Statistical analysis of NBA defensive responses to hot-hand streaks

Abstract

Understanding how humans adapt to streaky performance under uncertainty is central to research on belief updating, attention, and decision-making. Professional basketball provides a naturalistic setting in which these processes can be observed under real incentives and time pressure. Using NBA player-tracking data, we examine how defenders adjust their behavior in response to offensive shooting outcomes, shifting focus from shooter performance to defensive adaptation. We conduct three complementary studies relating multiple spatial defensive metrics to recent shooting history, including asymmetric responses to makes versus misses, cumulative success over short memory windows, and escalation during extended streaks. Across analyses, we estimate regression-based models with appropriate clustering and controls to isolate behavioral updating from contextual factors. The results show clear evidence of asymmetric updating: defenders tighten coverage more after made shots than they relax following misses, with effects strongest for the closest defender. Defensive adjustments are driven primarily by short recency windows of approximately three to five shots. Escalation during long hot streaks is limited, with defensive responses stabilizing rather than increasing indefinitely. Together, these findings suggest that defensive behavior reflects psychologically structured but bounded belief updating rather than full rational inference or indiscriminate overreaction to streaks.

Keywords

Hot-hand effect belief updating sports psychology basketball analytics statistical analysis

Introduction

Understanding how humans perceive, interpret, and respond to short-term fluctuations in performance is a central problem in psychology and decision-making. Competitive sports offer a uniquely rich natural setting for studying these processes, as athletes and opponents must continuously adapt their behavior under pressure and uncertainty. Among the most enduring and controversial phenomena in this domain is the hot-hand effect. This hypothesis revolves around the belief that recent success increases the likelihood of future success. While decades of research have debated whether the hot hand reflects a real performance effect or a cognitive illusion, far less attention has been paid to how beliefs about streaks shape the behavior of defenders tasked with responding to perceived offensive threats.

Beyond its traditional framing, the hot-hand effect can be understood as a specific, streak-based manifestation of the broader construct of psychological momentum. Psychological momentum refers to a dynamic process whereby initial success increases the subjective probability of future success, enhances perceived competence, and alters both individual and opponent behavior. As conceptualized by Iso-Ahola and Dotson (2016), momentum operates not only through cognitive inference but also through affective and behavioral channels, shaping how both performers and observers interpret unfolding events. From this perspective, defensive responses to perceived hot-hand streaks can be viewed as adaptive reactions to evolving momentum rather than isolated reactions to discrete outcomes.

The modern hot-hand debate originated with the influential work of Gilovich et al. (1985), who argued that apparent shooting streaks in basketball arise from random variation misinterpreted by observers. This conclusion aligned with broader psychological theories emphasizing representativeness heuristics and recency bias in sequential judgment (Bar-Hillel and Wagenaar, 1991; Kahneman and Tversky, 1972). For many years, this view dominated both psychology and sports analytics, framing the hot hand primarily as a cognitive fallacy rather than a behavioral regularity.

More recent work, however, has substantially revised this conclusion. Miller and Sanjurjo (2018) and Parsons and Rohde (2015) demonstrated that earlier statistical tests were systematically biased against detecting streak dependence, reopening the question of whether hot-hand effects exist at all. Building on this, several studies have shown that accounting for shot difficulty, defensive pressure, and contextual factors reveals meaningful streak-dependent performance changes (Bocskocsky et al., 2014; Yaari and Eisenmann, 2011). As a further extension, Kondur and Shen (2025) provide compelling evidence that once shot difficulty is properly controlled for, hot-hand effects in professional basketball are both statistically and practically significant. Their work underscores a critical methodological insight: observed outcomes are inseparable from the defensive and spatial context in which they occur.

Parallel work leans away from the traditional statistical frameworks developed and utilized in the aforementioned literature. Instead, an increasingly prominent approach models shooting performance through latent space dynamics. Rather than treating streaks as directly observable patterns, these models posit an unobserved performance state. This state evolves over time and can often be interpreted in the most basic sense as “hot” or “cold.” Discrete formulations, such as hidden Markov models, have been used to capture switching between latent performance regimes (Calvo et al., 2025; Sandri et al., 2020), allowing researchers to probabilistically characterize streaks as sequences of underlying states rather than realized outcomes. Related Bayesian approaches similarly compare constant-probability models to latent-state alternatives to test for streak dependence (Wetzels et al., 2016).

Complementing these discrete frameworks, continuous latent-variable models treat performance as a smoothly evolving process. For example, state-space formulations model shooting ability as a continuous stochastic process reflecting underlying form (Mews and Ötting, 2021). More recent work integrates offensive and defensive interactions into a joint latent momentum framework (Winkelmann and Michels, 2026). Together, these approaches provide a powerful statistical lens for detecting temporal dependence in performance, but they typically abstract away from the behavioral mechanisms and defensive adjustments that jointly generate observed outcomes. Notably, latent-variable approaches provide a complementary interpretation by formalizing “hotness” as an unobserved state rather than a directly measurable increase in success probability. While statistically powerful, such models remain agnostic about the cognitive and behavioral processes through which these states are perceived and acted upon by opponents.

The growing literature in this area, and the hot-hand space in general, has moved beyond simply asking only whether shooters become objectively better over a streak of made shots. Instead, it emphasizes how perceived momentum alters decision-making, attention, and risk assessment. Decisions from experience are known to overweight recent and salient outcomes, particularly in environments with rapid feedback and limited opportunity for deliberation (Barron and Erev (2003); Hertwig et al. (2004)). In sport, these dynamics are amplified by time pressure and role specialization. Defenders must rapidly infer threat levels based on sparse and noisy signals, making them especially susceptible to recency-weighted belief updating and asymmetric responses to success versus failure (Csapo and Raab (2014); Raab (2012)). Recent work further extends this perspective by emphasizing the interpersonal and biological foundations of momentum. Morgulev and Avugos (2020) argue that momentum is communicated through non-verbal cues and embodied signals, which can influence opponent behavior in real time. Drawing on concepts such as the “winner effect”, they suggest that perceived success can trigger physiological and behavioral changes that alter competitive interactions. In this framework, defenders are not merely updating beliefs from observed outcomes, but are also responding to subtle cues that signal confidence and threat.

Despite these insights, empirical work has largely focused on offensive behavior. This includes shot selection, efficiency, and streak persistence. Much of the current research landscape treats defensive responses as either static or exogenous. This omission is consequential, especially when considering the use of standard latent-variable approaches. They typically offer little insight into the behavioral tendencies of offenses and defenses alike. We must examine the hot-hand phenomenon as a task that inherently focuses on both sides of the game, not just one that models the offensive aspect. Indeed, if defenders adjust coverage based on perceived streaks, then observed shooting outcomes are jointly determined by offensive performance and defensive belief updating. Advances in player-tracking data and spatial analytics now allow these defensive adjustments to be measured directly. Prior work has established defender distance and spatial configuration metrics as meaningful indicators of defensive pressure and team coordination (Franks et al. (2015); Goldsberry (2012)), but these tools have rarely been integrated with psychological theories of memory, recency, and adaptive expertise. Some early work shows that players and coaches adjust decisions in response to perceived hotness using spatio-temporal data (Attali, 2013). However, such work still does not examine defensive spatial adjustments, especially at the level of granularity now possible with more modern tracking data.

Moreover, an important conceptual distinction remains underexplored: individual defenders and team defenses may operate on different cognitive and temporal scales. Psychological theories of skilled performance emphasize that individual actors rely heavily on short-term memory and cue-based heuristics, whereas team coordination reflects slower, distributed processes shaped by shared roles and tactical structure (Eccles and Tenenbaum (2004)). Further, Greve et al. (2019) provides tangential evidence that shows players are significantly more likely to commit fouls immediately after losing possession, particularly against the opponent responsible for the turnover. Although not directly tied to shooting streaks, the overarching idea highlights the fact that defensive actions reflect perceived responsibility. This reinforces the idea that individual responses may be driven by asymmetric psychological reactions rather than objective changes in opponent actions. Whether these levels of individual versus team response exhibit different sensitivity to recent success and whether they move in the same or opposite spatial directions remain open.

Our paper addresses these gaps by examining how NBA defenses adjust dynamically to recent shooting outcomes using high-resolution spatial defensive metrics and regression-based statistical modeling. Rather than asking whether shooters truly get “hot”, we focus on whether and how defenders believe they do. More importantly, we consider how those beliefs manifest behaviorally. Across three complementary studies, we analyze (i) asymmetric defensive responses to immediately preceding makes versus misses, (ii) cumulative defensive adjustments over short shot windows, and (iii) escalation patterns during sustained shooting streaks. Our results show that defensive updating is asymmetric, strongly recency-weighted, and sharply differentiated between individual and team-level behavior. Individual defenders tighten coverage quickly following recent success but do not exhibit runaway escalation. On the other hand, team defenses respond more gradually and often in the opposite spatial direction. Together, these findings position defensive adaptation as a psychologically grounded process shaped by perception, memory, and coordination under competitive pressure.

Materials and methods

In this section, we describe the data and analytic framework used to study how defenses adjust to offensive output in the NBA at both an individual and team-based level. The first subsection will describe the data collection and processing pipeline, while the second will detail the formulation of three complementary regression-based studies designed to capture aspects of history-based defensive behavior.

Data collection

Our data collection consists of merging NBA “moment” and play-by-play data taken from two different sources. Play-by-play data can be found publicly on Kaggle (Walsh, 2023) and moment data can be found publicly on GitHub (Seaward, 2018). Movement data, framed as moment data, has only been made publicly available for the early part of the 2015-2016 NBA season as the original host removed public access to player tracking data midway through the season. In specific, the timeframe of interest is regular season games from October 2015 to January 2016, as we have complete moment and play-by-play data during this interval.

The original moment dataset (Seaward, 2018) contains a collection of ZIP files, each of which corresponds to a single game in the 2015-2016 NBA season and contains a JSON object. This JSON object contains metadata of the game itself, such as a unique identifier, the date of the game, team identifiers, and a list of players who participated in the game. The information we are more interested in is labeled as “moment” information. The entirety of the game is split into “moments”, which are segments of $0.04$ seconds. For each moment, the JSON object stores the quarter number, how much time is left in the quarter, remaining time on the shot clock, as well as spatial coordinates for the ball and each of the 10 players on the court.

The original play-by-play dataset (Walsh, 2023) contains information for games from the inaugural 1946-1947 season and onwards. Specifically, it contains over 13 million rows and 37 columns. This includes information related to game metadata, codes for the type of event that took place in a certain play, and identifying information for the players involved in the play. Each play is allowed to have up to three players involved. Table 1 provides a more thorough description of each column in the play-by-play dataset.

Table 1.

Original play-by-play dataset column descriptions.

Column Name	Column Description
game_id	Unique ID for the game
eventnum	Sequential number of event within the game
eventmsgtype	Categorical code indicating type of event
eventmsgactiontype	Categorical code related to event message action type
period	Quarter number (1–4 for regulation, 5 for OT, 6 for 2OT, etc.)
wctimestring	Wall clock time, i.e. real-world time
pctimestring	Time remaining in the quarter on game clock
homedescription	Description of action taken by player on home team (potentially empty)
neutraldescription	Description of any neutral action (e.g. tip-off; potentially empty)
visitordescription	Description of action taken by player on visitor team (potentially empty)
score	Game score, with higher score first
scoremargin	Score margin, from home team’s perspective
person ${p}$ type	Type of player $p$ , for $p \in {1, 2, 3}$
player ${p}$ _id	Unique ID for player $p$ , for $p \in {1, 2, 3}$
player ${p}$ _name	Name of player $p$ , for $p \in {1, 2, 3}$
player ${p}$ _team_id	Unique ID for team of player $p$ , for $p \in {1, 2, 3}$
player ${p}$ _team_city	City for team of player $p$ , for $p \in {1, 2, 3}$ (e.g. Detroit, Houston)
player ${p}$ _team_nickname	Nickname for team of player $p$ , for $p \in {1, 2, 3}$ (e.g. Pistons, Rockets)
player ${p}$ _team_abbreviation	Abbreviation for team of player $p$ , for $p \in {1, 2, 3}$ (e.g. DET, HOU)
video_available_flag	Boolean value indicating if original play-by-play source has video available

The data processing pipeline works in seven steps and Figure 1 provides a visual of the ordering of these steps. Here, we provide a more detailed description of each data processing step. Since the moments dataset is stored as a folder of ZIP files, the primary processing step that we perform is extracting the underlying JSON file from each ZIP file and converting it into a tabular dataset with columns related to remaining clock time, player locations, and ball location. For empty JSON files, the corresponding game is skipped and not included in the final dataset. The moments dataset contains overlapping data, so these are dropped to avoid redundancy. This process results in data being stored for 632 games.

Figure 1.

Data processing pipeline.

Processing the play-by-play dataset is a bit more involved. The first step we take has two parts. We perform filtering to include only games in the 2015-2016 season between October 27, 2015 (i.e. the start of the season) and January 23, 2016. Recall that this timeframe is required since moment data only exists within this time range. We then perform additional filtering to only keep plays that correspond to shots (excluding free throws). This results in a dataset of shape $89478 \times 37$ . Much of the columns here are also irrelevant, so the next step is to keep only the relevant columns, with some renaming and data type conversions. This gives a dataset of shape $89478 \times 9$ .

The next step is to add shot history columns. This includes adding a shot number column that resets for each player after each game. We also add columns to track the result of the $k^{th}$ previous shot and the total number of made shots in the last $k$ shots (for $k \in {1, 2, \dots, 10}$ ). This data processing step is modeled by that which was done in Kondur and Shen (2025). We also add a column that tracks the number of consecutive made shots for that player within a game. These changes result in a dataset of shape $89478 \times 31$ .

The final step is to add player location and defender information. This is done by merging the moments dataset with the current play-by-play dataset using the moment corresponding to each shot in the latter. The complication is that there is usually not an exact match given the shot data. To find the exact moment corresponding to the shot release, we use the moment for the earliest time before the shot time in which the given shooter has possession of the ball and the ball height is monotonically increasing. Merging the moment and play-by-play dataset in this manner results in the final dataset of interest, which has shape $89478 \times 50$ . Table 2 provides a full description of each of the columns in this dataset. Summary statistics are provided in Table 3, with standard deviation omitted for binary indicator variables since it can be mechanically determined by the mean.

Table 2.

Final curated dataset column descriptions.

Column Name	Column Description
game_id	Unique ID for the game
game_date	Date on which game was played
event_id	Unique ID for the shot event within the game
player_id	Unique ID for the shooter
team_id	Unique ID for the shooter’s team
period	Quarter number (1-4 for regulation, 5 for OT, 6 for 2OT, etc.)
seconds_rem	Number of seconds remaining in the game
3pt	Boolean value representing if the shot was a 3-pointer
fgm	Boolean value representing if the shot was made
shot_number	Index of the shot taken by the player within the game
fgm ${k}$	Boolean value representing if the shot taken $k$ shots ago was made (for $k \in {1, 2, \dots, 10}$ )
tot ${k}$	Number of shots made in the last $k$ shots (for $k \in {1, 2, \dots, 10}$ )
streak	Number of consecutive shots made by the player before the current one
shot_dist	Shot distance (in feet)
shot_clock	Number of seconds left on the shot clock
close_def_id	Unique ID for the closest defender to the shooter
close_def_team_id	Unique ID for the closest defender’s team
close_def_dist	Distance (in feet) between the shooter and closest defender
avg_def_dist	Average distance (in feet) between all defenders and the shooter
def_hull_area	Area (in square feet) of the convex hull created by defender locations
shooter_x	Horizontal (x) coordinate of the shooter
shooter_y	Vertical (y) coordinate of the shooter
def ${d}$ _x	Horizontal (x) coordinate of defender $d$ (for $d \in {1, 2, \dots, 5}$ )
def ${d}$ _y	Vertical (y) coordinate of defender $d$ (for $d \in {1, 2, \dots, 5}$ )

Table 3.

Summary statistics for select columns in final curated dataset. Standard deviation is not reported for 3pt and fgm since they are binary indicator variables.

Column Name	Mean	Standard Deviation	Minimum	Maximum
seconds_rem	348.13	208.29	0.00	720.00
3pt	0.28	—	0.00	1.00
fgm	0.45	—	0.00	1.00
streak	0.66	1.04	0.00	12.00
shot_dist	14.62	9.26	0.00	47.53
shot_clock	13.83	6.44	0.00	24.00
close_def_dist	5.59	4.15	0.00	59.62
avg_def_dist	14.78	6.25	2.03	74.44
def_hull_area	200.18	150.41	0.78	2617.70

Study formulations

We outline the statistical models used to quantify defensive reactions to recent shooting outcomes. Specifically, our focus is on asymmetric belief updating, cumulative recent success, and escalation with streak length. Across these three studies, we estimate fixed-effects regression models that relate a specific defensive metric to shot covariate information. The models are built with controls for contextual game and player information. Each study uses data from the finalized dataset described in the previous section. This dataset contains all the game-level, shot-level, and player-level data needed to perform the studies as described below.

As our studies are focused on how defenses react to shooting performance, we consider a mixture of individual and team-based measurements. We use the closest defender’s distance to the shooter as our individual player defense metric. Although this is our primary focus, we supplement it with two team-based defense metrics: average defender distance to shooter and defender convex hull area. The convex hull of a set of five defenders can be easily defined from a computational geometric perspective as the smallest convex polygon that encloses the locations of those defenders.

The decision to include both individual and team-based defense metrics despite the prevalence of man-to-man defense in the NBA is an intentional one. While most teams do employ man-to-man defense, this description obscures the extent to which defensive responsibility is coordinated at the team level, especially when considering schemes that require help defense, switching, shading, and gap coverage. Indeed, recent work supports the line of thinking that defensive performance in team sports, especially basketball, must be thought of through collectivism rather than individualism (Franks et al., 2015; Gudmundsson and Horton, 2017). Capturing average defender distance provides a rudimentary scalar representation of defensive setup. On a more nuanced level, convex hull area captures a different spatial representation, which is important given that spatio-temporal data has been shown to impact individual and team-level defenses (Attali, 2013; Gudmundsson and Horton, 2017).

From a psychological perspective, distinguishing between individual and team-level responses is important in the context of belief updating and behavioral adaptation. If defenders respond to recent shooting outcomes purely at the individual level, we would expect adjustments to be localized to the closest defender. However, if perceptions of shooting streaks propagate through the defense, then systematic changes in team positioning should also be observable. Indeed, studies have already shown that team-level defense implementations and individual player defensive output respond differently to “hot” streaks (Csapo et al., 2015). Including both levels of measurement enables us to empirically assess whether defensive reactions are confined to individual matchups or reflect a more distributed response.

The regression models we use all follow a single, simple general form. For a shot $t$ taken by player $i$ , we estimate a defense metric $D_{i t}$ by the following general formula:

D_{i t} \sim f (r_{i t}) + X_{i t} β + α_{i} + ϵ_{i t} .

(1)

In Equation 1, $f (\cdot)$ captures how defenses respond to recent shooting outcomes $r_{i t}$ . We also let $X_{i t}$ be a vector of shot-based features with model coefficients $β$ , $α_{i}$ a vector of player fixed effects, and $ϵ_{i t}$ is an error term. Recall the list of features we presented in Table 2. Out of these, we use a subset to serve as the covariates $X_{i t}$ in our regression models (see Table 4). The general form of Equation 1 provides the basis for the questions we study in which we estimate specifications of $f (\cdot)$ . In the following subsections, we describe the studies of interest in more detail. We also omit the error term $ϵ_{i t}$ from practical formulas in order to simplify notation.

Table 4.

Basic feature descriptions for general OLS model.

Feature Name	Feature Description	Feature Type
player_id	Unique ID for the shooter	Categorical
period	Quarter number (1-4 for regulation, 5 for OT, 6 for 2OT, etc.)	Categorical
seconds_rem	Number of seconds remaining in the game	Numerical
shot_dist	Shot distance (in feet)	Numerical
shot_clock	Number of seconds left on the shot clock	Numerical

Study 1: Asymmetric responses to makes and misses

The first study we perform focuses on immediate recency bias. In this, we examine whether defenses respond asymmetrically to recent positive versus negative outcomes. Specifically, we wish to determine whether a made shot induces a stronger defensive reaction than a missed shot undoes. To do so, we add two columns to the initial shots dataset: make_last and miss_last. These are simply binary (numerical) features that take the value 1 if the immediately previous shot was made or missed, respectively, and 0 otherwise. With this, for each of the three defense metrics $D_{i t}$ (close_def_dist), avg_def_dist, def_hull_area), we fit an Ordinary Least Squares (OLS) model using Equation 1 with a linear combination of recent shooting outcomes $f (r_{i t}) = γ_{M} \cdot m a k e_l a s t + γ_{S} \cdot m i s s_l a s t$ . The full formula we use can be written as

\begin{matrix} D_{i t} \sim m a k e_l a s t + m i s s_l a s t + s h o t_d i s t \\ + s h o t_c l o c k + s e c o n d s_r e m + C (p e r i o d) \\ + C (p l a y e r_i d) \end{matrix}

(2)

This formulation allows defensive responses to makes and misses to differ in magnitude and direction. Psychological interpretations focus on whether defensive updating is symmetric, as would be implied by rational belief updating, or asymmetric, consistent with salience-driven or loss-averse attention mechanisms.

Study 2: Defensive responses to cumulative shooting success

To take this one step further, we expand the shot window in question to more than just the previously immediate shot. By focusing on multiple previous shots, our second study focuses on analyzing how and why defenses respond to cumulative shot successes in the recent past. To track the recent shooting success over a given window of $k$ previous shots, we use the $t o t k$ column. This acts as a rolling measure of recent shooting performance. Again, we use an OLS model using Equation 1, but this time with $f (r_{i t}) = C (t o t k)$ , since the measure for recent shooting performance is categorical and purely based on the total number of makes in the last $k$ shots. The full formula we use can be written as

\begin{matrix} D_{i t} & \sim C (t o t k) + s h o t_d i s t + s h o t_c l o c k \\ + s e c o n d s_r e m + C (p e r i o d) + C (p l a y e r_i d) \end{matrix}

(3)

This formulation allows us to determine two things. First, we can understand whether players and teams operate within a fixed shot “memory” timeframe. Second, we are able to investigate whether defenses respond to the quantity of successful shots rather than just individual outcomes.

As a point of consideration, we experimented with several different shot windows $k \in {1, 2, \dots, 10}$ and models outside of just an OLS model treating $t o t k$ as a categorical variable. The simplest alternative we considered was an OLS model treating $t o t k$ as a normal numerical variable. We also considered a Generalized Additive Model (GAM) framework with a B-spline (basis spline) basis of varying polynomial degree and degrees of freedom. Given that for each shot window, the AIC values of these models are relatively similar, as well as the simplicity of OLS models, we decided to move forward with the described OLS model with categorical recent shooting features. Note that we prefer AIC over BIC for model selection for two reasons. First, we are primarily interested in choosing the best predictive model and within models that perform similarly, interpretability is used as a secondary criteria. Second, with the implementations of our OLS and GAM models, BIC values operate on different scales, which makes one-to-one comparison difficult. Table 5 shows certain metrics for a select number of fitted models to close_def_dist for clarity in the decision to use the aforementioned model.

Table 5.

Metrics for models fit to predict close_def_dist from cumulative shot success data.

Model	Shot window k	AIC
OLS Categorical	3	285072
OLS Categorical	5	204708
OLS Categorical	10	75785
OLS Numerical	3	285069
OLS Numerical	5	204702
OLS Numerical	10	75786
GAM (deg $=$ 3, df $=$ 5)	3	285097
GAM (deg $=$ 3, df $=$ 5)	5	204730
GAM (deg $=$ 3, df $=$ 5)	10	75787
GAM (deg $=$ 4, df $=$ 6)	3	285097
GAM (deg $=$ 4, df $=$ 6)	5	204732
GAM (deg $=$ 4, df $=$ 6)	10	75779

Study 3: Defensive escalation during shooting streaks

The natural next step is to consider how defenses react during sudden hot shooting streaks. Our third and final study focuses on consecutive made shots to test whether defensive responses escalate smoothly or exhibit threshold-like behavior. Using the streak variable defined in our shot dataset, we fit several OLS models to the data using some transformation of streak. These transformations correspond to the function $f (r_{i t})$ . The transformations we consider are polynomials up to quartics, exponential, and basis splines with varying polynomial degrees and degrees of freedom. Table 6 shows certain metrics for each of the transformations used in the OLS model fit to the close_def_dist defense measurement. Similar tables were also created and analyzed for the team-based defense measurements, but omitted here for brevity and to avoid redundancy since the end output was largely the same as in Table 6.

Table 6.

Metrics for transformations in the OLS model fit to predict close_def_dist from shot streaks data.

Transformation	AIC
Linear	433264
Quadratic	433259
Cubic	433261
Quartic	433262
Exponential	433264
B-Spline (deg $=$ 2, df $=$ 4)	433261
B-Spline (deg $=$ 3, df $=$ 5)	433262
B-Spline (deg $=$ 4, df $=$ 6)	433264

Based on these metrics, as well as the interpretability of simple OLS models fitted using polynomials, we treat the poly_2 model as the best fit model for our study. This corresponds to the following function $f (r_{i t}) = γ_{1} \cdot s t r e a k + γ_{2} \cdot {s t r e a k}^{2}$ and the full formula we use can be written as

\begin{aligned} D_{i t} & \sim s t r e a k + {s t r e a k}^{2} + s h o t_d i s t + s h o t_c l o c k \\ + s e c o n d s_r e m + C (p e r i o d) + C (p l a y e r_i d) \end{aligned}

(4)

This formulation allows defenses to scale their response with the length of the streak, capturing whether longer streaks trigger disproportionately stronger reactions. The analysis is complemented by an assessment of whether observed defensive adjustments follow a linear pattern or exhibit nonlinear escalation consistent with “panic” or overreaction after extended streaks.

Together, these three studies provide a unified framework for examining how basketball defenses update their behavior in response to recent offensive performance across multiple time scales. The asymmetric make–miss analysis isolates immediate, outcome-specific reactions consistent with recency bias, while the cumulative success models test whether defenses condition on short-term shooting form rather than single events. The streak-based analysis extends this logic by examining whether defensive responses escalate as evidence of sustained success accumulates. Collectively, these approaches allow us to distinguish rational defensive adjustment from psychologically driven overreaction, offering insight into how perceptions of momentum shape real-time defensive decision-making.

Results

This section presents the empirical findings from the three complementary studies examining how NBA defenses adjust to recent shooting outcomes as described in the previous section. We split the presentation of these results into three subsections, one for each study. These are ordered in increasing relevance to the psychology of defenses in reaction to peak performance (i.e. hot shooting streaks). Together, these analyses characterize both individual and team-level defensive adaptation across short- and intermediate-term memory horizons.

Study 1: Asymmetric responses to makes and misses

Recall that our first study considers the most simplest of effects: how does an immediately previous make or miss by an offensive player result in defensive adjustments, if any occurs? We estimate an OLS regression model using Equation 2 and are primarily interested in the coefficients ascribed to the make_last and miss_last features. Denote these coefficients as $γ_{M}$ and $γ_{S}$ , respectively. It is not merely enough to just find these values. In order to make any meaningful interpretations, we must determine whether the coefficients themselves are statistically significant. To make this determination, we use an F-test across three different hypotheses.

We first want to know whether making or missing the immediately previous shot has some type of effect on defensive measures. This is easily captured by the null hypotheses $γ_{M} = 0$ and $γ_{S} = 0$ . We also want to find whether defenders react symmetrically based on shot outcomes. For this, we examine the third null hypothesis: $γ_{M} = γ_{S}$ . Table 7 shows the OLS model results, complete with the fit coefficients, excluding the coefficients for the categorical player_id variable. For the categorical period variable, we report a coefficient for each eligible value except the first period, which is treated as the baseline group. Table 8 shows the F-test p-values for the above null hypotheses related to the coefficients corresponding to the make_last and miss_last variables (i.e. $γ_{M}$ and $γ_{S}$ , respectively).

Table 7.

Coefficients for each non-player_id variable in the OLS model.

	avg_def_dist	close_def_dist	def_hull_area
Intercept	$4.747$	$0.747$	$49.431$
make_last	$2.377$	$0.314$	$25.428$
miss_last	$2.370$	$0.433$	$24.003$
shot_dist	$0.399$	$0.293$	$2.123$
shot_clock	$0.167$	$0.109$	$5.357$
seconds_rem	$- 0.003$	$- 0.002$	$- 0.018$
period=2	$- 0.053$	$- 0.020$	$1.554$
period=3	$0.077$	$0.008$	$4.559$
period=4	$0.004$	$- 0.132$	$11.772$
period=5	$- 0.108$	$- 0.391$	$24.884$
period=6	$- 0.397$	$- 0.340$	$- 1.745$
period=7	$1.033$	$0.209$	$42.861$
period=8	$0.119$	$- 0.570$	$2.839$

Each row corresponds to an independent variable in equation 2 and each column corresponds to a dependent variable (i.e. defense metric).

Table 8.

F-test p-values for three null hypotheses comparing $γ_{M}$ and $γ_{S}$ , which are the coefficients of make_last and miss_last in the OLS models separately fit to the three defense metrics of interest.

	avg_def_dist	close_def_dist	def_hull_area
$H_{0} : γ_{M} = 0$	$6.25 \times 10^{- 224}$	$1.849 \times 10^{- 21}$	$8.692 \times 10^{- 81}$
$H_{0} : γ_{S} = 0$	$3.744 \times 10^{- 232}$	$1.405 \times 10^{- 30}$	$1.509 \times 10^{- 74}$
$H_{0} : γ_{M} = γ_{S}$	$0.855$	$2.840 \times 10^{- 6}$	$0.184$

Interestingly, we see that there is a positive association across all metrics regardless of the previous shot’s success. Indeed, the F-test results for the single-outcome effects in Table 8 show that shot outcome does have a statistically significant effect across all defensive metrics. These results also indicate that only close_def_dist has a statistically significant asymmetric response between previously made and missed shots. Moreover, a made shot results in significant tightening of individual defenders, whereas the same claim cannot be made for team defenses as a whole.

While at first glance it may appear counterintuitive that defender distance increases following both made and missed shots, this pattern is consistent with the temporal structure of defensive possessions. Regardless of outcome, immediately following a shot, defenders often shift their overall shape and structure based on transition coverage, reassignment, and overall defensive strategy, rather than maintaining tight proximity to the original shooter. As a result, the observed increase in defender distance may reflect a short-lived reorganization of defensive coverage rather than a deliberate reduction in perceived threat.

Importantly, this interpretation does not contradict the presence of asymmetric updating. Although both makes and misses are associated with increased defender distance, the magnitude and statistical significance of these effects differ at the individual level. This is evidenced by the significance of the p-value in the F-test for the $H_{0} : γ_{M} = γ_{S}$ null hypothesis for the close_def_dist metric in Table 8. The main takeaway here is that individual defenders still respond more strongly to recent successes than failures when re-engaging with the shooter.

This asymmetry is consistent with a recency-biased updating process in which salient successes disproportionately influence defensive perception of threat. From a psychological perspective, made shots appear to elevate perceived shooter danger more rapidly than missed shots attenuate it, suggesting that defenders overweight recent successes when adjusting coverage. However, this would obviously only affect the closest defender to a shooter. Mentally, they are the sole player dealing with the offensive threat, which explains why team defensive metrics do not see much change based on previous shot success. The immediate impact of this study’s results is twofold. First, immediately previous shot outcomes have a measured impact on individual defensive adjustments rather than team-wide adjustments. Second, players psychologically assign higher importance to previously made shots than missed shots, and subsequently react with a higher magnitude.

Study 2: Defensive responses to cumulative shooting success

Moving past the single-shot framework, our second study considers variable-length shot windows. Recall that we again use OLS regression models given by Equation 3 in order to determine whether players react based on a certain shot window length and whether cumulative success plays a role in defensive adjustments. To answer the first question, we observe that based on Table 5, the best fit model appears to be that which uses a shot window of $k = 10$ . This occurrence is paralleled when considering the team-based metric avg_def_dist as well. Of course, this would indicate some type of subconscious long-term shot memory in which players use the cumulative shot success of the previous 10 shots in order to adjust defensively.

To understand whether this shot window is actually relevant, we first turn to a visual interpretation of the data. Figure 2 shows both the fit model coefficients for each number of made shots in the last $k = 10$ , as well as the predicted closest defender distance by the model. The shaded area represents the $95 %$ confidence interval.

Figure 2.

OLS model coefficients and predicted closest defender distance by number of made shots in the previous $k = 10$ shots; $95 %$ confidence interval.

With this, we are able to see that closest defender distance decreases sporadically while players have made up to 6 of their last 10 shots, whereas beyond that, closest defenders tend to separate away from the shooter. This initially may seem counterintuitive, but psychologically, we provide a different interpretation. Indeed, this visual finding may indicate the usage of a shorter shot window length. Logically, this also makes sense as keeping a running count of how many shots a player has made in their last 10 is quite difficult in the run of play. As such, even though our OLS model with $k = 10$ seems to be the best fit, it may be be a result of overfitting. This is consistent with the fact that as $k$ increases, models within a given class tend to be better fits in terms of AIC (see Table 5 for metrics).

From this perspective, the improved fit of the $k = 10$ model is best understood as capturing additional aggregation over recent possessions rather than evidence that defenders explicitly track ten-shot histories. Instead, the structure of the estimated effects is consistent with a shorter effective memory horizon, in which only a subset of the most recent outcomes meaningfully influences defensive adjustment. These results provide diagnostic evidence that defensive updating is not driven by a long cumulative count, but instead by a compressed and effectively short-horizon representation of recent shooting performance.

We thus turn to a model that uses a shorter shot window length of $k = 5$ . Figure 3 shows the model coefficients and predicted closest defender distance for this shot window length. Figure 4 shows the same for average defender distance. Again, both use a $95 %$ confidence interval. It becomes a lot clearer visually that there is a distinct effect on cumulative shot success on individual and team defensive adjustments.

Figure 3.

OLS model coefficients and predicted closest defender distance by number of made shots in the previous $k = 5$ shots; $95 %$ confidence interval.

Figure 4.

OLS model coefficients and predicted average defender distance by number of made shots in the previous $k = 5$ shots; $95 %$ confidence interval.

With this, we can make the claim that for individual defense, the strongest effects are observed at short memory windows. When shooters make more shots out of their previous five attempts, closest defender distance decreases significantly, indicating heightened defensive pressure. This effect weakens at longer windows, and for $k = 10$ , the relationship becomes non-monotonic, with defenders sometimes tightening even when cumulative success is low. This could be part of a broader preventative rather than reactive strategy.

While this answers the shot window aspect of our study, we are yet to show that cumulative success has an effect on defensive adjustments. Of course, the trend is clear to see visually, but we can strengthen the standard visual interpretation a bit more rigorously. We first run a joint significance ordered difference F-test on the OLS categorical model (with $k = 5$ shot window) to determine whether cumulative success has a statistically significant effect on defender metrics. For a shot window of $k = 5$ , there are $k + 1$ possible categories of cumulative shot totals (i.e. zero to five made shots out of five). This corresponds to six coefficients in the OLS model, which we denote by $γ_{0}, γ_{1}, \dots, γ_{5}$ . The F-test determines significance of the ordered differences and the null hypotheses are of the form $H_{0} : γ_{i + 1} - γ_{i} = 0$ for $i \in {1, 2, \dots, 5}$ . We go beyond this and perform a linear trend test to determine whether there is a monotonic linear trend on the differences found earlier. The p-values for both of these tests are listed in Table 9.

Table 9.

F-test and trend test p-values for OLS categorical model with $k = 5$ fit to cumulative shot success data.

Defensive Metric	F-test p-value	Trend test p-value
avg_def_dist	0.158	$3.750 \times 10^{- 111}$
close_def_dist	0.002	$9.431 \times 10^{- 38}$

Using an alpha level of $α = 0.05$ , we find that the F-test and trend test both give statistically significant results for the close_def_dist metric, but only the former is significant for the avg_def_dist metric. We interpret this to mean that cumulative shot success has a definite impact on how individual players adjust and that effect reveals itself in a monotonic decreasing manner during short window lengths. However, only the trend test reports significant results for the team-based defense metric (avg_def_dist). We take this to mean that team defenses shift monotonically, but the individual shifts in between shot success totals are not as meaningful.

Ultimately, this study provides two meaningful insights from a psychological perspective. First, individual defenders operate, perhaps subconsciously, based on short window lengths of past shot outcomes. Second, within these short sequences, defenders become increasingly close to shooters as past success increases. Tangentially, this overall trend is the opposite for overall team defense as average defender distance increases. This apparent loosening of the defense is not as significant and can potentially be attributed as a consequence of secondary defenders playing help defense as the primary defender tightens their defense.

Study 3: Defensive escalation during shooting streaks

The previous study focused on cumulative success over specific fixed length shot sequences. While this is a good start, in general, this does not properly encapsulate heat due to two main issues. First, it is hard for players to remember the number of shots an opponent has made over a certain window, especially when that window is of medium length (e.g. $k = 5$ ) or more. Second, the previous study involves comparisons between extremely cold streaks (zero makes in $k$ shots) and extremely hot streaks ( $k$ makes in $k$ shots). The formulation presented for our third study (see Equation 4) covers both of these issues by considering how defenses escalate action during increasingly longer hot streaks. Recall that the goal of this study is to determine whether such escalation even occurs, and if it does, at what rate or threshold it happens.

From Equation 4, the terms related to shot streaks are streak and ${s t r e a k}^{2}$ . Suppose in a given fit model that the corresponding coefficients are $γ_{1}$ and $γ_{2}$ . For our purposes, we need to make three checks. First, we must determine whether streaks matter at all for each defensive measure. This can be done using a joint significance F-test with a null hypothesis of $H_{0} : γ_{1} = γ_{2} = 0$ . If the null hypothesis is rejected, then we have proper evidence that a particular defensive measure varies with streak length. Second, we perform a general finite difference trend test to determine whether this variation is systematic. Note that this is different from what we did in the previous study since a general finite difference trend test does not assume linearity. Instead, we take differences in predicted defensive measures of varying streak lengths and the null hypothesis tests whether the average difference is equal to zero. Rejecting the null means a defensive measure varies systematically with streak length.

Finally, for directional monotonicity, we do not rigorously use any statistical test, but rather informal inferences of directional persistence and confidence interval bootstrapping. We define directional persistence as the rate of sign changes:

D P = \frac{1}{K - 1} \sum_{k = 2}^{K} 1 [d_{k} = d_{k - 1}]

where

K

is the maximum shot streak length,

d_{k} = sign (Δ_{k})

and

Δ_{k} = f (k) - f (k - 1)

is the difference in the predicted defensive measure using made shot streak lengths of

k

and

k - 1

. Persistence values closer to 1 indicate smooth (nearly monotonic) trends, whereas those less than 0.5 indicate oscillatory trends. As a robustness check, we also bootstrap

N = 1000

confidence intervals using predicted defensive measure values. If zero is excluded from the 95th percentile confidence interval, we infer that there is a directional trend. For each of the three defensive measures, we provide a summary of the results in Table 10.

Table 10.

Statistical and inference test results for OLS model with quadratic transformation fit to shot streak data.

Defensive Metric	F-test p-value	Trend test p-value	Directional persistence	95th percentile confidence interval
avg_def_dist	$0.190$	$2.331 \times 10^{- 4}$	1.0	$[0.063, 0.130]$
close_def_dist	$8.308 \times 10^{- 5}$	$0.025$	0.909	$[0.024, 0.128]$
def_hull_area	$0.352$	$1.306 \times 10^{- 8}$	1.0	$[0.474, 0.609]$

Immediately, we see that the F-tests’ null hypotheses cannot be rejected for the team-based defensive measure metrics. Although the visual trend displayed in Figure 5 shows an increase in team-based defensive measures, the lack of significant statistical evidence does not allow for a rigorous data-backed claim to be made here.

Figure 5.

Predicted defensive measures by OLS model fit to shot streak data; $95 %$ confidence interval.

However, we can make a strong claim regarding individual defensive adjustments. At the individual level, closest defender distance decreases following short streaks, indicating immediate defensive tightening after consecutive made shots. This effect does not scale beyond a certain streak length. For longer streaks, defenders begin to increase their distance relative to the baseline (streak = 0), suggesting regression to the standard defensive strategy rather than continued escalation. These findings do not suggest any such “panic” amongst individual defenders, but rather a gradual tightening of defense within short streak windows.

Across these three complementary analyses, we find consistent evidence that NBA defenses update dynamically in response to recent shooting outcomes, but in a manner that differs sharply between individual and team-level behavior. Defensive responses are asymmetric to makes and misses, reflect short to medium-length memory horizons for individual defenders, and escalate quickly over short streaks without runaway amplification. Together, these results highlight how recency and role specialization shape defensive decision-making under competitive pressure.

Discussion

In this paper, we examined how defenders adapt to offensive shooting performance in a dynamic environment. With this, professional basketball served as a naturalistic laboratory for studying belief updating, attention, and decision-making under uncertainty. Across three complementary analyses, we show that defensive adjustments are shaped not only by objective shooting outcomes, but by psychologically meaningful features such as outcome salience, recency, and perceived responsibility. Taken together, the results suggest that defensive behavior reflects bounded, asymmetric updating rather than fully rational inference or indiscriminate overreaction to streaks.

The first study provides clear evidence of asymmetric updating in response to recent outcomes. Defenders respond more strongly to made shots than they relax following misses, even when controlling for contextual factors. From a psychological perspective, this pattern aligns with extensive evidence that negative or costly outcomes carry greater informational and motivational weight than neutral or ambiguous feedback. Made shots are visually salient, socially reinforced events that directly signal defensive failure, whereas misses may be discounted as noise, luck, or poor shot selection. This asymmetry is consistent with loss-sensitive updating and recency bias, and suggests that defenders overweight confirmatory evidence of threat while underweighting disconfirming evidence. In essence, we concur with the purely psychological findings in Barron and Erev (2003) and Hertwig et al. (2004).

Importantly, this asymmetric response is most pronounced at the level of the closest defender and attenuates at the team level. This divergence highlights the role of perceived responsibility and attentional focus in shaping defensive cognition. Individual defenders are directly accountable for their matchup and therefore engage in sharper belief updating, while team-level positioning reflects a more distributed and indirect response. Psychologically, this suggests that defensive learning is localized: belief updates are strongest where responsibility is clearest and weakest where accountability is vague.

The second study investigates how defenders integrate information over short memory windows and reveals strong evidence for limited, recency-based updating. Individual defenders adjust most clearly based on the previous three to five shots, tightening defense as recent success accumulates. This window aligns closely with known constraints on working memory and temporal integration in fast-paced tasks. While longer memory windows yield statistically smoother relationships, they are less likely to reflect explicit or even implicit tracking by players. Instead, the results support the idea that defenders rely on a small, rolling sample of recent outcomes to guide behavior, consistent with heuristic-based updating rather than optimal Bayesian inference.

At the team level, cumulative success produces a different pattern: as shooters make more recent shots, average team defensive distance increases. This apparent paradox is psychologically interpretable. As the closest defender tightens coverage in response to perceived threat, other defenders can relax their help responsibilities and reallocate attention toward guarding their own assignments or protecting the paint. Thus, team-level spacing reflects second-order adjustments driven by individual reactions rather than a collective belief about shooter hotness. This distinction reinforces the idea that team defense emerges from coordinated but psychologically distinct individual decisions. This conclusion lies in agreement with Goldsberry (2012).

The third study examines whether defenders exhibit escalation during extended hot streaks. The results indicate that individual defenders respond rapidly at the onset of a streak but do not continue tightening indefinitely. Instead, defensive distance stabilizes or even reverts toward baseline at higher streak lengths. Psychologically, this pattern is consistent with threshold-based belief updating: once a defender is sufficiently convinced that a shooter is dangerous, additional confirming evidence produces diminishing behavioral change. This challenges popular narratives of runaway defensive overreaction and instead suggests bounded escalation.

More broadly, these findings contribute to debates in the psychology of performance about how humans adapt to streaky environments. Rather than fully ignoring streaks or irrationally overreacting to them, defenders appear to rely on salient cues, short memory windows, and responsibility-weighted updating. Basketball provides a rare setting in which these processes can be observed at scale, under real incentives, and with precise spatial measurement, offering insights that generalize beyond sport to other domains requiring rapid, adaptive decision-making.

Several limitations in our studies should be noted. Extreme streak lengths are rare, increasing uncertainty in estimates at the tails of the distribution. Such concerns have also been brought up in prior research (Miller and Sanjurjo, 2018). Additionally, while spatial defensive metrics capture behavioral adjustments, they cannot fully disentangle conscious strategy from automatic or habitual responses. Future work could incorporate defender identity, fatigue, or communication data to further isolate cognitive mechanisms, and extend this framework to other sports or competitive settings where belief updating under uncertainty plays a central role. Moreover, in the standard formulation of “heat” in a basketball setting, only three-pointers are considered. It would be interesting in future work to restrict similar studies to ours to only such long-range shots. This could effectively shed light onto whether defensive adjustments and behaviors differ when offenses are more centered around perimeter shooting. Finally, we also note that the data used is from a single NBA season. The game continues to evolve and it may be possible that gameplay now is based on different psychological and strategic factors than those presented in this paper.

Conclusion

This study used professional basketball as a naturalistic setting to examine how individuals and teams adapt to performance streaks under uncertainty. By combining spatial defensive metrics with regression-based inference, we show that defensive behavior reflects asymmetric, recency-weighted updating rather than indiscriminate reactions to streaks. Defenders respond more strongly to recent successes than failures, rely on short memory windows when adjusting behavior, and exhibit bounded escalation rather than panic during extended hot streaks. The results also highlight a clear psychological distinction between individual and collective responses. This separation suggests that what appears to be collective belief or strategy at the team level often emerges from coordinated but psychologically heterogeneous individual decisions. More broadly, the findings contribute to the psychology of peak performance by illustrating how belief updating, attention, and accountability shape behavior in fast-paced competitive environments. Beyond basketball, the framework and methods developed here offer a template for studying adaptive decision-making in other domains where outcomes are noisy, feedback is immediate, and performance depends on rapid interpretation of recent events.

Footnotes

ORCID iDs

Arya Kondur

Weining Shen

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