exGauss Distribution Analysis
Because the analyses of the IT and ITCV (Böer et al., 2023) indicated practice-related changes, ANOVAs with the factors day of practice (Days 1, 2, 3, 4, and 5), ISI (0 ms, 400 ms, 800 ms, and 1200 ms) and type of pass (pass with or without head fake) were performed for the distribution parameters mu, sigma, and tau.
Mu
An ANOVA with mu as dependent variable and type of pass (direct pass vs. head fake), day of practice (days 1, 2, 3, 4 and 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) as repeated measures indicated a reduction of mu with increasing ISI (F(3, 69) = 639.06; p < .001; ɳp2 = .96; ε = .496) (see figure 5). There was neither a main effect for day of practice nor for type of pass (both p > .05), however, the interaction of ISI with type of pass (F(3,69) = 12.03; p < .001; ɳp2= .343; ε = .650), as well as the interaction of ISI with day of practice (F(12,276) = 2.94; p < .05; ɳp2= .113; ε = .377) reached significance.
Single comparisons (paired t-tests adjusted to Holm-Bonferroni; Holm, 1979) of the interaction of the factors ISI and type of pass revealed higher values of mu for passes with head fakes compared to passes without head fakes at the ISI of 0 ms (581 vs. 558, t(23) = -4.135, p < .01, d = -.84). There were no significant differences in the values of mu between passes with and without head fakes at the ISI of 400ms, 800ms and 1200ms (all p > .05).
Single comparisons (paired t-tests adjusted to Holm-Bonferroni; Holm, 1979) of the interaction of the factors ISI and Day of practice revealed higher values of mu at the ISI 0 ms compared to the other 3 ISIs on every day of practice (all p < .001). Additionally, there was a significant difference in values of mu at the ISI 400 ms compared to the ISI 800 ms at the first day of practice, but not on the other days. All other comparisons did not reach significance (p > .05) (see figure 6). The table with the descriptive values as well as the results of the paired t-tests can be found in appendix A.
Additionally, the fake production costs in mu, quantified as difference from passes with minus passes without head fakes, for the ISI 0 ms and 400 ms are depicted in figure 7.
Sigma
An ANOVA with sigma as dependent variable and type of pass (direct pass vs. head fake), day of practice (days 1, 2, 3, 4 and 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) as repeated measures revealed a main effect for the factor ISI, indicating higher values of sigma at ISI 0ms compared to the other ISIs, F(3, 69) = 36.88; p < .001; ɳp2 = .61; ε = .582 (see figure 8). The ANOVA also revealed a significant main effect of the factor day of practice, shown by the reduction of sigma from Day 1 to Day 5, F(4,92) = 6.10; p < .01; ɳp2= .21; ε = .448 (see figure 9). All other main effects and interactions did not reach significance (p > .05).
Single comparisons (paired t-tests adjusted to Holm-Bonferroni; Holm, 1979) of the main effect for the factor Day of practice revealed significant reductions of sigma from day 1 to day 2 (40 vs. 31, t(23) = 3.319, p = .012, d = .678), day 1 to day 4 (40 vs. 29, t(23) = 3.005, p = .018, d = .613) and day 1 to day 5 (40 vs. 29, t(23) = 2.906, p = .018, d = .593). Interestingly, the reduction from day 1 to day 3 did not reach significance (p > .05).
Furthermore, paired t-tests were used to investigate the main effect for the factor ISI indicated significant differences in sigma between all four ISIs with higher values at the ISI 0 ms compared to the ISI 400 ms (44 vs. 24, t(23) = 7.834, p < .001, d = 1.599), ISI 800 ms (44 vs. 28, t(23) = 6.605, p < .001, d = 1.546) and ISI 1200 ms (44 vs. 32, t(23) = 4.590, p < .001, d = 1.348). The analysis also revealed significant differences for the ISI 400 ms with the other two ISIs, ISI 800 ms (24 vs. 28, t(23) = -3.466, p = .004, d = -.708) and ISI 1200 ms (24 vs. 32, t(23) = -5.286, p < .001, d = -1.079). Lastly, there was also a significant difference in sigma between the ISI 800 ms and the ISI 1200 ms (28 vs. 32, t(23) = -3.336, p = .004, d = -.681).
Tau
An ANOVA with tau as dependent variable and type of pass (direct pass vs. head fake), day of practice (days 1, 2, 3, 4 and 5), and ISI (0 ms, 400 ms, 800 ms, 1200 ms) as repeated measures revealed a significant main effect for the factor ISI, indicating a reduction of tau with increasing ISI (F(3, 69) = 35.09; p < .01; ɳp2 = .61; ε = .729). The ANOVA also revealed a main effect for the factor day of practice shown by a reduction of tau from day 1 to day 5 (F(4,92) = 9.3; p < .001; ɳp2= .210; ε = .401) (see figure 10) and a main effect for the factor type of pass with lower values of tau for direct passes (36.95) than head fakes (45.34) (F(1,23) = 9.4; p < .01; ɳp2= .029; ε = 1). The ANOVA also revealed an interaction of ISI with type of pass (F(3,69) = 3.81; p < .05; ɳp2= .097; ε = .645), as well as an interaction of ISI with day of practice (F12,276) = 2.28; p < .05; ɳp2= .048; ε = .479). The other interactions did not reach significance (p > .05).
Single comparisons (paired t-tests adjusted to Holm-Bonferroni; Holm, 1979) investigating the main effect of the factor day of practice revealed a reduction of tau for from day 1 to day 2 (59 vs. 41, t(23) = 3.54, p = .020, d = .72), day 1 to day 3 (59 vs. 37, t(23) = 3.3, p = .024, d = .68), day 1 to day 4 (59 vs. 35, t(23) = 3.3, p = .024, d = .59) and from day 1 to day 5 (59 vs. 33, t(23) = 3.4, p = .02, d = .69). The other comparisons did all not reach significance (p > .05).
The paired t-tests investigating the interaction between the factors day of practice and ISI indicated significant reductions of tau at the ISI 0 ms with increasing practice, from day 1 to day 2 (76 vs. 55, t(23) = 3.75, p = .014, d = .76), day 1 to day 3 (76 vs. 47, t(23) = 3.55, p = .020, d = .72), day 1 to day 4 (76 vs. 39, t(23) = 5.89, p < .001, d = 1.20) and day 1 to day 5 (76 vs. 44, t(23) = 4.56, p = .002, d = .93). There was also a significant reduction in tau at the ISI 400 ms, but only from day 1 to day 2 (73 vs. 45, t(23) = 3.65, p = .017, d = .74), while all other comparisons did not reach significance (p > .05). Mean values of tau as a function of day of practice and ISI are shown in figure 11.
The single comparisons also indicated lower values of tau for passes without than for passes with head fakes at the ISI 0ms (45 vs. 59, t(23) = -2.63, p < .05, d = -.53) and at the ISI 400ms (43 vs. 58, , t(23) = -3.7, p < .01, d = -.76). The were no significant differences at the ISI 800ms and 1200ms (all p > .05). The mean values of tau as a function of type of pass and ISI are shown in figure 12. Additionally, the fake production costs in tau, quantified as difference from passes with minus passes without head fakes, for the ISI 0 ms and 400 ms are depicted in figure 13.
Mixed effect analyses
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 showed a mixture effect for all the 5 days of practice 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 did show 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 only in 68%, 63%, 57%, and 61% of times where they performed a pass with a head fake at the ISI of 0 ms did they show fake production costs. All other participants have an aggregated mixture proportion > 0.75, which means that the majority of participants 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.
Table 1
Analysis of Mixture Effects for ISI 0 ms
Notes. The values are provided separately for each day of practice (Days 1-5). In the column on the most right, the aggregated values across all sessions are provided. Presented are from left to right: Observed fake production costs (in ms), class of the effect, estimated mixture proportion (P), observed χ2 value of the likelihood ratio test, and corresponding p value. Aggregate P is the mean P across the days of practice. Aggregate χ2 is the total χ2 across all days of practice. Each observed χ2 has one degree of freedom, and the aggregate χ2s have five degrees of freedom.
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 analyze 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 showed a mixture effect for all the 5 days of practice and the number of participants showing a mixture effect didn’t 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 did show fake production costs, the mixture proportion (P, see Table 1). 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.
Table 2
Analysis of Mixture Effects for ISI 400 ms
Notes. The values are provided separately for each day of practice (Days 1-5). In the column on the most right, the aggregated values across all sessions are provided. Presented are from left to right: Observed fake production costs (in ms), class of the effect, estimated mixture proportion (P), observed χ2 value of the likelihood ratio test, and corresponding p value. Aggregate P is the mean P across the days of practice. Aggregate χ2 is the total χ2 across all days of practice. Each observed χ2 has one degree of freedom, and the aggregate χ2s have five degrees of freedom.