How does practice modulate fake-production costs in a basketball task? Analyses of frequency distributions and mixture effects

In recent years, researchers began to investigate the cognitive costs associated with performing deceptive actions, the so-called fake-production costs (Böer et al., 2024; Güldenpenning et al., 2023; Kunde et al., 2019; Wood et al., 2017). Fake-production costs for the head fake in basketball are evident in increased initiation times (ITs) and higher error rates (ERs) when executing a pass with a head fake as compared to a pass without a head fake. These costs can be reduced when participants are given time to mentally prepare the movement with Interstimulus Intervals (ISIs) of 800 ms and more (Güldenpenning et al., 2023). Fake-production costs for the head fake are suggested to reflect response-response incompatibility-costs, as they are also known from bimanual actions (Hazeltine, 2005; Heuer, 1995; Peterson, 1965) and similarly have been shown to be reduced by mental preparation time (Heuer et al., 1998). Here, response-response incompatibility-costs occur, for example, when moving two fingers simultaneously, but one vertically and one horizontally (incompatible action), as compared to moving two fingers simultaneously in the same direction with the same trajectory (compatible action, cf. Hazeltine et al., 2003). The head fake also requires the execution of two movements which are spatially incompatible, namely a head turn to one side (e.g., to the left) and a passing movement to the other side (e.g., to the right).

Until now, fake-production costs have been largely disregarded in sport psychology research (for some exceptions, see Böer et al., 2024; Güldenpenning et al., 2023; Kunde et al., 2019; Wood et al., 2017), even though they are of great relevance for sport practice. Specifically, the costs when performing a head fake can determine its benefit, as an increase in the initiation time of the attacking player by 100 ms has been shown to increase the chance that the opponent would anticipate a deceptive movement by 10% (Kunde et al., 2019). Accordingly, it seems worthwhile investigating how and why fake-production costs can be decreased.

One factor that can reduce fake-production costs is extensive practice (i.e., five consecutive days of practice). In a recent study, we showed that novice participants, who elicited fake-production costs when given no (ISI 0 ms) or only little time (ISI 400 ms) to mentally prepare performing the passes with head fakes stabilized their performance [reduction of the variation coefficient of initiation times (IT_cv)], especially from the first day of practice to the second day of practice, and reduced fake-production costs (Böer et al., 2024). Moreover, the study revealed that practice with the task led to a general improvement in ITs, independent of the type of pass, but also showed a greater improvement in ITs for passes with compared to passes without head fakes, as indicated by a reduction of the fake-production costs from day 1 to day 5 (see Fig. 5, p. 22). While participants were even able to eliminate these costs when given time to mentally prepare the movement (for an ISI of 400 ms), we only found a non-significant, descriptive reduction of fake-production costs through practice when participants had no mental preparation time (ISI 0 ms). Taken together, results suggested different effects of practice on the fake-production costs for the ISIs 0 ms and 400 ms. These effects of practice are most pronounced from day 1 to day 2, which was reflected by the steep learning curve well known for motor tasks (Dayan & Cohen, 2011; Hong et al., 2019). Thus, one day of practice substantially improves the ITs in passes with and without head fakes, however, as the learning curve flattens, significantly more practice might be necessary (i.e., over a longer period than 5 days) to achieve further visible increases in performance. However, it might also be that smaller effects of practice during day 2 and day 5 differ between individuals and/or are not visible in the mean value of the group. We therefore re-analysed the individual data of the participants of the study mentioned above (Böer et al., 2024) regarding so-called mixture effects (Miller, 2006), focusing on the ISIs 0 ms and 400 ms, where we found significant fake-production costs on the first day of practice.

The likelihood ratio test of mixture effects (Miller, 2006) is an analysis of the individual IT distributions. With this analysis at an individual level, we explore whether the fake-production costs from each participant stem from a relatively uniform slowing of all ITs when producing passes with head fakes compared to passes without head fakes (uniform effect), or from a combination of a minority of significantly slowed responses and a majority of equivalent ones (mixture effect). A uniform effect would suggest that response-response incompatibility-costs occur in most, if not all, trials (e.g., Compton & Logan, 1991; Osman et al., 2000). Conversely, a mixture effect would suggest that participants are capable of successfully executing passes with head fakes in most trials, and only sporadically exhibit the fake-production costs, but when they do, the extent is considerably larger. The analysis could help us understand if the reduction of fake-production costs through practice shown in our previous work (cf. Böer et al., 2024) differs between participants. The analysis may also uncover how cognitive processing evolves over the course of practice, for example, from a uniform effect where participants elicit fake-production costs in all trials to a mixed effect, where participants are able to overcome the fake-production costs in some trials.

Moreover, since the original study could not clarify whether the general improvement of participants’ ITs, and especially the reductions in fake-production costs at the two short ISIs (0 ms, 400 ms) were based on the effects of practicing the passes or attributable to test–retest effects, we collected data post-hoc from a control group. The control group performed exactly the same experiment (i.e., participants performed 320 trials of passes with and without head fakes with randomized ISIs of 0 ms, 400 ms, 800 ms, 1200 ms), but were only tested on day 1 and day 5 without practicing in between. The data from both groups was analysed with the between-subjects factor group (control group, practice group) and the within-subjects factors type of pass (pass without head fake, pass with head fake), ISI (0 ms, 400 ms, 800 ms, 1200 ms), and day (day 1, day 5). If the reductions seen in the mean ITs of participants in the practice group are (at least partly) connected to practice with the task, the control group should display less improvements between day 1 and day 5, reflecting test–retest effects. Therefore, we expected significantly greater reductions of ITs in the practice group compared to the control group from day 1 to day 5, both for passes with and without head fakes. Also, we expected a greater reduction of the fake-production costs in the practice group at the ISIs 0 ms and 400 ms compared to the control group. The comparison between the practice and control groups was further analysed through repeated measures ANOVAs and distribution analyses, as described below.

The analysis of the distribution of the initiation times was performed with the data of both groups, to further disentangle differences between them. Typically, reaction times (RTs)/ITs are not normally distributed but can be fitted to an ex-Gaussian distribution (Whelan, 2008), which is a mixture of a normal distribution and an exponential distribution. While most studies analyse mean RTs, this only reflects a precise value for the central tendency in the case of normally distributed distributions. However, RTs in experimental studies are often not normally distributed but also consist of an exponential part (Hockley, 1984; Hohle, 1965; Osmon et al., 2018; for more details see Luce, 1986). Therefore, an analysis of the whole RT distribution can yield valuable information about differences between conditions that may be overlooked in analyses focused solely on mean RTs.

The ex-Gaussian distribution can be described by the parameters mu and sigma, the mean and standard deviation of the Gaussian part, and the parameter tau, which represents the right tail of the distribution. The analysis of the ex-Gaussian parameters provides a succinct method to determine how the experimental manipulation impacts participants’ ITs. Specifically, this distribution analysis reveals whether the experimental manipulation shifts the IT distribution (i.e., a change in mu and/or sigma), elongates the tail of the distribution (i.e., a change in tau), or both. It has been proposed that the parameters of the ex-Gaussian distribution reflect cognitive processes, with some authors suggesting that the Gaussian part of the distribution (mu and sigma) encapsulate the aggregate of perceptual and motor processes (Kane & Engle, 2003; Luce, 1986; Mewhort et al., 1992). Tau, the exponential part of the distribution, has been discussed to reflect lapses of attention in some trials as evident in especially slow responses (i.e., the right tail of the distribution) and has been shown to be higher for cognitively demanding tasks (Heathcote et al., 1991; Hervey et al., 2006; Vasquez et al., 2018). However, some researchers have critically noted that it can be misleading to assume the parameters reflect specific cognitive processes and one has to be careful not to overinterpret them (Matzke & Wagenmakers, 2009; Osmon et al., 2018). But, when treated with some caution, the analysis of RT distributions can expand the knowledge generated by simple analysis of mean RTs and even uncover differences between experimental condition which are not visible in the analysis of mean values (Ratcliff, 1979).

Two key aspects of the ex-Gaussian parameters are relevant to this study:

Accordingly, the following predictions were made for the distribution analyses: Both parameters of the distribution variation (spread (sigma), skew (tau)) should reduce from day 1 to day 5 for both groups, reflecting a general familiarization with the task. We also expect greater reductions of sigma and tau for the practice group than for the control group, reflecting practice-related performance improvements exceeding simple test–retest effects. As we previously argued, we believe performing head fakes is connected to higher cognitive demands and should therefore be reflected in higher values of tau for passes with compared to passes without head fakes. We expect no modulation of mu by the interaction of the factor day and the factor group, as we believe the reduction of ITs with increasing practice reported for the practice group in Böer et al. (2024) reflects mainly a stabilization of performance through decreasing variance (i.e., familiarization) and a reduction of the number of trials with extremely slow responses.

The analysis of mixture effects (Miller, 2006) performed on the data of the practice group should complement this analysis of the IT distribution parameters by allowing insight into individual differences between participants. A uniform effect, where participants consistently elicit fake-production costs, should be visible in greater values of the distribution mean (mu). Conversely, a mixture effect, where participants only show fake-production costs in some trials, should be visible in greater values of the distribution skew (tau).

Participants stood 250 cm away from a screen wall, positioned at a purpose-built apparatus engineered for executing and measuring basketball passes, both with and without a head fake. The apparatus consisted of two steel holders flanking a desk, each equipped with buttons (height: 1.20 m; button separation: 1.25 m). To signify a pass to the left or right, participants utilized a basketball to depress these buttons. Additionally, a button on the desk in front of participants (height: 1 m; cf. Figure 1) was incorporated.

The static visual stimuli featured two distinct basketball players, one clad in a red shirt and the other in blue. The depictions captured the players executing a defensive manoeuvre towards one side, obstructing a potential pass with their body orientation and a raised hand on that side. Conversely, the opposite side remained unguarded, implying an opportunity for a pass (cf. Figure 1). These visual stimuli were projected onto a screen wall (height: 140 cm, width: 200 cm) before the participants, employing an Optoma X320 projector. On the first day of practice, participants were shown four brief videos of a professional basketball player executing a pass with or without a head fake to the left or right side. This was done to familiarize them with the movements they would be required to perform. Subsequently, participants were handed a basketball and instructed to place it onto a button on the desk in front of them, referred to henceforth as the starting position. Additionally, participants were equipped with a black cap adorned with a white stripe along the middle of the visor (posterior to anterior). This cap, coupled with a camera positioned above the participants' starting point, facilitated the analysis of head movements for each trial. Participants were then instructed to perform a passing action to the side that was not defended by the basketball player presented in the visual stimuli. The pass, contingent on the color of the basketball player's shirt, was to be executed with or without a head fake. Participants received instructions to initiate both head and basketball movements simultaneously for both pass types. The allocation of passes with or without head fakes to the corresponding shirt colors was counterbalanced across participants. Execution of the pass was to occur only upon the presentation of an auditory GO-signal (300Hz, jigsaw soundwave), with participants initiating the passing action only after planning their reaction. This GO-signal was either presented simultaneously with the visual cue (ISI 0 ms) or 400 ms, 800 ms or 1200 ms after the stimulus onset.

Each trial initiated with an instruction displayed on the screen, prompting the participant to place the basketball on the start button. This action signalled the commencement of the respective trial. Initially, a white fixation cross appeared for 500 ms at the centre of the screen, succeeded by a 500 ms blank screen before the display of the basketball player target stimulus. The target stimulus persisted on the screen until participants utilized the basketball to press one of the response buttons (on the left or right side of the basketball apparatus). In case participants responded before the auditory GO-signal, the German words “Zu schnell” (too fast) were presented for 1000 ms. Following the participants' passing movement to the left/right, the instruction to place the basketball on the start button reappeared, marking the commencement of the next trial. The trial sequence is illustrated in Fig. 2.

On the first day of practice, participants began the experiment with a practice block of 32 trials in a fixed order and received feedback on their pass direction and head movement after each trial from the instructor. If a participant struggled with performing the correct head turning and passing movement even at the end of the practice block (4 or more wrong answers in the last 8 trials), the participant had to complete the practice block again. After the practice block was completed, the participants performed 4 training blocks of 80 trials each. The trials varied with regard to type of pass (pass with head fake vs. pass without head fake) and ISI (0 ms, 400 ms, 800 ms, 1200 ms), and were presented in randomized order. Thus, each condition was repeated ten times within a block. The direction of the pass (i.e., left vs. right) was evenly distributed and not manipulated as an experimental factor. On the second through fifth days of training, participants of the practice group exclusively undertook a brief practice block of 8 trials in a fixed order before engaging in the four subsequent training blocks of 80 trials each. Participants of the control group did not practice on days 2, 3 and, 4 but performed the same four blocks of 80 trials each at day 5.

Part I is dedicated to the analysis of mixture effects (Miller, 2006) in ITs to evaluate if practicing the task affects performance on an individual level. IT is the time interval between presentation of the GO-signal and the point in time when participants lifted the ball from the starting position. The data for this analysis is taken from the study of Böer et al. (2024). An overview of the ITs, separately for the practice and the control group, in dependence of day¹ (day 1, day 5), ISI (0 ms, 400 ms, 800 ms, 1200 ms), and type of pass (pass with head fake, pass without head fake) is shown in Fig. 5 (section Results part II; Comparison of the mean ITs between the practice and the control group).

While Miller’s MixTest does not require a specific distribution for the reaction times, it is designed to work with the typical right-skewed distributions observed in RT/IT data, which are often well-described by an ex-Gaussian distribution (Miller, 2006; Whelan, 2008). Ex-Gaussian distributions are a mixture of a normal distribution and an exponential distribution. Before computing the MixTest, we first evaluated whether a Gauss function or an ex-Gauss function describes the IT data better (see section below). By doing so, we also extracted the values for the probability distribution family (here ex-Gaussian distribution), of which at least estimates for the parameter values mu, sigma, and tau were required to run the MixTest.

Table 1 summarizes the results of the mixture analysis for the ISI of 0 ms calculating whether there was a uniform or significant mixture effect. For 14 of 24 participants (participants 2, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 21, 22, 23), the assumption of a uniform effect could not be rejected (see Table 1). The other 10 participants (1, 3, 4, 7, 8, 11, 15, 16, 17, 24) showed statistically significant mixture effects by the likelihood ratio test aggregated across all sessions. None of the participants consistently exhibited mixture effects across all practice days and the number of participants showing a mixture effect even reduced from day 1 to day 5 (4, 3, 4, 2, 0). Interestingly some participants did not show fake-production costs in some sessions, with participants 7, 14 and 23 having higher ITs for passes without than for passes with head fakes on multiple days of practice.

The mixture model analysis also provides an estimate of the proportion of the experimental trials in which participants showed fake-production costs, the mixture proportion (P, see Table 1). Participants 11, 12, 17, and 24 had an aggregated mixture proportion below 0.75, which means that they showed fake-production costs in only 68%, 63%, 57%, and 61% of the times, respectively, when they performed a pass with a head fake at an ISI of 0 ms. All other participants have an aggregated mixture proportion > 0.75, which means that the majority of participants showed fake-production costs slowing their ITs down in over 75% of trials where they played a pass with compared to a pass without a head fake.

Table 2 summarizes the results of the mixture analysis for the ISI of 400 ms. Here for only 8 of 24 participants (participants 4, 7, 9, 10, 14, 17, 19, 23), the assumption of a uniform effect could not be rejected (see Table 2). One participant did not show fake-production costs at any day of practice at the ISI 400 ms and therefore the mixture analysis could not analyse the data properly (no categorization to mixture or uniform effect possible). The other 15 participants (1, 2, 3, 4, 5, 6, 11, 12, 13, 15, 16, 18, 20, 21, 22, 24) showed statistically significant mixture effects by the likelihood ratio test aggregated across all sessions. None of the participants consistently exhibited mixture effects across all practice days and the number of participants showing a mixture effect did not change greatly from day 1 to day 5 (6, 6, 5, 7, 5). The analysis also revealed an increasing number of bad effects with increasing practice from day 1 to day 5 (3, 7, 11, 12, 12). This increase in bad effects in the mixture model indicates that there might not be a consistent difference between passes with and without head fakes at the ISI 400 ms since an increasing number of participants showed bad effects, meaning they were able to overcome the fake-production costs.

The mixture model analysis also provides an estimate of the proportion of the experimental trials in which participants showed fake-production costs, the mixture proportion (P, see Table 2). Here, 17 participants had an aggregated mixture proportion below 0.75, which means most participants showed fake-production costs in under 3 out of 4 cases. The other 6 participants have an aggregated mixture proportion > 0.75, which means they did show fake-production costs slowing their initiation times down in over 75% of trials where they played a pass with compared to a pass without a head fake.

Overall, these findings suggest that mental preparation time significantly impacts whether fake-production costs occur consistently or sporadically across trials. The short preparation time at ISI 400 ms seemed to be sufficient to reduce the percentage of trials in which participants elicited fake-production costs, with some even overcoming them. In contrast, without mental preparation time (ISI 0 ms) participants showed consistent fake-production costs even on day 5.

Part II focusses on the comparison of mean ITs and ex-Gaussian distribution parameters of the practice group with the newly collected control group to analyse whether the improvement of participants’ ITs in the practice group (cf. Böer et al., 2024) were based on the effects of practicing the passes or merely reflected test–retest effects.

A mixed ANOVA with mean IT as dependent variable and the between-subjects factor group (control group, practice group) and the within-subjects factors type of pass (pass without head fake, pass with head fake), day (day 1, day 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) revealed a main effect for the factor type of pass: initiation times were significantly slower for passes with head fakes (M = 442 ms) than for passes without head fakes (M = 425 ms), F(1,54) = 20.16; p < 0.001; ɳ_p² = 0.272; ε = 0.993. Also, the ANOVA indicated a main effect for the factor ISI, pointing to a reduction of initiation times with increasing ISI length, F(1,54) = 1102.07; p < 0.001; ɳ_p² = 0.953; ε = 1. The factor day also reached significance, with decreasing mean ITs from day 1 (M = 444 ms) to day 5 (M = 423 ms), F(1,54) = 4.81; p = 0.033; ɳ_p² = 0.082; ε = 0.577. We found significant interactions between the factors day and ISI, F(3,162) = 20.94; p < 0.001; ɳ_p² = 0.279; ε = 1, between type of pass and ISI, F(3,162) = 56.06; p < 0.001; ɳ_p² = 0.509; ε = 1, as well as an interaction between type of pass and day, F(1,54) = 19.85; p < 0.001; ɳ_p² = 0.269; ε = 0.992. The ANOVA also revealed a three-way interaction between type of pass, ISI, and day, F(3,162) = 5.04; p = 0.002; ɳ_p² = 0.085; ε = 0.913, see Fig. 5.

To analyse the three-way interaction of the factor day with the factor type of pass and the factor ISI we performed paired t-tests of the changes in ITs from day 1 to day 5. The analysis indicated a significant reduction of ITs for passes with head fakes at the ISI 0 ms from day 1 (M = 648 ms) to day 5 (M = 600 ms), t(55) = 3.976, p = 0.002, d = 0.531, as well as for passes without head fakes from day 1 (M = 596 ms) to day 5 (M = 567 ms), t(55) = 3.548, p = 0.005, d = 0.474. Also, we found a significant reduction of ITs for passes with head fakes at the ISI 400 ms from day 1 (M = 416 ms) to day 5 (M = 378 ms), t(55) = 2.860, p = 0.036, d = 0.382. All other comparisons did not reach significance (p > 0.05).

We further evaluated how fake-production costs (passes with head fakes minus passes without head fakes) changed at ISI 0 ms and at ISI 400 ms from day 1 to day 5. Paired t-tests showed that fake-production costs significantly decreased at ISI 0 ms from day 1 (M = 52 ms) to day 5 (M = 33 ms), t(55) = 2.803, p = 0.007, d = 0.375, and at ISI 400 ms from day 1 (M = 35 ms) to day 5 (M = 4 ms), t(55) = 4.824, p < 0.001, d = 0.645.

Regarding the main factor of interest, namely the between-subjects factor group, there was no main effect (p > 0.05). However, there were significant interactions between the factors group and ISI, F(3,162) = 8.84; p < 0.001; ɳ_p² = 0.141; ε = 0.995, modulated by the generally higher ITs of the practice compared to the control group at ISI 0 ms (53 ms) and at ISI 400 ms (29 ms) which wasn’t of further interest for our study (see Fig. 5) and between the factors group and day, F(1,54) = 6.53; p = 0.013; ɳ_p² = 0.108; ε = 0.709, indicating group differences in the improvement of ITs from day 1 to day 5.

To evaluate the differences in improvements of ITs between day 1 and day 5 between the practice and the control group a post-hoc t-test was conducted. First, the overall changes in the groups ITs (aggregated over the factors ISI and type of pass) were calculated by subtracting the IT of day 5 from the IT of day 1. Then the changes in ITs between both groups (see Fig. 6) were compared by an independent samples t-test, which revealed significantly higher reductions of ITs for the practice (M = 46 ms) compared to the control group (M =  – 3 ms), t(54) =  – 2.557, p = 0.013, d =  – 0.691. More precisely, further t-tests (one-sample t-test for each group) were conducted to determine whether mean IT improvements from day 1 to day 5 differed significantly from zero. The one-sample t-test for the IT improvement of the practice group (M = 46 ms, SD = 95 ms) showed a significant difference from zero, t(23) = 2.375, p = 0.026, d = 0.485, while the control group showed no significant differences from zero (p > 0.05).

Taken together, the post-hoc analyses of the interaction between day, type of pass, and ISI indicated reduced fake-production costs at ISI 0 ms and at ISI 400 ms. The interaction between the factors group and day showed that only the practice group improved their performance significantly from day 1 compared to day 5. In contrast, the control group showed no change between their performance on day 1 and day 5. We therefore suggest that, although there was no four-way interaction with group, reduced fake-production costs from day 1 to day 5 are driven by practice and not solely by test–retest effects. We further analyse this not entirely clear data pattern in the later sections.

The results of the analysis of the mean ITs indicate that practice leads to a general reduction of ITs, but it is unclear whether practice affects passes with head fakes in a greater extent than passes without head fakes. To gain a more comprehensive understanding of potential differences in cognitive processing through practice between both groups, we conducted an ex-Gaussian analysis of IT distributions in the next section. This approach allows for a more nuanced examination of the IT data than focusing solely on measures of central tendency. Therefore, mixed ANOVAs with the factors day (day 1, day 5), ISI (0 ms, 400 ms, 800 ms, and 1200 ms), type of pass (pass without head fake, pass with head fake), and the between-subjects factor group (control group, practice group) for the distribution parameters mu, sigma, and tau were performed.

An ANOVA with the distribution mean (mu) as dependent variable, the between-subjects factor group (control group, practice group), and the within-subjects factors type of pass (pass without head fake, pass with head fake), day (day 1, day 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) was conducted. A significant main effect was observed for the factor ISI, F(3, 162) = 841.14; p < 0.001; ɳ_p² = 0.940; ε = 1, but no main effects were found for day, type of pass, and for the between-subjects factor group (all p > 0.05).

However, the interaction of ISI with day, F(3,162) = 15.25; p < 0.001; ɳ_p² = 0.220; ε = 0.995, as well as the interaction of ISI with type of pass, F(3,162) = 26.54; p < 0.001; ɳ_p² = 0.330; ε = 1, reached significance. Additionally, a significant interaction was observed between the factors type of pass with day, F(1,54) = 7.49; p < 0.01; ɳ_p² = 0.122; ε = 0.767, as well as a significant three-way interaction between type of pass, day, and ISI, F(3,162) = 6.52; p < 0.01; ɳ_p² = 0.108; ε = 0.921.

To analyse the three-way interaction of the factor day with the factor type of pass and the factor ISI we performed paired t-tests of the changes in mu from day 1 to day 5. The analysis indicated a significant reduction of mu for passes with head fakes at the ISI 0 ms from day 1 (M = 591) to day 5 (M = 544), t(55) = 3.944, p < 0.001, d = 0.527, but not for the other pairs (p > 0.05). Figure 7 illustrates the changes in mu from day 1 to day 5 for passes with and without head fakes at ISI 0 ms.

We further evaluated if this reduction for head fakes significantly decreased fake-production costs at ISI 0 ms. A paired t-tests showed that fake-production costs significantly decreased at ISI 0 ms from day 1 (M = 51 ms) to day 5 (M = 16 ms), t(55) = 3.727, p < 0.001, d = 0.498.

The only significant interaction with the between-subjects factor group was found with the factor ISI, F(3,162) = 4.94; p < 0.05; ɳ_p² = 0.084; ε = 0.742. Independent sampled t-tests of the interaction of the between-subjects factor group with the factor ISI did not reach significance after the Holm-Bonferroni correction (all p > 0.05).

Taken together, the results suggest that the distribution mean (mu) does not reflect the different reductions between the groups found in the mean ITs, as the absence of group differences for mu suggests that the improvements are not enhanced by practice and are likely attributable to test–retest effects. Mu (distribution mean) was significantly higher for passes with head fakes than for those without head fakes at ISI 0 ms. This result highlights the motor programming demands of response-response incompatible movements, when no time is available to mentally prepare the movement.

An ANOVA with the distribution spread (sigma) as dependent variable, the between-subjects factor group (control group, practice group), and the within-subjects factor type of pass (pass without head fake, pass with head fake), day (day 1, day 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) revealed a main effect for the factor ISI, F(3, 162) = 39.49; p < 0.001; ɳ_p² = 0.422; ε = 1.0. The ANOVA also revealed a significant main effect of the factor day shown by the reduction of sigma from day 1 (M = 38) to day 5 (M = 29), F(1,54) = 18.19; p < 0.001; ɳ_p² = 0.252; ε = 0.987. This reduction indicates less variability in participants’ ITs from day 1 to day 5. The other main effects did not reach significance (p > 0.05).

Additionally, we found significant interactions of the factors ISI and day, F(3, 162) = 4.03; p = 0.008; ɳ_p² = 0.069; ε = 0.834, of the factors ISI and type of pass, F(3, 162) = 7.60; p < 0.001; ɳ_p² = 0.123; ε = 0.986, and of the factors type of pass and day, F(1, 54) = 7.12; p = 0.010; ɳ_p² = 0.117; ε = 0.746.

Paired t-tests of the interaction of the factor day with the factor ISI revealed a significant reduction of sigma at the ISI 0 ms from day 1(M = 57) to day 5 (M = 41), t(55) = 4.915, p < 0.001, d = 0.657, and at the ISI 400 ms from day 1 (M = 32) to day 5 (M = 21), t(55) = 4.206, p < 0.001, d = 0.562. These results indicate that participants showed less variance in ITs at shorter ISIs as they became more familiar with the task over the course of practice. The other comparisons did not reach significance (p > 0.05).

Paired t-tests of the interaction of the factor type of pass with the factor ISI revealed a significantly higher distribution spread (sigma) for passes with (M = 54) compared to passes without head fakes (M = 45) at the ISI 0 ms, t(55) = 3.623, p < 0.001, d = 0.484. This result highlights that fake-production costs increase response variability when participants cannot mentally prepare their movements (see Fig. 8). The other comparisons did not reach significance (p > 0.05).

Furthermore, a single comparison (paired t-test) was used to investigate the interaction of the factors type of pass and day. The difference between day 1 and day 5 were calculated by subtracting the values of day 5 from day 1 for both types of passes (higher values indicating a greater reduction). The post-hoc test revealed a greater reduction of sigma from day 1 to day 5 for passes with head fakes (M = 12) than for passes without head fakes (M = 5), t(55) = 2.670, p = 0.010, d = 0.357, see Fig. 9.

Similar to the results of the ANOVA for mu, the only significant interaction of the between-subjects factor group was with the factor ISI, F(3, 162) = 3.67; p = 0.024; ɳ_p² = 0.064 ε = 0.699. All other interactions did not reach significance (p > 0.05). To test the interaction of the between-subjects factor group and the factor ISI, single comparisons (independent samples t-tests adjusted to Holm-Bonferroni) were used. None of the comparisons were significant after correction (p > 0.05).

Together, these results suggest that the distribution spread (sigma) reflects a familiarization of participants with the task instead of practice effects, as both groups showed significant reductions of sigma from day 1 to day 5. These findings indicate that one day of practice is sufficient to stabilize performance, allowing participants to respond more consistently. The results also confirm that performing a pass with or without a head fake without any mental preparation time beforehand (ISI 0 ms) is more complex, leading to greater variability in performance compared to longer ISIs.

An ANOVA with the distribution skew (tau) as dependent variable, the between-subjects factor group (control group, practice group), and the within-subjects factors type of pass (pass without head fake, pass with head fake), day (day 1, day 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) revealed a significant main effect for the factor ISI (F(3, 162) = 36.23; p < 0.001; ɳ_p² = 0.402; ε = 1). The ANOVA also revealed a main effect for the factor day, caused by a reduction of tau from day 1 (M = 53) to day 5 (M = 40), F(1,54) = 11.67; p = 0.001; ɳ_p² = 0.178; ε = 0.919. The factor type of pass also reached significance, with lower values of tau for passes without (M = 39) than for passes with head fakes (M = 54)) F(1,54) = 14.97; p < 0.001; ɳ_p² = 0.217; ε = 0.967.

The ANOVA also revealed an interaction of ISI with type of pass, F(3,162) = 3.16; p = 0.026; ɳ_p² = 0.097; ε = 0.645, as well as a significant three-way interaction of ISI with day and type of pass, F(3,162) = 4.35; p = 0.006; ɳ_p² = 0.075; ε = 0.864 (see Fig. 10).

Paired t-tests investigating the three-way interaction of the factors ISI, type of pass, and day revealed only a significant reduction of tau for passes with head fakes at the ISI 400 ms from day 1 (M = 87) to day 5 (M = 58), t(55) = 3.639, p < 0.001, d = 0.486. No other comparisons were significant (p > 0.05).

The only interaction with the between-subjects factor group, which reached significance, was with the factor day, F(1,54) = 5.31; p = 0.025; ɳ_p² = 0.090; ε = 0.619. All other interactions did not reach significance (p > 0.05). To analyse the three-way interaction, we calculated the difference of tau from day 1 to day 5 (aggregated over the factors ISI and type of pass) for both groups. An independent samples t-tests was used to compare these reductions of tau between both groups, which revealed a significantly greater reduction from day 1 to day 5 for the practice group (M =  – 21) compared to the control group (M =  – 4), t(55) =  – 2.304, p = 0.025, d =  – 0.486, see Fig. 11. Further t-tests (one-sample t-test for each group) were conducted to determine whether mean improvements of tau from day 1 to day 5 differed significantly from zero. The one-sample t-test for the improvement of tau of the practice group (M =  – 21) showed a significant difference from zero, t(23) =  – 3.210, p = 0.004, d = -0.655, while the control group (M =  – 4) showed no significant differences from zero (p > 0.05).

Together the results for the analysis of the distribution skew (tau) revealed higher values of tau for passes with compared to passes without head fakes, indicating increased cognitive demands when performing movements with response-response incompatibility. Tau was the only distribution parameter clearly modulated by practice with the task, with significant reductions in the practice group while the control group did not improve from day 1 to day 5.

The analysis of mixture effects (Miller, 2006) for the ISI of 0 ms revealed that most participants (20 of 24) showed a uniform effect for fake-production costs, observed in over 75% of trials where they played a pass with compared to a pass without a head fake. None of the participants showed a consistent mixture effect for all 5 days of practice and the number of participants showing a mixture effect decreased from the first day of practice (4 participants) to the last day of practice (0 participants). This indicates that participants stabilized their performance but also that there are general cognitive costs associated with the production of passes with head fakes without mental preparation time (ISI 0 ms), which cannot be overcome through practice.

In contrast, the analysis for the ISI of 400 ms revealed stronger evidence for a mixed effect as 15 out of 24 participants showed significant mixture effects across the aggregated sessions and most participants (17 of 24) elicited the fake-production costs in less than 75% of trials. It also must be noted that the analysis indicated that the number of participants, who were able to completely overcome the fake-production costs at the ISI of 400 ms, increased from day 1 (3 participants) to day 5 (12 participants). This pattern of results is represented by the number of bad effects in the MixTest (Miller, 2006). Our finding aligns with the previous analysis of mean ITs (see Böer et al., 2024), which indicated that participants were able to overcome the fake-production costs at the ISI of 400 ms with increasing practice.

The findings of persistent cognitive costs with no mental preparation time (i.e., ISI 0 ms) and of reduced, or even eliminated, fake-production costs with some mental preparation time (i.e. ISI 400 ms), are of interest for the applied field: The uniform effect at ISI 0 ms means that players always show fake-production costs when performing head fakes without mentally preparing the action beforehand. This could be a disadvantage for the attacking player, as research suggests that an increase in the ITs of a fake action increases the chance that the opponent expects a deceptive movement (Kunde et al., 2019). Conversely, the mixed effect at the ISI 400 ms indicates that participants only show fake-production costs in some trials, but in those cases the fake-production costs are significantly higher. The attacking player therefore only has a disadvantage in those few trials, in which ITs are substantially slowed when performing a pass with head fake, as the opponent would have a better chance to correctly anticipate the fake action (cf. Kunde et al., 2019). However, in the trials where ITs are not slowed, this disadvantage vanishes. Overall, the analysis confirms our previous conclusion in Böer et al. (2024), that it is helpful to mentally prepare the execution of a deceptive action shortly beforehand. Furthermore, practicing fake movements and reducing the time needed to mentally prepare the movement is essential to gain an advantage over the defending player.

The results of this analysis revealed that the practice group exhibited significantly greater reductions in ITs for both passes with and without head fakes compared to the control group, who did not improve at all. This finding suggests that the improvements observed in the practice group are not merely simple test–retest effects but reflect genuine performance enhancements due to practice with the task.

Interestingly, the current analysis did not indicate a specific reduction of fake-production costs with practice as the four-way interaction of the between-subjects factor group with the within-subjects factors ISI, type of pass, and day did not reach significance. This finding contrasts with the results of the practice group reported in Böer et al. (2024). However, the ANOVA indicated a significant interaction between the three within-subjects factors, with post-hoc tests revealing significant reductions of fake-production costs at the ISI 0 ms as well as at the ISI 400 ms. The reduction of fake-production costs appears to be mediated primarily by the performance improvements of the practice group, as the control group did not show any significant improvements from day 1 to day 5.

This result was surprising as one day of practice resulted in great improvements of participants' ITs in the practice group on the following day (day 1 to day 2; cf. Böer et al., 2024), which assumes that also the control group should show this performance improvement. As this is not the case, this indicates that these effects of only one day of practice may instead be short-lived, as the control group did not show any significant improvements on day 5 compared to day 1 (i.e., potential effects of practice on day 1 did not persist to day 5).

The significant main effect of the factor type of pass, with slower ITs for passes with head fakes, underscores the increased cognitive load and decision-making complexity associated with performing deceptive actions (Güldenpenning et al., 2023; Kunde et al., 2019; Wood et al., 2017). Similarly, the main effect of ISI, showing reduced ITs with increasing ISI length, supports the notion that longer mental preparation times facilitate better performance. This finding is consistent with theories of motor preparation in optimizing response execution (Welford, 1980).

The analyses of the ex-Gaussian distribution parameters (mu, sigma, tau) offer a sophisticated way of analysing IT data by decomposing response latencies into distinct components. It should be noted, however, that the interpretation of these parameters is debated in the literature (Matzke & Wagenmakers, 2009; Osmon et al., 2018). When used cautiously, it can be a helpful tool to describe IT data (Fitousi, 2020) and uncover differences between experimental conditions that weren’t visible in the analysis of mean values (Ratcliff, 1979).

Mu, the distribution mean, reflects the general response speed and is suggested to encapsulate motor processes (i.e., response planning, response selection; Luce, 1986). Sigma, the distribution variance, reflects the consistency of participants' responses, which has been linked to attentional control (Schmiedek et al., 2007). Tau describes the skew of the distribution, indicating the presence of long ITs in some trials or attentional lapses, which are associated with high task complexity and increased cognitive demands, e.g., in working memory (Engle et al., 1999; Heathcote et al., 1991; Logan, 1988; Tse et al., 2010). Together, these parameters provide a multidimensional view of performance, offering insights that might remain hidden in simple mean IT analyses, particularly in tasks involving varying levels of cognitive and motor demands.

Mu was significantly higher for passes with head fakes than for those without head fakes at the ISI of 0 ms, confirming its role as a marker of motor programming costs (Heuer, 1995). The lack of differences for passes with head fakes and passes without head fakes at ISI 400 ms suggests that even brief mental preparation time can mitigate these programming demands for response-response incompatible movements. As the analysis of mu showed, there were no practice improvements beyond test–retest effects, which indicates that motor planning processes may require prolonged or domain-specific practice to improve significantly. Future studies might explore whether mu values are lower among basketball experts, who likely refine motor programming over years of deliberate practice (Ericsson et al., 1993).

While sigma decreased from day 1 to day 5 for both the practice and control group, these reductions did not differ significantly between groups, supporting the conclusion that they reflect general test–retest effects. This suggests that even brief exposure to the task enables participants to stabilize their performance by reducing response variability, a finding consistent with broader RT research (Heathcote et al., 1991).

Tau consistently reflected the increased cognitive demands of performing passes with compared to passes without head fakes. Across all conditions, tau was higher for passes with head fakes, supporting the hypothesis that it captures the attentional costs arising from resolving the response-response incompatibility. This indicates that response-response incompatibility elongates the distribution’s tail by increasing attentional lapses in certain trials. Tau seems to be a good general parameter for the increased complexity of the response-response incompatibility associated with performing passes with head fakes, reflecting the increased cognitive demands even at the longer ISIs of 800 ms and 1200 ms which were not evident in simple mean analyses (cf. Böer et al., 2024). Notably, reductions in tau from day 1 to day 5 were only significant in the practice group, suggesting that practice decreases attentional lapses by increasing task automatization. This aligns with the notion that repeated practice leads to performance transitions from algorithmic processing to fast memory retrieval, resulting in reduced cognitive load and improved task performance (Logan, 1988).

The distribution analyses highlighted potentially distinct cognitive processes as the source of fake-production costs at different ISIs. At the ISI 0 ms, mu was the primary source of fake-production costs, which we take as a strong indicator that mu reflects higher motor programming costs, connected to planning response-response incompatible movements (cf. Heuer, 1995). We therefore argue that mu encapsulates motor processes (i.e., response planning, response selection) as also suggested by Luce (1986). In contrast, at the ISI 400 ms, tau became the dominant factor, indicating that some trials remained cognitively demanding despite reduced programming costs. These findings suggest that brief mental preparation allows participants to address motor demands effectively but does not eliminate attentional lapses, which persist in some trials and elongate the distribution’s tail (Schmiedek et al., 2007). The parameter sigma seemingly did not reflect any distinctions in the cognitive processes associated with producing passes with head fakes or without head fakes, but was merely modulated by test–retest effects, indicating reduced response variability.

These results, together with the analysis of mixture effects for the ISI 0 ms and 400 ms in the practice group, indicate that there might be distinct cognitive processes involved when performing a pass with head fake without (ISI 0 ms) vs. with mental preparation time (ISI 400 ms). The uniform effect observed at ISI 0 ms indicates consistent fake-production costs affecting most trials, which is reflected in significantly higher values of the distribution mean (mu) for passes with head fakes. This pattern at ISI 0 ms persisted throughout the practice sessions. Conversely, at ISI 400 ms we found a mixed effect with participants showing fake-production costs only in some, but not all trials. This is reflected in the distribution parameters, as we only found differences between passes with head fakes and passes without head fakes in the distributions skew (tau) but not in mu.

If the exact same cognitive process were involved, an increase in preparation time should not change the shape of the distribution, but only shift the distribution to the left. In that case we only should have found a decrease in the mu with increasing ISI, as the processes could be started and finished before participants had to initiate the movement (in line with the widely accepted notion that specific cognitive processes generally take the same amount of time in an individual; cf. Sternberg, 1969). In contrast, the tail of the distribution (tau) should not have differed between different ISIs when the same cognitive processes would have taken place just at different points of time (as long as no real data were removed by the outlier filtering at the short ISIs). But our analyses indicated that the sources of fake-production costs differ between the two ISIs (i.e., mu at ISI 0 ms & tau at ISI 400 ms) as also shown by the different patterns found in the analysis of mixture effects and in their modulation by practice. These results emphasize the necessity of analysing the full IT distribution for a more detailed picture of the effects of different task demands.

The first limitation specifically refers to the distributional analyses conducted here. The number of trials for each condition (ISI, type of pass, day) for each participant (n = 40) was rather small compared to the best-case recommendations for distribution analysis methods detailed in previous research. Increasing the number of trials decreases the chance that the estimated distribution deviates from the actual underlying IT distribution (Lacouture & Cousineau, 2008). However, it should be noted though that the ideal number of trials might not be necessary in this case, as research has shown that for distinct RT distributions like the ex-Gaussian distribution good fitting distributions can be found with as little as 20 reaction times per participant and condition (Ratcliff, 1979).

Further limitations refer to the experimental setup, which has also been discussed in detail in the previous paper (Böer et al., 2024): Participants did not have to decide on their own when and how to play the pass. Instead, they just had to react as fast as possible to a fixed auditory cue. While this oversimplification of a real situation in basketball enabled us to investigate the characteristics of fake-production costs caused by response-response incompatibility, it limits the transferability of our findings to basketball practice. Specifically, it seems that fixed reactions to a specific cue (i.e., always playing a pass to the left side if the right side is blocked by defending player) are easier to execute, than reactions in situations in which the participant could decide for themselves which action to perform (cf. Weller et al., 2018). To improve our understanding of these additional cognitive costs, future research should examine the fake-production costs in a setting where the attacking player him/herself must decide under time constraints which action he/she wants to perform.

Therefore, participants might elicit higher fake-production costs (i.e., longer initiation times) in real game situations in which players must decide under time pressure whether to play a pass with or without a head fake. Also, participants did not have to deceive an opponent, which, however, has been shown to increase cognitive costs potentially as the learnt social rule not to cheat has to be violated (Foerster et al., 2017). Of course, the absence of an opponent in our study limits the transferability of this knowledge to the basketball court. To address this limitation, we started data collection in a social interaction setting where two participants interact with each other, one acting as a defending and the other as an attacking player. This future research hopefully provides more insight into differences of head fake-production costs and might enable us to provide specific recommendations for basketball players and their training routines.