A total of 160 participants were recruited through Prolific (www.prolific.co) in exchange for monetary payment ($6.00 USD). The two volition conditions were run consecutively. The first group of 80 participants was run in the direction assigned condition, and the second group of 80 participants was run in the direction selected condition. These sample sizes were chosen to match the sample size used in Experiment 2 of Gibson et al. (2023) study. Note that power analyses are not well developed for mixed-effects models (Maxwell et al., 2018); this is especially true for estimating the power of random factors because the null value (population variance equal to zero) is also the minimum possible value of this parameter, and the sampling distributions of such “boundary values” are not well understood by statisticians (Verbeke and Molenberghs, 1997). To be included in the experiment, participants were required to (1) self-report that they were a fluent English speaker; (2) self-report normal or corrected-to-normal visual acuity; and, (3) finish the experiment with an overall percent error rate on the visual search task that was ≤ 30%. The Institutional Review Board at the University of Notre Dame approved all procedures reported in this manuscript. These data were collected in the fall of 2021.

Both volition conditions were programmed using PsychoPy Experiment Builder (Peirce et al., 2019), and virtual data collection was hosted through PsychoPy's open science website Pavlovia. The sizing of stimuli in PsychoPy are specified in 'height units' which are relative to the height of participants' computer screen while the ratio of the height to width of the stimuli remain absolute. The use of these units in PsychoPy ensure that stimuli are presented consistently without restricting participation based on screen-size or OS requirements. In the following description, we report the size of the stimuli in terms of height units, but for the sake of clarity, we also report their size in terms of centimeters (cm) based on a 13-inch widescreen display.

As shown in Figure 1 Top, each trial in the direction assigned condition consisted of three displays which were presented against the black background of the screen: a fixation display, a cue display, and a target display. The fixation display contained a small white fixation dot in the center of the display; the fixation dot measured 0.015 units (0.3 cm) in diameter. Four boxes were presented 0.16 units (2.80 cm) above, below, left of, or right of central fixation. Each box appeared as a square, 0.07 units (1.30 cm) tall and 0.07 units (1.30 cm) wide, and had a black fill and gray outline. The fixation dot was replaced by a white number between 1 and 4 that indicated the direction of one of the four peripheral boxes. The “1” cue referred to the above location; the “2” cue referred to the right location; the “3” cue referred to the below location; and the “4” cue referred to the left location. The number cues were 0.04 units (0.50 cm) at their widest point and 0.06 units (1.10 cm) tall. The target display contained a single white target letter (E or U) along with three non-target letters (A, P, and S). Each letter was 0.04 units (0.50 cm) tall and 0.04 units (0.50 cm) wide and appeared in one of the four gray boxes; the target was equally likely to appear in any of the four directions (above, below, left, or right).

As shown in Figure 1 Bottom, the sequence of displays in the direction selected condition was identical to the sequence of displays in the direction assigned condition with the sole exception being the insertion of a selection display that appeared at the start of each trial. In the selection display, one of the four number cues was randomly assigned (as in the direction assigned condition); however, participants in the direction selected condition were allowed to change the direction of the cue on each trial by pressing the right arrow key on the keyboard. This arrow key advanced through a random sequence of the number cues, and the cycle could be repeated until a cue was selected. Participants locked in their choice by pressing the space bar which then triggered the appearance of the fixation display followed by the cue display (which contained the selected cue) and then the target display.

The design of the direction assigned condition was identical to the design of the two experiments reported in Gibson et al. (2023) with the sole exception being that number cues were used to convey direction in the present study. Four levels of a conditional target entropy (or cue validity) factor [1.00-bit (100% valid), 2.36-bit (70% valid), 3.00-bit (25% valid), and no cue] were presented within the context of a repeated measures design. Note that although the cue was absent from the display in the no cue context, we treated this context as a separate level of the conditional target entropy factor for the purposes of balancing the design. Each of the four conditional target entropy conditions was presented in a separate block of 40 trials, and this group of four blocks was repeated four times for a total of 16 blocks (640 total experimental trials). The order of the four conditional target entropy conditions was randomized separately within each repetition group for each participant.

The design of the direction selected condition was identical to the design of the direction assigned condition with the sole exception being the elimination of the no-cue condition, which was omitted out of necessity because there is no cue to be selected in the no-cue condition. Each of the remaining three conditional target entropy conditions was presented in a separate block of 40 trials, and this group of three blocks was repeated four times for a total of 12 blocks (480 total experimental trials). The order of the three conditional target entropy conditions was randomized separately within each repetition group for each participant.

At the beginning of each block in the direction assigned condition, participants were informed of the presence and validity of the cues; the direction of the cues was described as “always accurate” in the 1.00-bit (100%-valid) cue condition, “mostly accurate” in the 2.36-bit (70%-valid) cue condition, and “rarely accurate” in the 3.00-bit (25%-valid) cue condition. Note that when the cue was invalid, the target was equally likely to appear at each of the three uncued locations. In the 2.36-bit (70%-valid) cue condition, the target appeared at each of the uncued locations 10% of the time; and, in the 3.00-bit (25%-valid) cue condition, the target appeared at each of the uncued locations 25% of the time. On each trial within a block, a fixation display appeared first for 500 ms followed by the cue display. After 600 ms, the target display was added to the cue display and the two displays remained on screen until a response was made. On each trial, the target letter was equally likely to be an E or U. Observers always pressed the “E” key with their left hand to discriminate the identity of the E target and the “U” key with their right hand to discriminate the identity of the U target.

At the end of each block, participants were instructed to rate the level of agency they felt. Participants were told that individuals are thought to have a positive sense of agency when they consider themselves to be the initiator of their actions, along with the following instructions, provided at the beginning of the experiment:

“We are interested in how much agency you feel in these different visual search contexts. Please always use the cue to try to find the target, regardless of how useful or accurate it is. At the end of each block, a rating scale will appear on the screen, and you will be asked to rate the extent to which you considered yourself to be in control of finding the target. A rating of ‘1′ will correspond to ‘no control' whereas a rating of ‘7′ will correspond to ‘full control.' You will respond by using the corresponding number keys on your keyboard to report the magnitude of your positive sense of agency in each block.”

The procedure in the direction selected condition was identical to the procedure in the direction assigned condition with two exceptions. First, cues were always present in each block. Second, a selection display appeared first on each trial; the selection display remained on the screen until participants made their direction choice, at which point they were instructed to press the space bar and then the trial proceeded as in the direction assigned condition.

We began by examining valid and invalid response times (RTs) in each of the two volition conditions to ensure that standard spatial cueing effects were obtained in this study. The top panel of Figure 2 shows mean correct valid and invalid RTs as a function of conditional target entropy in each of the direction assigned and selected conditions; the bottom panel of Figure 2 shows the corresponding percent error rates. First, a 2 × 3 repeated measures analysis was conducted on mean correct valid RTs, with volition condition (direction assigned vs. direction selected) and conditional target entropy (1.00 bit vs. 2.36 bits vs. 3.00 bits) as the two within-subjects factors. These analyses were conducted using the multivariate approach to avoid violating the sphericity assumption. As can be seen in Figure 2, mean correct valid RTs were significantly slower overall (by 176 ms) in the direction selected condition than in the direction assigned condition, F_{(1, 158)} = 28.26, p < 0.001, ηp2 = 0.15, for the main effect of volition condition. Although this main effect of volition condition was not explicitly predicted, it may reflect a greater contribution of intentional processes, which are known to be slower (Wolfe et al., 2000), in the direction selected condition. In addition, as expected, mean correct valid RTs also increased significantly as a function of conditional target entropy, F_{(2, 78)} = 213.86, p < 0.001, ηp2 = 0.85, for the main effect of conditional target entropy. However, the effect of conditional target entropy was very similar across the two volition conditions resulting in a non-significant interaction between volition condition and conditional target entropy, F_{(2, 78)} = 1.49, p = 0.23, ηp2 = 0.04. Likewise, an identical analysis of percent error rates revealed no significant main effects or interaction (all p's > 0.26 or more).

Second, we also examined the relation between valid and invalid trials in the 2.36-bit (70%-valid) and 3.00-bit (25%-valid) cue conditions (recall that the 1.00-bit (100%-valid) cue condition did not include any invalid trials). A 2 × 2 × 2 Split-Plot ANOVA was conducted on mean correct RTs, with volition condition as the sole between-subjects factor, and with conditional target entropy (2.36 bit vs. 3.00 bit) and cued location (valid vs. invalid) as the two within-subjects factors. This analysis was conducted using the univariate approach because the sphericity assumption could not be violated with only two levels of each factor. Most importantly, as expected, there was a significant two-way interaction between conditional target entropy and cued location, F_{(1, 158)} = 64.72, p < 0.001, ηp2 = 0.29, indicating that the 113-ms spatial cueing effect observed in the 2.36-bit (70%-valid) cue condition was larger than the 40-ms spatial cueing effect observed in the 3.00-bit (25%-valid) cue condition. In addition, there was also a significant two-way interaction between volition condition and cued location, F_{(1, 158)} = 3.86, p = 0.051, ηp2 = 0.02, indicating that the 93-ms spatial cueing effect observed in the direction selected condition was larger than the 61-ms spatial cueing effect observed in the direction assigned condition. This interaction may reflect the manifestation of a greater effect of agency on RTs in the direction selected condition. The three-way interaction between volition condition, conditional target entropy and cued location did not attain significance, F_{(1, 158)} = 0.31, p = 0.58, ηp2 = 0.002. An identical analysis was also conducted on percent error rates. Although the pattern of error rates mirrored the pattern of RTs, the two-way interaction between conditional target entropy and cued location was only marginally significant, F_{(1, 158)} = 2.92, p = 0.089, ηp2 = 0.02. Furthermore, the interaction between volition condition and cued location did not attain significance, F_{(1, 158)} = 0.42, p = 0.52, ηp2 = 0.003.

A mixed-effects (growth-curve) modeling approach was used in the present study to examine potential individual differences in voluntary attention control. Mixed-effects models explicitly distinguish between fixed factors and random factors in the analysis of repeated measures designs (see Singer and Willett, 2003, for an introduction to these methods), where a fixed factor refers to an independent variable whose levels have been pre-selected by the researcher and a random factor refers to an independent variable whose levels have been chosen randomly. In this way, each participant in a repeated measures design is a random level of an independent variable labeled “subject;” in this context, a significant effect of the random “subject” factor is reflected by significant variation across the different levels of the subject factor (constituting individual differences). One of the main advantages to using mixed-effects models is that they afford greater flexibility in how the random subject factor is allowed to interact with the fixed factor (see, Kliegl et al., 2011; Barr et al., 2013; Matuschek et al., 2017; Oberaurer, 2022 for recent discussion).

To help understand the advantage of the mixed-effects model approach, consider the standard univariate ANOVA approach to analyzing repeated measures designs. In this standard approach, the fixed factor—conditional target entropy—must be treated as a categorical variable whereas it can be treated as a continuous variable in the mixed-effects model approach. Figures 3A, B show both the individual (light gray lines) and average (dark black symbols and lines) agency ratings plotted as a function of conditional target entropy in each of the direction assigned and direction selected conditions, respectively. In these designs, the subject X conditional target entropy interaction serves as the error term (i.e., the denominator) in the F-test that evaluates the main effect of conditional target entropy within each volition condition. However, this interaction is assumed to be zero in the population within the context of the univariate ANOVA approach; this assumption is called the “sphericity assumption.”

From the perspective of mixed-effects models, sphericity corresponds to a model that includes conditional target entropy as the fixed factor along with a random intercept factor that allows each participant to have a unique intercept. However, this model assumes that the individual difference in intercepts is the only difference between individuals' true trajectories which are all assumed to be parallel. For instance, Figures 3C, D show the individual trajectories that would be predicted by a random intercept model in each of the direction assigned and direction selected conditions. The discrepancy that is apparent between the actual and predicted individual trajectories reflects nothing more than error according to the univariate approach.

However, if the actual individual trajectories depicted on the left-hand side of Figure 3 reflect the existence of slopes (and intercepts) that are truly different for each participant, then this would represent a violation of the sphericity assumption, and the univariate approach to analyzing repeated measures designs would no longer be statistically appropriate because the type I error rate could be drastically higher than the nominal alpha value (see Maxwell et al., 2018). Fortunately, there are three potential solutions to a violation of the sphericity assumption: (1) use various correction factors (such as Greenhouse-Geisser) that adjust the degrees of freedom and thus the critical values associated with the univariate F-tests; (2) use the multivariate approach to analyzing repeated measures designs; and (3) use the mixed-effects approach to analyzing repeated measure designs. Options 2 and 3 have been deemed the most appropriate because they take into consideration all the error that results when the subject X fixed factor interaction is non-zero (Maxwell et al., 2018). The main difference between options 2 and 3 is that the multivariate approach does not attempt to explicitly model the random factors whereas the mixed-effects model does.

Accordingly, we used a growth-curve modeling approach in the present study that treated the two levels of volition condition (direction assigned vs. direction selected) as a categorical variable and the three levels of conditional target entropy as a continuous variable. The statistical model included fixed main effects of volition condition and conditional target entropy as well as the interaction between these two fixed factors. In addition, the statistical model also included both random slope and intercept factors that were fit across the three levels of conditional target entropy. Note that the conditional target entropy scale was shifted by −1.00 bit in the analysis so that the value of the intercept estimate corresponded with the value predicted in the 1.00-bit (100%-valid) cue condition. These analyses were conducted using SAS PROC MIXED. Model parameters were estimated using restricted maximum likelihood (REML), and accompanying p-values were calculated based on a Wald test, both of which are the default setting in SAS PROC MIXED.

Figure 4 depicts the individual (light gray lines) and average (dark black symbols and lines) agency slopes (and intercepts) that were fit across the three levels of conditional target entropy in each of the direction assigned (panel A) and direction selected (panel B) conditions. There were significant fixed main effects of volition condition, F_{(1, 158)} = 15.04, p = 0.0002, η² = 0.09, and conditional target entropy, F_{(1, 158)} = 64.19, p < 0.0001, η² = 0.29. More importantly, as expected, there was also a significant volition condition X conditional target entropy interaction, F_{(1, 158)} = 26.23, p < 0.0001, η² = 0.14, suggesting that conditional target entropy had a larger effect on agency ratings in the direction selected condition than in the direction assigned condition.

The two-way interaction between volition condition and conditional target entropy was examined further by analyzing the fixed effect of conditional target entropy, along with the random intercept and slope factors, within each volition condition separately. With respect to the direction assigned condition, we expected to replicate the main findings reported by Gibson et al. (2023). Consistent with their findings, the average slope was found to be slightly negative (-0.21 units of agency/bit), and it was also found to be significant in this experiment, F_{(1, 79)} = 3.98, p = 0.049, η² = 0.05, 95% CI [−0.42, −0.0006]. In addition, the variance associated with the random slope factor was found to be significant, Var = 0.79, z = 5.55, p < 0.0001, 95% CI [0.57, 1.16], as was the variance associated with the random intercept factor, Var = 1.96, z = 5.70, p < 0.0001, 95% CI [1.43, 2.86]. The fact that the variance associated with the random slope factor was significant suggests that the individual differences in the slope estimates represented true slope differences; therefore, contrary to the sphericity assumption underlying the standard univariate ANOVA approach to analyzing repeated measures designs, these findings suggest that a statistical model which only allows individuals to differ in their intercepts should be rejected. In addition, there was also significant covariation between these two random factors, Cov = −0.79, z = −4.22, p < 0.0001, 95% CI [−1.15, −0.42], which corresponds to a correlation (r) of −0.632. This finding suggests that the slope estimates tended to vary inversely with the intercept estimates.

With respect to the direction selected condition, the average slope was strongly negative (-0.95 units of agency/bit), and it was also found to be significant in this experiment, F_{(1, 79)} = 90.65, p < 0.0001, η² = 0.53, 95% CI [−1.15, −0.76]. In addition, the variance associated with the random slope factor was found to be significant, Var = 0.55, z = 4.10, p < 0.0001, 95% CI [0.36, 0.95], as was the variance associated with the random intercept factor, Var = 0.47, z = 2.72, p =0.003, 95% CI [0.26, 1.13]. As in the direction assigned condition, the fact that the variance associated with the random slope factor was significant suggests that a statistical model which only allows individuals to differ in their intercepts should be rejected. However, the covariation between these two random factors did not attain significance, Cov = −0.10, z = −0.86, p = 0.39, 95% CI [−0.40, 0.13], which corresponds to a correlation (r) of −0.198.

One way to help visualize the treatment effects associated with the random slope and intercept factors is to depict the corresponding theoretical population distributions that were extracted by the growth-curve models in this experiment. The distributions associated with the direction assigned and direction selected conditions are depicted in Figure 5; as required by the mixed-effects modeling approach, these distributions are assumed to have a normal shape.

With respect to the direction assigned condition (solid lines), the mean value of the slope distribution was −0.21 units of agency/bit, and it had a standard deviation of 0.89 units of agency/bit (see Figure 5, Left); the mean value of the intercept distribution was 5.53 units of agency, and it had a standard deviation of 1.40 units of agency (see Figure 5, Right). Based on the random slope distribution, it was estimated that a slight majority (60%) of individuals would be expected to exhibit the expected pattern in which they felt the greatest agency in the 1.00-bit (100%-value) cue context and the least agency in the 3.00-bit (25%-valid) cue context—i.e., a slope < 0, though this also means that a substantial percentage of individuals (40%) would also be expected to exhibit the opposite pattern. These findings corroborate the findings reported by Gibson et al. (2023) using a number cue that should not have elicited involuntary (or automatized) shifts of attention.

With respect to the direction selected condition (dotted lines), the mean value of the slope distribution was −0.95 units of agency/bit, and it had a standard deviation of 0.74 units of agency/bit (see Figure 5, Left); the mean value of the intercept distribution was 6.30 units of agency, and it had a standard deviation of 0.68 units of agency (see Figure 5, Right). Based on the random slope distribution, it was estimated that a substantial majority (90%) of individuals would be expected to exhibit the expected pattern in which they felt the greatest agency in the 1.00-bit (100%-value) cue context and the least agency in the 3.00-bit (25%-valid) cue context—i.e., a slope < 0. These findings support the conclusion that allowing participants to voluntarily choose (or not) the direction of the number cue on each trial increased the proportion of participants who generated negatively sloped growth curves.

The percentage of positive and negative slopes estimated by the mixed-effects modeling approach requires an assumption of normality, which can be difficult to prove, especially when relatively small samples are used. To allay these concerns, we also used the Ordinary Least Squares (OLS) method to estimate the observed agency slopes for each participant in order to count the relative number of positive and negative slopes observed in our sample, as well as to help visualize how the direction of the slope estimates varied inversely with the intercept estimates, at least in the direction assigned condition.

With respect to the direction assigned condition, there were 48 individuals (60%) with negative slopes, and 24 individuals (30%) with positive slopes; eight individuals (10%) provided the same rating across all three conditional target entropy conditions and were assigned a slope of zero. These percentages are very similar to the percentages estimated by the corresponding growth curve model. Likewise, with respect to the direction selected condition, there were 65 individuals (81.25%) with negative slopes, and 10 individuals (12.5%) with positive slopes; five individuals (6.25%) provided the same rating across all three conditional target entropy conditions and were assigned a slope of zero. These percentages were also very similar to the percentages estimated by the corresponding growth curve model. Thus, the percentage of positive and negative slopes observed in the empirical frequency distribution of OLS slopes was similar to the percentage inferred from the theoretical distributions suggesting that the assumption of normality was justified.

It is also worth pointing out that, within the context of classical test theory, reliability is conceptualized as the ratio of true variance to observed variance. In the present study, the estimated variance of the random slope and intercept factors can be interpreted as the true variance and the estimated variance of the individual OLS slopes and intercepts can be interpreted as the observed variance. In the direction assigned condition, the ratio of these two variances was found to be 0.89 for slopes and 0.91 for intercepts; in the direction selected condition, the ratio of these two variances was found to be 0.68 for slopes and 0.49 for intercepts.

Figures 6A, B show the relation between the direction of the OLS slope estimates and the magnitude of the OLS intercept estimates in the direction assigned and direction selected conditions, respectively. With respect to the direction assigned condition (see Figure 6A), in the 1.00-bit (100%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 5.97, SE = 0.18) were significantly higher than the agency ratings reported by the group with positively sloped growth curve estimates (M = 4.19, SE = 0.25), F_{(1, 70)} = 33.85, p < 0.001, η² = 0.33. In contrast, in the 3.00-bit (25%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 4.40, SE = 0.19) were now significantly lower than the agency ratings reported by the group with positively sloped growth curve estimates (M = 5.86, SE = 0.27), F_{(1, 70)} = 20.08, p < 0.001, η² = 0.22. In the 2.36-bit (70%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 4.99, SE = 0.16) were more similar to the agency ratings reported by the group with positively sloped growth curve estimates (M = 5.41, SE = 0.22) and the two groups did not differ significantly in this condition, F_{(1, 70)} = 2.41, p = 0.12, η² = 0.03.

In addition, we also conducted a repeated measures analysis on agency ratings in the direction assigned condition with cue presence (no cue vs. 3.00 bits) as the sole within-subjects factor in both the negatively sloped and positively sloped growth curve groups separately (see the black and white triangles in Figure 6A, respectively). This analysis was conducted using the univariate approach because the sphericity assumption could not be violated with only two levels of the factor. As expected, agency ratings were significantly higher in the no cue condition (M = 5.30, SE = 0.21) than in the 3.00-bit (25%-valid) cue condition (M = 4.40, SE = 0.21), for those individuals with negatively sloped growth curve estimates, F_{(1, 47)} = 20.33, p < 0.001, d = 0.92. Likewise, agency ratings were also significantly higher in the no cue condition (M = 6.55, SE = 0.11) than in the 3.00-bit (25%-valid) cue condition (M = 5.86, SE = 0.18), for those individuals with positively sloped growth curve estimates, F(1,23) = 12.33, p = 0.002, d = 1.01. Thus, these findings also replicate the findings reported by Gibson et al. (2023), and suggest that individuals had a stronger sense of agency when they searched for the target without a cue, even though neither context provided any top-down information about the location (or identity) of the target.

Turning now to the direction selected condition (see Figure 6B), in the 1.00-bit (100%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 6.22, SE = 0.11) were now similar to the agency ratings reported by the group with positively sloped growth curve estimates (M = 5.88, SE = 0.29), and the two groups did not differ significantly in this condition, F_{(1, 73)} = 1.82, p = 0.18, η² = 0.02. In contrast, in the 2.36-bit (70%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 4.97, SE = 0.12) were now significantly lower than the agency ratings reported by the group with positively sloped growth curve estimates (M = 5.98, SE = 0.31), F_{(1, 73)} = 8.71, p = 0.004, η² = 0.11. Likewise, in the 3.00-bit (25%-valid) cue condition, the agency ratings reported by the group of individuals with negatively sloped growth curve estimates (M = 3.69, SE = 0.16 units of agency/bit) were also significantly lower than the agency ratings reported by the group with positively sloped growth curve estimates (M = 6.30, SE = 0.48), F_{(1, 73)} = 25.54, p < 0.001, η² = 0.26.

For the sake of comparison, Figure 6C shows the growth curves of those individuals who generated negatively sloped growth curves from each of the two volition conditions, and Figure 6D shows the growth curves of those individuals who generated positively sloped growth curves from each of the two volition conditions. As can be seen, those who generated negatively sloped growth curves were very similar across the two volition conditions whereas those who generated positively sloped growth curves tended to have higher agency ratings in the direction selected condition than in the direction assigned condition across the three levels of conditional target entropy, and especially in the 1.00-bit (100%-valid) cue conditions.

We also examined the extent to which participants exercised their freedom to choose the direction of the cue in the direction selected condition. Figure 7 shows the proportion of trials that participants chose the randomly assigned direction in the direction selected condition as function of conditional target entropy. The results of the linear growth curve analysis revealed that the average slope was close to zero (0.007 units of agency/bit), and non-significant, F_{(1, 79)} = 1.26, p = 0.27, η² = 0.02. In addition, only the variance associated with the random intercept factor was found to be significant, Var = 0.10, z = 5.99, p < 0.0001, 95% CI [0.08, 0.15]. The variance associated with the random slope factor was found to be non-significant, Var = 0.0004, z = 0.71, p =0.24, 95% CI [0.00009, 0.40], as was the covariation between these two random factors, Cov = −0.001, z = −0.53, p =0.60, 95% CI [-0.005, 0.003], which corresponds to a correlation (r) of −0.17.

Given that the average slope did not differ from zero and given that individual slopes did not differ significantly around this average, we averaged the proportion of trials that participants chose the randomly assigned direction across the three levels of conditional target entropy. The modal proportion was 1.00 (N = 18) and a total of 41 participants (51.25%) had proportions that were 0.90 or above. Thus, a substantial number of participants chose not to choose the direction of the cue in the direction selected condition. Of course, this finding may not be completely unexpected given that a similar number (48) of individuals were able to align their experience of agency with the top-down information that was conveyed by the cues in the direction assigned condition. However, upon closer inspection, those participants who chose not to choose the direction of the cue in the direction selected condition did not always align their experience of agency with the top-down information conveyed by the cues. Specifically, seven of these participants generated positively sloped growth curves and three of these participants generated a zero slope. Thus, a small number of participants seemed to resist exercising their freedom to select the direction of the cues as well as the opportunity to align their agency with the top-down information conveyed by the cues.

In addition, we also examined how participants distributed their choices across the four possible directions in the direction selected condition. We measured the distribution of direction choices by comparing the observed proportion of times each participant chose each direction with a model that predicted an equal distribution (0.25) of choices across the four directions. The magnitude of the deviation of the observed proportion from the predicted proportion was then squared, and we summed these squared deviations across the four directions to get a measure of error.

Figure 8 is a scatterplot that relates this error to the average proportion of trials that each participant chose the randomly assigned direction. As expected, the error was essentially zero on y-axis when the average proportion of accepting the randomly assigned direction was near 1.00 on the x-axis because these participants chose not to change the cued direction, and the experiment was programmed to randomly assign the cued direction across the four directions. However, most individuals (N = 68) managed to distribute their choices equally across the four directions even when they exercised their freedom to choose the cued direction on nearly all the trials. We identified this large group of participants as the “small bias” group (see the circle symbols clustered on or near the x-axis) in Figure 8. However, we also identified two smaller groups of participants who exercised their freedom to choose the cued direction on nearly all the trials, but who also seemed to consistently chose only one or two directions throughout the duration of the experiment. For instance, we identified one group of four participants as the “medium bias” group (see the triangle symbols), and another group of eight participants as the “large bias” group (see the diamond symbols) in Figure 8 based on the magnitude of their error from the random distribution model.

Figure 9 shows more clearly how these three groups of participants distributed their direction choices (or not) across the four cued directions. As can be seen in Figure 9, when there was a direction bias (as in the “medium bias” and “large bias” groups), participants tended to prefer the above direction. Note that there is nothing inherently wrong with this pattern of choices, as participants in the direction selected condition were allowed to choose any direction they desired. In fact, it is perhaps surprising that only a total of 12 participants (15%) opted to cue the same direction on every (or nearly every) trial.