of games are already decided, there is a very long tail of games where scoring a point can have a 20% swing on the win probability of each team, implying that a two- or three-pointer can easily change which team wins. These moments, while few and far between, are what makes the end of college basketball games so memorable and exciting.
The pattern we observe here is overall what we expect. Averaging the regression slope across each minute can tell us which minutes, in aggregate, tend to be more important and more exciting.
Surprisingly, there is little to no discernable pattern here. The rate at which games come down to nail-biting finishes is almost exactly counterbalanced by the rate at which games enter garbage time all too soon. And even in the middle of the game, from the 10- to 30-minute mark, each minute has approximately the same importance to the outcome of the game.
This means that whether you witness a bucket right after the tip-off or right before the buzzer, those buckets on average added the same amount of win probability to the team that scored them. So if you have a limited amount of time, it doesn’t matter when you tune into your favorite team’s game—only that you do.
1 From hoopR, 5827 games analyzed