Despite wide recognition that a moving object is perceived to last longer, scientists do not yet agree as to how this illusion occurs. In the present study, we conducted two experiments using two experimental methods, namely duration matching and reproduction, and systematically manipulated the temporal frequency, spatial frequency, and speed of the stimulus, to identify the determinant factor of the illusion. Our results indicated that the speed of the stimulus, rather than temporal frequency or spatial frequency per se, best described the perceived duration of a moving stimulus, with the apparent duration proportionally increasing with log speed ( Experiments 1 and 2). However, in an additional experiment, we found little or no change in onset and offset reaction times for moving stimuli ( Experiment 3). Arguing that speed information is made explicit in higher stages of visual information processing in the brain, we suggest that this illusion is primarily mediated by higher level motion processing stages in the dorsal pathway.

*SD*= 2 deg; the visible extent was approximately 6 deg in radius), a vertically oriented sinusoidal luminance modulation was drifted rightward or leftward (with the direction chosen at random) at a predetermined temporal frequency (0, 1, 2, 4, 8, and 16 Hz) and spatial frequency (0.5, 1, 2, and 4 c/deg). The peak contrast was nominally 100%, and the mean luminance was 41.6 cd/m

^{2}. The background was maintained at the mean luminance level. The spatial frequency of the stationary patch was always 1 c/deg. The physical durations of the moving stimuli were 0.45, 0.64, or 0.91 s, with the duration for each trial chosen at random. We will refer to these as the “standard durations” hereafter.

*F*(2, 14) = 6.46,

*p*< 0.05, but this effect disappeared when we performed the same analysis on the duration ratio,

*F*(2, 14) = 0.603,

*p*= 0.56. Therefore, we chose the ratio of PSE to physical duration as the better index of the illusion. Because the ratio did not systematically vary across standard durations, we calculated the average ratio for each spatiotemporal frequency condition.

*DM*

_{ i}and PSE

_{ i}indicate the physical duration of the

*i*th moving stimulus (of the three standard durations) and its PSE, respectively.

*β*

_{0}+

*β*

_{1}

*S*+

*β*

_{2}

*T,*where

*S*indicates log2(spatial frequency) and

*T*indicates log2(temporal frequency). The best-fit function, which defines a slanted plane above the two-dimensional spatiotemporal frequency axes, was plotted as a mesh plot (see Figure 4). The corresponding equation is given as follows:

*S*or the

*T*axis alone were the determinant factor of the illusion, we should obtain a slanted plane passing through the

*T*or

*S*axis, respectively ( Figure 3a).

*k*+

*aV*=

*k*−

*aS*+

*aT,*where

*V*indicates log2(speed) and

*k*and

*a*are constants ( Figure 3b). The best-fit function supported the second prediction, i.e., that speed governs the illusion. The coefficients of

*S*and

*T*were both significant,

*t*(153) = −2.99,

*p*< 0.05 for

*S*;

*t*(153) = 2.98,

*p*< 0.05 for

*T*. This finding confirms that both frequency components contribute to the magnitude of the illusion. Furthermore, the values of the two coefficients were similar in size but with opposite signs. In fact, the value of each coefficient fell within the confidence limits of the other, reflecting no significant difference between them. Since

*S*and

*T*are logarithmic,

*T*−

*S*provides a logarithmic measure of stimulus speed, or

*V*. Therefore, Equation 2 can be rewritten as

*S*was unclear and did not reach statistical significance.

*β*

_{0}+

*β*

_{1}

*S*+

*β*

_{2}

*T*+

*β*

_{3}

*S*

^{2}+

*β*

_{4}

*T*

^{2}+

*β*

_{5}

*ST,*and compared the goodness of fit in terms of Bayesian Information Criterion (BIC)

^{1}. The best-fit parameters were (1.283, −0.006, 0.028, −0.020, 0.007, −0.017), and BIC = 71.19, whereas BIC = 58.64 in the linear fit. We next tested a linear plane model with a floor nonlinearity, RO = max(

*β*

_{0}+

*β*

_{1}

*S*+

*β*

_{2}

*T, β*

_{6}), but the best-fit parameters (1.258, −0.062, 0.051, 1.185) resulted in a poor fit, namely BIC = 63.01. Therefore, we concluded that the linear plane model without nonlinearity was the most reasonable one.

*β*

_{7}

*V*

^{2}+

*β*

_{8}

*V*+

*β*

_{9}. The best-fit parameters were (0.006, 0.033, 1.247), but only

*β*

_{8}and

*β*

_{9}were significantly different from 0 (

*p*< 0.05), which confirmed the claim that the RO increased proportionally with log speed.

*p*< 0.05).

*F*(3,21) = 0.43,

*p*> 0.05). The reason for this is not clear, but we think that there might be two possible reasons. One possibility is that the speed dependence of the illusion that we propose here exists only in the frequency area higher than 1 Hz. Since the aforementioned linear regression analysis on the data excluding 8 and 16 Hz showed a shallower slope, we cannot reject this possibility. The other possible reason is the noisy nature of the data. We noticed that even at 1 Hz, some tendency of speed dependence was seen for the data at 1, 2, and 4 c/deg, but the point at 0.5 c/deg was located irregularly against the speed dependence scenario. At this low spatial frequency, subjects might be able to track the displacement of one of luminance stripes in the Gabor patch to judge duration based on displacement information. In any event, currently we cannot resolve whether the conclusion of speed dependence is applicable to all data points with a conditional statement about random noise, or whether the conclusion is applicable to a subset of parameter space without 1 Hz conditions. We do not emphasize that duration dilation should strictly obey a linear function of log speed at every spatiotemporal frequency point; a rising function with some nonlinearity and floor/ceiling effects would be more realistic. However, since the present study lacks strong evidence for these claims, it is concluded that linear speed dependence is the best summary of the present data.

*RepT*

_{ Mi}indicates the reproduction time for the moving stimulus having the

*i*th duration (of the four standard durations), and

*RepT*

_{ Si}indicates the reproduction time for the stationary stimulus having the same duration.

*t*(117) = −3.05,

*p*< 0.05 for

*S*;

*t*(117) = 3.35,

*p*< 0.05 for

*T*; and there was no significant difference between their absolute values. Thus, as in Experiment 1, this equation can be rewritten as

*β*

_{0}+

*β*

_{1}

*S*+

*β*

_{2}

*T*+

*β*

_{3}

*S*

^{2}+

*β*

_{4}

*T*

^{2}+

*β*

_{5}

*ST,*and a linear plane model with a floor, RO = max(

*β*

_{0}+

*β*

_{1}

*S*+

*β*

_{2}

*T, β*

_{6}). Then again we compared the goodness of fit in terms of BIC. The best-fit parameters of each model and BIC were (1.050, 0.003, 0.025, −0.015, −0.004, −0.001), BIC = −236.20 and (1.042, −0.020, 0.018, 0.995), BIC = −240.69, whereas BIC = −245.48 in the original linear model. Therefore, we concluded that the linear plane model without nonlinearity was the most reasonable.

*S*+ 0.020

*T*(both coefficients were significant,

*p*< 0.05).

*r*= 0.55,

*t*(18) = 2.82,

*p*< 0.05, indicating consistency across the two experiments in what was measured.

*F*(2,4) = 11.89,

*p*< 0.05. Post hoc analysis showed a significant difference between slower (1 c/deg, 16 Hz condition) and faster (0.5 c/deg, 16 Hz condition) motions (

*p*< 0.05). RT to the onset of the 0.5 c/deg moving stimulus tended to be shorter than RTs to other stimuli. This may be due to spatial frequency, for it is known that subjects react more quickly to the onset of stimuli having lower spatial frequencies (Breitmeyer, 1975; Gish, Shulman, Sheehy, & Leibowitz, 1986; Lupp, Hauske, & Wolf, 1975).

*F*(2,4) = 4.63,

*p*> 0.05. Although the RT to the offset of the moving stimulus appeared to be longer than that to the offset of the stationary stimulus, the difference was very small.

*event*and the perception of

*duration*are not always integrated experiences. Correct perception of stimulus change and distorted perception of stimulus duration can occur simultaneously.