S.-W. Wu, M. F. Dal Martello, and L. T. Maloney (2009) evaluated subjects' performance in a visuo-motor task where subjects were asked to hit *two* targets in sequence within a fixed time limit. Hitting targets earned rewards and Wu et al. varied rewards associated with targets. They found that subjects failed to maximize expected gain; they failed to invest more time in the movement to the more valuable target. What could explain this lack of response to reward? We first considered the possibility that subjects require training in allocating time between two movements. In Experiment 1, we found that, after extensive training, subjects still failed: They did not vary time allocation with changes in payoff. However, their actual gains equaled or exceeded the expected gain of an ideal time allocator, indicating that constraining time itself has a cost for motor accuracy. In a second experiment, we found that movements made under externally imposed time limits were less accurate than movements made with the same timing freely selected by the mover. Constrained time allocation cost about 17% in expected gain. These results suggest that there is no single speed–accuracy tradeoff for movement in our task and that subjects pursued different motor strategies with distinct speed–accuracy tradeoffs in different conditions.

*that constrained time allocation in itself might reduce accuracy of the sequential movements*. If this were the case, it would challenge the idea that there is a simple tradeoff between time and accuracy embodied in an SAT function. We return to this point in the Discussion.

*constrained timing task*and the

*choice timing task,*respectively. We were interested in whether subjects who had received training in the constrained timing task would later vary their time allocation in the test condition so as to increase their expected reward.

*s*is a motor strategy,

*S*is the set of all possible strategies,

*R*

_{ i}is the value of the

*i*th target,

*H*

_{ i}is the event of the

*i*th target being hit, and

*C*is the event that the trial is eligible for rewards. As described previously, a trial was eligible for rewards if both reaches fell within the outer rings of the targets and the second movement was completed before the time limit.

*s*is equivalent to the selection of movement time in the sequential reach, more particularly, the ratio of the movement time for the first reach,

*t*

_{1}, to the total movement time,

*T*. The underlying basic idea is to trade off between the movement times of the two reaches. With the ideal mover reduced to an “ideal time allocator,” we fit the model of Wu et al. (2009) to the data after testing its assumptions.

*t*

_{ i}is the planned movement time for the

*i*th reach,

*T*is the total movement time,

*P*(

*V*) is the probability of completing both movements before the time limit,

*P*(

*H*

_{ i}∣

*t*

_{ i}) is the probability to hit the

*i*th target given the

*i*th movement time is

*t*

_{ i}, and

*P*(

_{i}∣

*t*

_{i}) is the probability to hit within the outer ring of the

*i*th target when the

*i*th movement time is

*t*

_{ i}.

*T, P*(

*V*), and

*R*

_{i}are constants. To express expected gain purely as a function of time allocation,

*t*

_{1}/

*T,*we need to determine the nature of

*t*

_{2},

*P*(

*H*

_{i}∣

*t*

_{i}), and

*P*(

_{i}∣

*t*

_{i}).

*T*of a sequential movement consists of three parts: the first movement time

*t*

_{1}, the dwell time on the first endpoint

*t*

_{dwell}, the second movement time

*t*

_{2}. We found that we could readily predict dwell time: the ratio of dwell time to total time is a linear function of

*T*/

*t*

_{1}:

*m*and

*k*are parameters estimated from the data separately for each subject. Assuming that the subject chooses the same timing across trials of a condition, we compute the mean

*t*

_{dwell}/

*T*and

*T*/

*t*

_{1}for each of the four timing conditions in the training session and three value conditions in the test session to estimate

*m*and

*k*. Then we could write

*t*

_{2}as:

*P*(

*H*

_{i}∣

*t*

_{i}) and

*P*(

_{i}∣

*t*

_{i}), we adopted the following steps. First, we obtained the relation of the standard deviation of a movement's endpoint to its movement time. We model the standard deviation of the

*i*th movement separately for the directions parallel and perpendicular to the movement based on Schmidt, Zelaznik, Hawkins, Frank, and Quinn (1979):

*d*

_{i}is the distance of the

*i*th movement, and

*b*

_{∥},

*b*

_{⊥},

*c*

_{∥}, and

*c*

_{⊥}are estimated parameters. We assume that the subject has the same timing plan throughout a condition and the first and second movements have the same parameters. We compute

*σ*

_{∥},

*σ*

_{⊥}, and

*d*

_{i}/

*t*

_{i}for the four timing conditions in the training session to estimate the parameters.

*i*th reach,

_{ i}, is distributed as a bivariate Gaussian random variable whose mean is the center of the target,

*f*

_{ i}(

*t*

_{i}) over the target or outer circle using the integration method of DiDonato and Jarnagin (1961):

*r*and 4

*r*are respectively the radius of the target and outer circle. Although the subject is supposed to aim at the center of the target, there can be constant error in their movements as found in other studies (Wright & Meyer, 1983). We compute the constant errors for the first and second targets in the parallel and perpendicular directions as an average across all trials of the experiment. As noted above, this error had negligible effect on subjects' expected gain.

*t*

_{1}/

*T,*for each timing condition and regressed it against the required ratio.

*t*

_{1}/

*T*was 180/600, S01 spent a larger share of time on the first movement than required, while when required

*t*

_{1}/

*T*was 340/600 or 420/600, the reverse. That is, mean observed

*t*

_{1}/

*T*in both higher and lower required

*t*

_{1}/

*T*conditions approached to a middle value. This pattern repeated itself with all subsequent subjects. Figure 3B shows the fitted regression line for all the eight subjects and the average across subjects.

^{1}(Efron & Tibshirani, 1993), resampling the movement times for each timing condition with 10,000 runs. At the 95% confidence level, all the slopes were greater than zero except for one subject, demonstrating that subjects were able to voluntarily vary their timing in sequential movements. However, all slopes were lesser than one, implying that subjects did not do the constrained time division perfectly and instead contracted toward a

*preferred t*

_{1}/

*T ratio*. The mean slope across subjects was 0.52. Consistent with the above individual analysis, the mean slope, by two-tailed Student's

*t*-tests, was significantly greater than zero,

*t*(7) = 5.72,

*p*< .001, and significantly smaller than one,

*t*(7) = −5.20,

*p*= .001.

*t*

_{1}/

*T*ratio to required

*t*

_{1}/

*T*ratio for each group. We examined the training effect by calculating the regressive slope of the abstract difference to the group number. The last group was not included, for it was performed immediately before the test session, typically on the next day of the training session. There was no evidence of improvement.

*t*

_{1}/

*T*across subjects against trial group for each timing condition. Improvement in timing performance should have resulted in a negative slope. At the 95% confidence level (Bonferroni corrected for four conditions), only one slope of one subject was significantly different from zero. Neither did the timing performance worsen after the interval between the training session and the test session. A one-tailed Student's

*t*-test for each subject in each timing condition revealed few differences between the mean absolute deviation of the movement time of the 9th group and that of the 10th group. Averaged across all the eight subjects, only in 0.5 out of 4 conditions

^{2}was the mean observed

*t*

_{1}/

*T*of the 10th trial group further from the required

*t*

_{1}/

*T*than that of the 9th trial group at the 95% confidence level (Bonferroni corrected for four conditions).

*P*(hit2∣miss1) is plotted against

*P*(hit2∣hit1) in Figure 4 for each subject (in a unique color). According to Pearson's

*χ*

^{2}test on the number of hits or misses, at the 95% confidence level (Bonferroni corrected for seven conditions),

*P*(hit2∣miss1) differed from

*P*(hit2∣hit1) only for two data points of two different subjects in two different conditions (circled in Figure 4). Put together, these two analyses demonstrate that the two movements within a sequential movement can be treated as independent, in agreement with the conclusions of Wu et al. (2009).

*t*

_{dwell}/

*T*−

*T*/

*t*

_{1}pairs and fitted line of subject S01. The

*R*

^{2}of the eight subjects (in descending order) were .95, .89, .86, .72, .67, .66, .64, and .61. The median across subjects was .70.

*t*

_{1}/

*T*with the optimal

*t*

_{1}/

*T*that led to maximal expected gain. We used a bootstrap method (Efron & Tibshirani, 1993) as follows to estimate the 95% confidence interval (Bonferroni corrected for three conditions) of the observed-optimal

*t*

_{1}/

*T*difference. We ran a simulated experiment for 10,000 runs. In each run, we resampled data for each condition in each group, then estimated the parameters in the dwell time and SAT functions, and finally searched for the maximum of the subject's expected gain function and estimated the optimal

*t*

_{1}/

*T*. We used bootstrap methods based on 10,000 simulations (Efron & Tibshirani, 1993) to calculate 95% confidence intervals for these estimates.

*t*

_{1}/

*T*ratio at all.

*t*

_{1}/

*T*ratio in each value condition is plotted in Figure 6B against the subject's model-predicted optimal

*t*

_{1}/

*T*ratio, with “optimal” data points in black and “suboptimal” ones in red. An observed

*t*

_{1}/

*T*ratio is labeled

^{3}“optimal” if it did not significantly deviate from optimal

*t*

_{1}/

*T*at the 95% confidence level (Bonferroni corrected for three conditions) according to the bootstrap test; otherwise, “suboptimal.”

*t*

_{1}/

*T*range as shown in the training session are also presented. Two points should be highlighted for Figure 6B: First, all but one subject did not vary their time allocation (the three points fall on a horizontal line). Second, most subjects' observed

*t*

_{1}/

*T*ratios were close to their preferred ratio in training. The remaining subject S02 (upper row, center) was the subject who was partially aware of the hypothesis under test. He did vary time allocation but two of his three time allocations are significantly different from optimal.

*P*(

*V*) was computed for each subject and each value condition as the proportion of trials in the test session in which the time limit was not exceeded. For each subject, we computed the 95% confidence interval (Bonferroni corrected for three conditions) of efficiency using the method for computing observed-optimal

*t*

_{1}/

*T*difference confidence interval as described earlier. Figure 7 shows the data. To our surprise, almost no efficiencies were significantly smaller than one, and some were even significantly larger than one.

*constrained timing task*before completing a decision task similar to that of Wu et al. (2009) in which they could pick whatever allocation of time they wished (

*choice timing task*). The constrained timing task demonstrated subjects' ability to divide up movement time arbitrarily and should have given them opportunity to observe how their own accuracy varied with the duration of each movement. However, we found that, even after the 800 trials of training, subjects did not vary their timing in the choice timing decision task.

*preferred ratio*. A second interesting finding is that the dwell times (the time subjects spent in contact with the first target before initiating the second movement) had a simple reciprocal relation to the proportion of time allocated to the first movement.

*F*(3, 9) = 0.56,

*p*= .66. For Experiment 1, the effect of condition is significant,

*F*(6, 42) = 15.12,

*p*< .001, but as a Tukey's HSD test shows, the significant differences are either between two constrained timing conditions, or between choice and constrained conditions with the standard deviation of a choice timing condition significantly

*less*than that of a constrained timing condition, exactly the reverse of what we might expect given previous work. Subjects achieve higher spatial accuracy without detectable decreases in temporal accuracy (Experiment 2) or even with increases in temporal accuracy (Experiment 1).

^{1}The confidence limits on linear regression parameters are typically calculated in closed form based on the assumption that the distribution of errors is Gaussian (Draper & Smith, 1998, p. 34ff). Examination of QQ plots (Gnanadesikan & Wilks, 1968) of the observed

*t*

_{1}/

*T*values separately for each timing condition and each subject indicates that the distribution of errors, in many cases, deviated from Gaussian. Accordingly, we calculated confidence limits on regression slope estimates using bootstrap (resampling) methods (Efron & Tibshirani, 1993) since these methods are less sensitive to failures of distributional assumptions. We also repeated all analyses of hypotheses concerning regression slopes, computing the confidence limits in the usual way, and reached the same conclusions as we reached using bootstrap methods.

*Psychonomic Bulletin & Review*, 12, 911–916. [PubMed] [CrossRef] [PubMed]

*Journal of Neuroscience*, 27, 6984–6994. [PubMed] [CrossRef] [PubMed]

*Statistical decision theory and Bayesian analysis*(2nd ed.). New York: Springer.

*Theory of games and statistical decisions*. New York: Wiley.

*Spatial Vision*, 10, 433–436. [PubMed] [CrossRef] [PubMed]

*Human Movement Science*, 27, 771–798. [PubMed] [CrossRef] [PubMed]

*Journal of Motor Behavior*, 26, 43–50. [PubMed] [CrossRef] [PubMed]

*Journal of Vision*, 7, (5):10, 1–12, http://www.journalofvision.org/content/7/5/10, doi:10.1167/7.5.10. [PubMed] [Article] [CrossRef] [PubMed]

*Mathematics of Computation*, 15, 375–382. [CrossRef]

*Journal of Neurophysiology*, 101, 283–288. [PubMed] [CrossRef] [PubMed]

*Applied regression analysis*(3rd ed.). New York: Wiley-Interscience.

*An introduction to the bootstrap*. New York: Chapman & Hall.

*Journal of Experimental Psychology*, 47, 381–391. [PubMed] [CrossRef] [PubMed]

*PLoS Computational Biology*, 4, 1–9. [PubMed] [Article] [CrossRef]

*Psychological Review*, 89, 449–482. [PubMed] [CrossRef] [PubMed]

*Spatial Vision*, 10, 437–442. [PubMed] [CrossRef] [PubMed]

*Psychological Review*, 86, 415–451. [PubMed] [CrossRef]

*Information processing in motor control and learning*. (pp. 183–203). New York: Academic Press.

*Quarterly Journal of Experimental Psychology*, 49, 379–397. [PubMed] [CrossRef] [PubMed]

*Vision and action*(pp. 332–349). Cambridge, UK: Cambridge University Press.

*Nature Neuroscience*, 7, 907–915. [PubMed] [CrossRef] [PubMed]

*Nature Neuroscience*, 5, 1226–1235. [PubMed] [CrossRef] [PubMed]

*Journal of Neuroscience*, 25, 7169–7178. [PubMed] [CrossRef] [PubMed]

*Psychological Science*, 17, 981–988. [PubMed] [CrossRef] [PubMed]

*Trends in Cognitive Sciences*, 12, 291–297. [PubMed] [CrossRef] [PubMed]

*Journal of the Optical Society of America A, Optics, Image Science, and Vision*, 20, 1419–1433. [PubMed] [CrossRef]

*Spatial Vision*, 16, 255–275. [PubMed] [CrossRef]

*Flat maxima in linear optimization models*(Tech Rep. No. 011313- 4-T). Engineering Psychology Laboratory, University of Michigan.

*Proceedings of the Royal Society of London B: Biological Sciences*, 277, 1667–1674. [PubMed] [CrossRef]

*Quarterly Journal of Experimental Psychology*, 35, 279–296. [PubMed] [CrossRef] [PubMed]

*Journal of Experimental Psychology: Human Perception and Performance*, 14, 221–230. [CrossRef]