Figure 2 shows the rotational speed thresholds (top) and temporal frequency thresholds (bottom) across all six conditions for participants separated by gender and age groups. As expected, thresholds decreased systematically with increasing number of targets. Rotational speed thresholds were lower for patterns with 10 dots/ring compared to five dots/ring, and this difference was virtually eliminated when thresholds were expressed in terms of temporal frequency (Holcombe & Chen,
2013). Comparing performance across age and gender groups revealed higher overall thresholds in men compared to women, and in younger participants compared to older participants. Importantly, the magnitude of the age group differences appeared to increase with the number of targets.
We analyzed the results using linear mixed-effects models both on raw and on log-transformed rotational speed and temporal frequency thresholds. Given that the pattern of results was similar for raw and log-transformed values, we report results for log-transformed analyses only. Note that effects observed on log-transformed values reflect differences in the ratios between groups and conditions as opposed to differences in means. The data were fit with the following linear mixed-effects model:
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\bf{\alpha}}\)\(\def\bupbeta{\bf{\beta}}\)\(\def\bupgamma{\bf{\gamma}}\)\(\def\bupdelta{\bf{\delta}}\)\(\def\bupvarepsilon{\bf{\varepsilon}}\)\(\def\bupzeta{\bf{\zeta}}\)\(\def\bupeta{\bf{\eta}}\)\(\def\buptheta{\bf{\theta}}\)\(\def\bupiota{\bf{\iota}}\)\(\def\bupkappa{\bf{\kappa}}\)\(\def\buplambda{\bf{\lambda}}\)\(\def\bupmu{\bf{\mu}}\)\(\def\bupnu{\bf{\nu}}\)\(\def\bupxi{\bf{\xi}}\)\(\def\bupomicron{\bf{\micron}}\)\(\def\buppi{\bf{\pi}}\)\(\def\buprho{\bf{\rho}}\)\(\def\bupsigma{\bf{\sigma}}\)\(\def\buptau{\bf{\tau}}\)\(\def\bupupsilon{\bf{\upsilon}}\)\(\def\bupphi{\bf{\phi}}\)\(\def\bupchi{\bf{\chi}}\)\(\def\buppsy{\bf{\psy}}\)\(\def\bupomega{\bf{\omega}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\begin{equation}{\rm{log}}2{(Y)_{ij}} = {\beta _0} + {\beta _g}{\rm{gender}} + {\beta _a}{\rm{age}} + {\beta _t}{\rm{targets}} + {\beta _d}{\rm{dpr}} + {\beta _{td}}{\rm{targets}}\mbox{:}{\rm{dpr}} + {\beta _{at}}{\rm{age}}\mbox{:}{\rm{targets}} + {\beta _{at}}{\rm{age}}\mbox{:}{\rm{dpr}} + {\beta _{atd}}{\rm{age}}\mbox{:}{\rm{targets}}\mbox{:}{\rm{dpr}} + {b_{0i}} + {b_{ti}} + {b_{di}} + {b_{tdi}} + {\epsilon _{ij}}\end{equation}
where
Y represents i-th subject's temporal frequency or the rotational speed threshold for j-th condition,
β are the coefficients for the fixed effects terms, the
b terms represent random effects for the by-subject intercept and slopes of the two within-subjects factors (targets and dots/ring), and
ϵij represents the normally-distributed residual errors. We specified the model's random effects terms with correlated intercepts and slopes for targets and dots/ring because this random-effects structure was a better fit than a model with uncorrelated slopes, χ
21 = 17.0,
p < 0.001, and a model with random by-subject intercepts only, χ
28 = 26.4,
p < 0.001. The fixed-effect terms included in the model were determined based on the effects of interest and a model comparison process. The terms for the age and gender interaction and higher order interactions containing age and gender effects were not included, due to the low power to detect these interactions with our sample size (the power to detect a two-way Age × Gender interaction was 0.28 and 0.09 for a medium and small effect size, respectively). The two- and three-way interactions between gender and the two within-subjects factors—target number and dots/ring—were also not included, because a comparison of the above model with a model containing these additional terms did not result in a significantly improved fit,
F(5, 56) = 0.84,
p = 0.53, which indicates that the effect of gender did not vary as a function of target number or dots/ring.
We examined the significance of the fixed-effects terms of the complete model using the Kenward-Roger degree of freedom approximation with type-III sums-of-squares. This analysis revealed significant main effects of gender, F(1, 33) = 10.3, p = 0.003; age group, F(1, 33) = 15.2, p = 0.004; and target number, F(2, 33) = 198.9, p < 0.001, for rotational speed and temporal frequency thresholds. The effect of dots/ring was significant for rotational speed thresholds only, F(1, 34) = 317.0, p < 0.001, and not for temporal frequency thresholds, F(1, 34) = 0.05, p = 0.81. Results of the main effects confirmed our earlier observations that men showed higher (better) thresholds than women, thresholds of older participants were generally lower (worse) than those of younger participants, thresholds decreased with increasing number of targets, and the number of dots/ring had no effect on temporal frequency thresholds. However, there were also significant two-way interactions between targets and dots/ring, F(2, 65) = 9.51, p < 0.001; age and target number, F(2, 33) = 14.4, p < 0.001; age and dots/ring, F(1, 34) = 7.12, p = 0.01; as well as a three-way interaction between age, target number, and dots/ring, F(2, 66) = 11.8, p < 0.001, for both measures. These interactions indicated that the effect of dots/ring varied with the number of targets and that the effect of aging depended on the number of targets and dots/ring.
To decompose the interactions, we fit separate mixed-effects submodels at each level of target number. The submodels contained terms for gender, age, dots/ring, and the interaction between age and dots/ring as fixed effects and by-subject random intercepts.
Table 2 shows the estimated fixed-effects coefficients, their significance results, and the associated semi-partial
R2 for the models fits to log-transformed temporal frequency data. The effect of gender was large and statistically significant for all target number conditions (one target:
β = 0.51,
p = 0.001,
R2p = 0.24; two targets:
β = 0.63,
p = 0.003,
R2p = 0.22; three targets:
β = 0.71,
p = 0.003,
R2p = 0.18). In contrast, the effect of age was small and not statistically significant in the one target condition (
β = 0.29,
p = 0.07,
R2p = 0.05) and increased as the number of targets increased. In the two targets condition, the age effect was statistically significant but smaller than the effect of gender (
β = 0.56
, p = 0.008,
R2p = 0.10), whereas for the three targets condition, the effect of age was larger and of a similar magnitude as the gender effect (
β = 0.88,
p = 0.001,
R2p = 0.16). Thus, while the gender difference in thresholds was large and constant across all target number conditions, the age difference in thresholds was not present with one target and progressively increased with additional targets.
With regards to the effect of dots/ring, temporal frequency thresholds were slightly higher in the 10 dots/ring condition for one target in both groups (β = 0.22, p = 0.001, R2p = 0.03). In the two targets condition, there was no main effect of dots/ring or any interaction between age and dots/ring. In the three targets condition, the main effect of dots/ring was not significant, but there was a trend for an Age × Dot/Ring interaction (β = 0.60, p = 0.06, R2p = 0.04). Whereas temporal frequency thresholds tended to be higher in the 10 dots/ring condition in the younger group (β = 0.22, p = 0.17, R2p = 0.03), the effect was reversed in the older group, (β = −0.38, p = 0.23, R2p = 0.04), although neither difference was statistically significant. Thus, when expressed in terms of objects/second, tracking thresholds were only slightly better in displays with 10 dots/ring when tracking one target compared to five dots/ring. With increasing number of targets, performance in the older group declined more for stimuli containing 10 dots/ring relative to five dots/ring. It is also important to remember that several older participants were unable to reliably track three targets even at slower speeds, resulting in two and six thresholds missing in the five and 10 dots/ring conditions, respectively. Results of the analysis of rotational speed thresholds were identical, except for the main effects of dots/ring, which was large and statistically significant for all target numbers (one target: β = −0.78, p < 0.001, R2p = 0.24; two targets: β = −0.97, p < 0.001, R2p = 0.24; three targets: β = −1.39, p < 0.001, R2p = 0.27), consistent with lower rotational speed thresholds for displays with greater number of dots/ring.
In sum, participants showed similar temporal frequency thresholds for both dots/ring conditions, which were highest when tracking one target and lowest when tracking three targets. Comparison of performance across age group and gender revealed that men had higher thresholds than women for all target number conditions, while the effect of age group was small and not statistically significant with one target and increased progressively with additional targets.