Hitting a baseball, one of the most difficult skills in all of sports, requires complex hand-eye coordination, but its link with basic visuomotor capabilities remains largely unknown. Here we examined basic visuomotor skills of baseball players and demographically matched nonathletes by measuring their ocular-tracking and manual-control performance. We further investigated how these two capabilities relate to batting performance in baseball players. Compared to nonathletes, baseball players showed better ocular-tracking and manual-control capabilities, which remain unchanged with increasing baseball experience. Both, however, become more correlated with batting accuracy with increasing experience. Ocular-tracking performance is predictive of batting skill, accounting for ≥ 70% of the variance in batting performance across players with ≥ 10 years of experience. A simple linear additive-noise cascade model with shared front-end visual noise that limits batting performance can explain many of our results. Our findings show that fundamental visuomotor capabilities can predict the complex, learned skill of baseball batting.

*SD*: 27 ± 8 years) with baseball experience in the range of 3 to 30 years (mean ±

*SD*: 11 ± 6 years) participated in this study. Although none of them were professional baseball players, all had competition experience in major international baseball contests such as the Hong Kong Baseball Open, the Phoenix Cup Hong Kong International Women's Baseball Tournament, and/or the Women's Baseball World Cup. Many also played softball before starting playing baseball and joining the National Team. Based on their field positions in major baseball contests, there were 16 infielders (9 males, 7 females), 12 outfielders (3 males, 9 females), 12 pitchers (3 males, 9 females), and 4 catchers (2 males, 2 females) in our baseball player participant group. According to the World Baseball Softball Confederation, the Hong Kong men's national baseball team was ranked 30th out of 85 nations and the Hong Kong women's team was ranked 10th out of 20 nations in 2019.

*SD*: 24 ± 6 years) participated in the experiment as the control group. All were staff or students at the University of Hong Kong and reported having no previous competitive ball-sports experience.

*SD*: 28 ± 8 years) with baseball experience in the range of 3 to 18 years (mean ±

*SD*: 9 ± 4 years) volunteered to participate in the batting performance test on a different day.

^{2}) was displayed in the center of a black background (2.2 cd/m

^{2}) on a computer screen (39°H × 30°V). Participants were asked to fixate the central target and press a mouse button to trigger the start of each trial. After a random time delay drawn from a truncated exponential distribution (mean: 700 ms; minimum: 200 ms; maximum: 5,000 ms), the target would jump 3.2° to 4.8° away from the fixation point and immediately move back at a constant speed toward the center of the screen and then onward for a random amount of time from 700 to 1,000 ms before disappearing (see Figure 1a). To minimize the likelihood of an initial catch-up saccade, the target always crossed the center of the screen at 200 ms after its motion onset. Both the target speed and moving direction were randomly sampled from a range (speed range: 16–24°/s; direction range: 0–358° in 2° increment without replacement) to minimize expectation effects. Participants were instructed to keep their eyes on the target in the center without blinking once they initiated the trial and then to follow it as best as they could once it started moving until it disappeared.

*SE*) for the baseball players and 0.68° ± 0.03° for the nonathletes. The ocular-tracking task took less than 30 min to finish.

- 1. Pursuit initiation as measured by
*latency*(the median across trials of the time between target motion onset and the initiation of smooth pursuit eye movements) and*open-loop acceleration*(the median across trials of the mean radial acceleration along the target direction of smooth pursuit in the 100-ms interval immediately following pursuit onset). - 2. Steady-state tracking as measured by
*steady-state gain*(the median across trials of the mean speed of smooth pursuit in the steady-state tracking interval 400 to 700 ms after motion onset, projected along the direction of target motion and divided by the target speed),*proportion smooth*(the median across trials of the proportion of time that tracking within the steady-state tracking interval was smooth pursuit, as a metric of how much pursuit is contributing to steady-state tracking),*saccadic rate*(the total number of both forward and backward catch-up saccades made in the 400- to 700-ms period of steady-state tracking divided by the total steady-state tracking time, i.e., 300 ms),*saccadic amplitude*(the median amplitude of the forward catch-up saccades occurring in the steady-state tracking interval with forward saccades classified as within 180° of the direction of target motion), and*saccadic dispersion*(the standard deviation of the distribution of directions across forward catch-up saccades). - 3. Direction tuning as measured by
*direction noise*and*direction asymmetry and anisotropy*. Direction noise is the average across trials of the local standard deviations taken across the measured pursuit directions at a given direction and its two nearest neighbor directions corrected for the 2° expected differences.*Direction asymmetry and anisotropy*refer to vertical-horizontal asymmetry and oblique-cardinal anisotropy, respectively, which are the best-fitting first and second polar harmonic modulations of the direction gain (i.e., the local linear regression slope of the pursuit- versus target-direction curve within a 30° window; see details in Krukowski & Stone, 2005, and Liston & Stone, 2014). - 4. Speed tuning as measured by
*speed noise*(the standard deviation across trials of the difference between the actual radial pursuit speed and the best linear regression estimate for a given target speed divided by the mean target speed) and*speed responsiveness*(the best-fitting linear regression slope of the mean radial pursuit speed vs. target speed). These two measures capture how well pursuit can discriminate between small differences in target speed as opposed to how*effective*the pursuit response is in general, which is captured by steady-state gain.

*SE*) out of 180 trials were used for the pursuit-initiation and direction-tuning analysis, and 168 ± 3 out of 180 trials were used for the steady-state tracking and speed-tuning analysis. For the nonathletes, on average, 136 ± 4 out of 180 trials were used for the pursuit initiation and direction-tuning analysis, and 163 ± 3 out of 180 trials were used for the steady-state tracking and speed-tuning analysis.

*M*) and standard deviation (

*σ*):

^{−1}is the inverse of the normal cumulative distribution function. We then converted each raw oculometric measure into a

*Z*value metric (ω) relative to the normative standard for each baseball player (ω

_{baseball}) and nonathlete (ω

_{control}):

*Baseball*ω

*i*) and nonathlete (

*Control*ω

*i*):

*Baseball*ω

*i*vectors across our baseball player population to yield a baseball vector (

*Baseball*

*vector*):

*N*is the number of the baseball players. Because the

_{Baseball}*Baseball*ω

*i*vectors are “normalized,” each element of the

*Baseball vector*gives the distance (in

*z*values) between the average baseball player participant and the average of the control population for a specific oculometric measure, and the larger distances weight the oculometric measures with higher discrimination power. To quantify the scalar magnitude of ocular-tracking performance along the

*Baseball vector*, we took the dot product between an individual's 12-element normalized oculometric vector (

*Baseball*ω

*i*or

*Control*ω

*i*) and the

*Baseball vector*to yield a projection-based scalar metric (i.e., ocular-tracking performance index):

*Control*ω is the matrix containing all oculometric vectors in the control population of nonathletes, COV is the covariance matrix, and CHOL is the Cholesky decomposition. Applying the Cholesky decomposition of the covariance matrix of

*Control*ω to the

*Baseball vector*produces a sample baseball player vector with the covariance properties of the 12 oculometric measures in the control population.

*Scaling factor*in the denominator thus normalizes the ocular-tracking performance index by taking the magnitude of the correlation between oculometric measures into consideration.

^{2}) was displayed on a uniform black background (0.14 cd/m

^{2}) on a rear projected large screen (110°H × 94°V) at 60 Hz (see Figure 1b). Its horizontal position was updated by a perturbation function

*u*consisting of the sum of seven harmonically unrelated sinusoids, given as:

*a*represents the amplitude and

_{i}*f*represents the frequency of the

_{i}*i*th sine component (see Table 1). ρ

_{i}is a random phase offset drawn from –π to π on each trial.

*D*is the disturbance gain, which was set to 8.1° and led to an average uncorrected perturbation speed of 25.1°/s (peak: 95.7°/s). This sum-of-sinusoids perturbation series made the target movement appear random and allowed for a frequency-based analysis of the control response.

^{2}.

*root mean square*(RMS) of the time series of the target position error relative to the center of the screen, (b) control response amplitude and (c) delay, as measured by

*gain*and

*phase Iag*from the frequency response (Bode) analyses on the recorded time-series data. Specifically, we performed Fourier transform of the time series of both the joystick control output (in percentage of maximum displacement) and the target position error (in degrees of visual angle or deg). We computed the control response amplitude (i.e., gain in percentage of max/deg) by taking the ratio of the Fourier coefficients of the joystick displacement and the target position error at each perturbation frequency, and the response delay (i.e., phase lag in degrees of sinusoidal phase or °) by taking the phase difference between the Fourier components of the joystick displacement and the target position error at each perturbation frequency.

*Baseball*ω

*i*) and nonathlete (

*Control*ω

*i*):

*hit rate*(total number of hits divided by 30). Due to the tight training schedule, participants were tested on two to three pitches per week, and the entire test lasted about 3 months.

*t*tests showed that the baseball players were superior to the nonathletes in all four aspects of ocular-tracking performance, as indicated by 8 out of 12 oculometric measures: shorter latency (

*t*(84) = 2.84,

*p*= 0.0056, Cohen's

*d*= 0.61), larger open-loop acceleration that reached borderline significance (

*t*(

*84*) = 1.98,

*p*= 0.051, Cohen's

*d*= 0.43), larger steady-state gain (

*t*(84) = 4.12,

*p*< 0.001, Cohen's

*d*= 0.89), smaller catch-up saccade amplitude (

*t*(84) = 2.98,

*p*= 0.0038, Cohen's

*d*= 0.64) and dispersion (

*t*(84) = 2.78,

*p*= 0.0068, Cohen's

*d*= 0.60), smaller direction noise (

*t*(84) = 3.48,

*p*< 0.001, Cohen's

*d*= 0.75) and vertical-horizontal asymmetry (

*t*(84) = 2.49,

*p*= 0.015, Cohen's

*d*= 0.54), and larger speed responsiveness (

*t*(84) = 2.50,

*p*= 0.014, Cohen's

*d*= 0.54). In summary, despite considerable within-group variance observed for each oculometric measure, the baseball players systematically showed overall better ocular-tracking performance than did the demographically matched nonathletes across most (but not all) metrics.

*t*test showed that our baseball players had better binocular Freiburg visual acuity (Bach, 1996) than the nonathletes (

*t*(84) = 3.47,

*p*< 0.001, Cohen's

*d*= 0.75). However, across both the baseball player and the nonathlete participants in the current study, visual acuity was not significantly correlated with any of the 12 oculometric measures (Pearson's

*r*(86) ≤ 0.25,

*p*≥ 0.22 after Holm's sequential Bonferroni correction for multiple correlations; see also Liston & Stone, 2014). This indicates that visual acuity is not a contributing factor to the better ocular-tracking performance observed in baseball players. Furthermore, it has been reported that there is no significant difference in the static visual acuity among professional Japanese baseball players at different performance levels (Hoshina et al., 2013), indicating that visual acuity is also not a predictor of baseball performance level.

*t*test showed that the values of the ocular-tracking performance index were significantly higher for the baseball players than for the nonathletes (

*t*(84) = 4.66,

*p*< 0.001, Cohen's

*d*= 1.00), indicating that the baseball players had overall better ocular-tracking capabilities than did the nonathletes. In addition, the ROC area for the ocular-tracking performance index (0.79) was larger than that for each of the 12 oculometric measures (see Figure 3) and exceeded 0.70, indicating that the combined ocular-tracking performance index has strong discrimination power to separate baseball players from nonathletes.

*F*(3, 43) < 2.55,

*p*> 0.069, η

^{2}< 0.16). A one-way ANOVA with player position as a categorical variable on the ocular-tracking performance index also did not reveal any significant effect of player position (

*F*(3, 43) = 1.21,

*p*= 0.32, η

^{2}= 0.083). These results indicate that the baseball player's position does not have a significant relationship with ocular-tracking performance.

*t*test revealed that the RMS error in degrees of visual angle (deg) was significantly smaller for the baseball players than for the nonathletes (mean ±

*SD*: 31.15 ± 2.30 deg vs. 34.88 ± 3.16 deg,

*t*(88) = 6.36,

*p*< 0.001, Cohen's

*d*= 1.34), indicating that the baseball players’ overall control performance was better than the nonathletes.

*F*(1, 88) = 39.80,

*p*< 0.001, η

^{2}= 0.31 and

*F*(6, 528) = 301.32,

*p*< 0.001, η

^{2}= 0.77, respectively), and so was their interaction effect (

*F*(6, 528) = 19.29,

*p*< 0.001, η

^{2}= 0.18). Gain increased with perturbation frequency at lower frequencies and then decreased with perturbation frequency at higher frequencies, which is a typical response characteristic for acceleration control dynamics (Li et al., 2005, 2016). Newman-Keuls post hoc tests revealed that while the baseball players did not differ from the nonathletes in gain at the four lower frequencies, they showed larger gain in the three highest frequencies (0.74–2.19 Hz:

*p*< 0.001), indicating that the baseball players were more responsive to high-frequency motion signals. An independent-samples

*t*test showed that the mean gain averaged across seven input perturbation frequencies was also significantly larger for the baseball players than for the nonathletes (9.3 ±1.3 dB vs. 6.8 ± 2.3 dB,

*t*(88) = 6.31,

*p*< 0.001, Cohen's

*d*= 1.33).

*F*(1, 88) = 58.71,

*p*< 0.001, η

^{2}= 0.40 and

*F*(6, 528) = 4764.15,

*p*< 0.001, η

^{2}= 0.98, respectively), but their interaction effect was not significant (

*F*(6, 528) = 1.30,

*p*= 0.25, η

^{2}= 0.010). As expected, phase lag increased with perturbation frequency. An independent-samples

*t*test showed that the mean phase lag averaged across seven input perturbation frequencies was also significantly smaller for the baseball players than for the nonathletes (77.7 ± 7.7° vs. 91.0 ± 8.7°,

*t*(88) = 7.66,

*p*< 0.001, Cohen's

*d*= 1.62), indicating that the baseball players initiated manual-control responses sooner than did the nonathletes across all frequencies.

*t*test showed that the values of the manual-control performance index were significantly higher for the baseball players than for the nonathletes (

*t*(88) = 9.51,

*p*< 0.001, Cohen's

*d*= 2.01), showing that baseball players showed overall better manual-control capabilities than did the nonathletes. In addition, the ROC area for the manual-control performance index (0.92) was larger than that for each of the three manual-control measures (see Figure 5), indicating that the combined manual-control performance index also has excellent discrimination power to separate baseball players from nonathletes.

*F*(3, 42) < 1.45,

*p*> 0.24, η

^{2}< 0.10). A one-way ANOVA with player position as a categorical variable on the manual-control performance index also did not reveal a significant effect of player position (

*F*(3, 42) = 0.54,

*p*= 0.66, η

^{2}= 0.040). These results indicate that the baseball player's position does not have a significant relationship with manual-control performance.

*r*(43) = –0.45 and –0.47,

*p*= 0.030 and

*p*= 0.019, respectively, after Bonferroni correction), and no other significant correlations were found. For the nonathletes, no significant correlation was found between any oculometric and manual-control measures. This shows that for the baseball players, higher gain or speed responsiveness in ocular tracking can stochastically predict better performance in manual control, but no such prediction exists for the nonathletes.

*r*(43) = 0.45,

*p*= 0.0025) but not for the nonathletes (Pearson's

*r*(42) = 0.11,

*p*= 0.48). This shows that ocular-tracking and manual-control capabilities are highly linked in the baseball players but not in the nonathletes. That is, while better ocular-tracking performance is associated with better manual-control performance for the baseball players, better ocular-tracking performance does not imply better manual-control performance for the nonathletes.

*r*(44) = 0.077,

*p*= 0.31 and Pearson's

_{one-tailed}*r*(43) = 0.18,

*p*= 0.13, respectively). One-tailed testing is justified because of the a priori assumption that the correlation would be positive. As expected, these two basic visuomotor skills were not changed by playing baseball. On the other hand, hit rate of batting performance (Figure 6c) showed a trend of improvement with years of experience (Pearson's

_{one-tailed}*r*(23) = 0.32,

*p*= 0.070), perhaps failing to reach full significance because of a smaller sample size—we only had access to the batting performance of the female players, with only three of them having more than a decade of experience, thus limiting the

_{one-tailed}*x*-axis range. Note that although years of experience may correlate with age, there was no correlation between hit rate and age for these female baseball players (Pearson's

*r*(23) = –0.15,

*p*= 0.49). This was expected because increased age

*per se*does not correlate with improved baseball hitting performance (e.g., Ng, 2017).

*r*) of the ocular-tracking and manual-control performance indices on hit rate as a function of experience level (Figure 6d), with the cohort of baseball players with experience level

^{2}*N*defined as those with

*N*or more years of experience in playing baseball (Table 2). The predictive power for both indices on hit rate shows a significant linear increase with experience level (ocular tracking: Pearson's

*r*(23) = 0.92,

*p*< 0.001; manual control: Pearson's

_{one-tailed}*r*(22) = 0.84,

*p*= 0.0048). However, ocular-tracking performance indices showed systematically larger power than did manual-control performance indices in predicting hit rate. In fact, the correlation between hit rate and ocular-tracking performance index was significant at all experience levels, whereas that was never the case for manual-control performance index (Table 2).

_{one-tailed}*. This is supported by the findings that the noise in pursuit speed (Kowler & McKee, 1987) and direction (Stone & Krauzlis, 2003) provides indirect yet reliable measures of the noise in the visual perception of speed and direction, with little additional noise added by the oculomotor system. The small noise source in the oculomotor system (η*

_{v}*) is given by η*

_{o}_{o}

*=*ξ

*× η*

_{o}*, with ξ*

_{v}*the noise scalar for the oculomotor system.*

_{o}*) that adds to the overall noise observed in manual-control performance, which is given by η*

_{m}*ξ*

_{m}=*× η*

_{m}_{v}, with ξ

*the noise scalar for the motor system. Lastly, the model's third assumption is that there is additional independent batting noise (η*

_{m}*) added to the batting performance that decreases monotonically with baseball experience and affects batting accuracy, which is given by η*

_{b}*= (ξ*

_{b}*– ξ*

_{i}*× experience level) × η*

_{s}*, where ξ*

_{v}*represents the intercept and ξ*

_{i}*represents the slope of the linear trend in batting noise.*

_{s}*, ξ*

_{o}*, ξ*

_{m}*ξ*

_{s},*) by a best fit to the 16 data points in Figure 6d using a least squares procedure with 1,000 Monte Carlo simulations of the different sources of noise. The reduced χ*

_{i}^{2}of the model fit was 0.8, indicating that it can explain the high predictive power of ocular-tracking over manual-control performance on hit rate that increases with experience level (see Figure 6d). With the same fitted parameters, the model also independently quantitatively reproduces the observed significant correlation between ocular-tracking and manual-control performance indices in baseball players (mean and 95% confidence intervals of Pearson's

*r*: 0.43 [0.17, 0.65]; see Figure 4c). Last, the model can also explain the observed insignificant correlation in nonathletes shown in Figure 4c simply by fitting a new value of η

*to the nonathlete data to capture the fact that nonathletes in general have higher manual motor system output noise (η*

_{m}*= 8.37 × η*

_{m}*) than do baseball players (η*

_{v}*= 2.06 ×*

_{m}*η*), with all other parameters remaining the same. The higher η

_{v}*in nonathletes then dominates over η*

_{m}*to conceal any correlation. The lower η*

_{v}*in baseball players is presumably due to selection and training. Note that although this simple linear additive-noise cascade model with no nonlinearities or interactions explains our data well, we do not rule out the possibility that a more complex model with added nonlinear features, interactions, or nonindependent or multiplicative noise sources might fit our data better (at the expense of added complexity). What our model simulations capture is the fact that the data of this study can be successfully accounted for when ocular tracking, manual control, and baseball batting depend on significant shared noise in upstream visual processing of motion signals.*

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