Results and discussion

The mean accuracy for the duration judgment task was 80.35% (SEM =1.76), indicating that participants successfully encoded duration information in a large proportion of trials. Figure 1c displays psychometric curves separately for the TT and DT conditions based on whether the prior duration was short or long. Notably, there is a small but clear difference in the curves for both TT and DT conditions concerning prior duration. When the prior duration was long, the curve appears shifted leftward compared to when it was short, indicating an attractive sequential effect toward the prior duration. Specifically, participants tend to judge the current comparison durations as “Longer than one second” more often when the prior duration was long compared to when it was short.
We then calculated the corresponding PSE for each psychometric curve (as illustrated in Figure 1d). In the TT condition, the PSE for prior long and short durations were 0.766 ± 0.047 and 0.828 ± 0.051 s, respectively. In the DT condition, the PSE for prior long and short durations were 0.770 ± 0.049 and 0.818 ± 0.054 s, respectively. In general, the PSE is smaller than the standard one-second stimulus, indicating an overall underestimation of current duration. A two-way repeated measures ANOVA with “prior stimuli” (short vs. long) and “task relevance” (TT vs. DT) revealed a main effect of the prior stimuli on current duration judgment, F(1,23) = 6.324, p = .019, \(\eta_{p}^{2}\) = 0.013. However, there were no significant main effect of task relevance (F(1,23) = 0.033, p = .857,\(\eta_{p}^{2}\) = 0.000) nor an interaction between the two factors (F(1,23) = 0.098, p = .757,\(\eta_{p}^{2}\) = 0.000), with comparable sequential effects for the task-relevant (TT) and task-irrelevant (DT) conditions in the current duration discrimination task. These findings indicate that prior duration had a noticeable impact on current duration judgment, leading to shifted psychometric curves in both TT and DT conditions. This shift reflects an attractive bias towards the previous duration, resulting in a higher proportion of judgments of the current duration as longer than the standard one-second stimulus when the prior duration was long compared when it was short. However, sequential effects in the task-relevant (TT) and task-irrelevant (DT) conditions did not differ significantly.
To examine whether current duration judgments were influenced by preceding reports (decisional carry-over effect), we grouped all TT trials based on participants’ reports (“Short” or “Long”) in the previous trial. In Figure 1e (left panel), we present psychometric curves based on these prior reports. Notably, a distinct difference is evident in the curves concerning prior reports. When participants reported “Long” in the previous trial, the curve shifted leftward compared to when they reported “Short”, indicating an attractive sequential effect toward the prior report. Specifically, participants tended to judge the current comparison durations as “Longer than one second” more frequently after reporting “Long” compared to “Short” reports. The PSE for “Long” and “Short” reports in the previous trial were 0.741 ± 0.045 and 0.898 ± 0.051 s, respectively (Figure 1e, right panel). A two-tailed paired sample t-test on the PSE revealed a significant decisional carry-over effect. The PSE was significantly shifted leftward when participants reported “Long” compared to “Short” decisions in the previous trial (t(23)= 3.740, p =.001, d = 0.671).
These findings demonstrate that current duration judgments were influenced by both preceding durations and the decisions made in the previous trials. Specifically, durations presented immediately after long intervals tend to be perceived as longer, while durations following short intervals are perceived as shorter, showing an attractive sequential effect. However, there was no significant difference between trials following the timing discrimination task and the direction adjustment task, suggesting that post-perceptual processes may not be involved in the sequential effect underlying duration discrimination tasks. Furthermore, participants exhibited a significant decisional carry-over effect, meaning they were inclined to continue making “Long” judgments in subsequent duration discrimination tasks when they had reported a stimulus as “Long” in the previous trial, and vice versa for “Short” judgments. However, it is important to note that in the time discrimination task, decisions are categorical, involving dichotomous judgments as either “shorter” or “longer” than one second. It remains uncertain whether these findings from Experiment 1 can be generalized to tasks involving continuous critical dimensions. Therefore, in Experiment 2, we employed the time reproduction task, where participants were required to reproduce the duration of the presented stimulus.

Experiment 2

Method

Participant

24 participants were recruited in Experiment 2 (13 females; age 18 - 27, mean ± SD: 20.75 ± 2.45 years), all of them were right-handed, with normal or corrected-to-normal vision and color vision. Before the experiment, participants provided written informed consent and received compensation of 9 Euros/hour for their participation. The study was approved by the ethics committees of the Psychology Department at LMU Munich.

Stimuli and procedure

Experiment 2 closely followed the design of Experiment 1, with two notable modifications. In this experiment, participants had to reproduce the duration of the target stimuli for the timing task (see Figure 2a). Besides, the target duration was randomly sampled from 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, and 1.8 s.
After the post-cue display, when the task involved duration reproduction, a display of static green RDK (15 dots, each dot diameter of 0.4°; the luminance of 45.8 cd/m2) was presented on the center of the screen. Participants initiated the reproduction process at their own pace by pressing and holding the down arrow key on the keyboard, then released the key when they believed the elapsed duration matched that of the coherent motion of the green dots during the encoding phase. Immediately after pressing the down arrow key, the static green dots transitioned into a random motion pattern (velocity of 6°/s) to minimize inter-trial bias. The key holding duration was recorded as the reproduced duration. Participants would receive visual feedback if the relative error exceeded 30%, with “Too short” for relative errors below -30% and “Too long” for relative errors exceeding 30%. The procedure for the direction adjustment task remained identical to that used in Experiment 1.

Data analysis

The response error for each duration reproduction trial was calculated as the difference between the reproduced duration and the actual duration. We eliminated the first trial of each block and then filtered out trials with response errors exceeding three standard deviations from the participant’s mean error to account for accidental button presses or lapses in attention. These outliers were rare, comprising only 0.39% of the duration reproduction trials (ranging individually from 0 to 4 outlier trials). The remaining duration reproduction trials were categorized into two conditions based on the task relevance: “Time to Time” (TT) as the task-relevant condition and “Direction to Time” (DT) as the task-irrelevant condition.
Previous research has demonstrated that subjective timing reproduction is susceptible to contextual factors, with the primary bias known as the “central tendency effect”. This effect results in the underestimation of long durations and the overestimation of short durations, reflecting a tendency toward the mean value of the stimulus distribution (Burr et al., 2009; Jazayeri & Shadlen, 2010; Nakajima et al., 1992). Another bias arises from the sequential effect, where subjective reproduction is influenced by preceding durations (Dyjas et al., 2012; Glasauer & Shi, 2022). We assumed that the response error at trial n depended on both the current duration (\(T_{n}\)) and the previous duration (\(T_{n-1}\)(Glasauer & Shi, 2022). The variation in the current duration primarily contributes to the central tendency effect, while the variation in the previous duration causes the sequential bias. These two effects were combined into a single model expressed as:
\[\text{Error}_n=a*T_n+b*T_{n-1}+c.\]
To quantify the central tendency and sequential effects in the time reproduction task, we performed multiple linear regressions using response errors (\(\text{Error}_{n}\)) as the dependent variable, with the current duration (\(T_{n}\)) and the previous duration (\(T_{n-1}\)) as predictors. In this equation, the slope (\(a\)) was computed for the current duration (\(T_{n}\)), and \(-a\) represented the central tendency index, where 0 indicated no central tendency and 1 indicated a strong central tendency (Cicchini et al., 2012; Jazayeri & Shadlen, 2010; Shi et al., 2013). The slope (b ) of the linear fit on the previous duration (\(T_{n-1}\)) served as the sequential bias index (Cicchini et al., 2014, 2018; e.g., Glasauer & Shi, 2022). A positive slope indicates that the current estimation is attracted towards the previous duration, denoted as the “assimilation” or “attractive sequential effect”, while a negative slope indicates that the current time estimation is repelled from the previous duration. The statistical significance of both the central tendency effect and the sequential effect was assessed individually using two-tailed one-sample t-tests against the null hypothesis of zero effect for each condition (task-relevant: TT and task-irrelevant: DT) to determine the presence and strength of these effects. Subsequently, a two-tailed paired sample t-test was conducted to compare the difference between conditions (TT vs. DT).
Furthermore, to compare sequential effects between the two experiments, we categorized the reproduced durations into two groups: “Longer” for reproductions exceeding 1.2 s, and “Shorter” for reproductions shorter than 1.2 s, considering that the target durations were centered around 1.2 seconds. Sequential effects of prior stimuli and prior reports were then calculated separately, following the same methodology as in Experiment 1. First, we grouped previous durations into “Short” (0.6, 0.8, and 1.0 s) and “Long” (1.4, 1.6, and 1.8 s), omitting 1.2 seconds. Consequently, there were two factors: task relevance (TT vs. DT) and previous durations (short vs. long), resulting in four conditions mirroring Experiment 1. For each condition, we fitted psychometric functions and calculated the PSE individually for each participant. A two-way repeated measures ANOVA was applied to the PSE to assess main effects and the interaction effect. To investigate the decisional carry-over effect, we grouped all TT trials based on preceding “Short” or “Long” reports, fitted psychometric functions for each group, and calculated the PSE individually for each participant. We then performed a two-tailed paired sample t-test on the PSE to assess the decisional carry-over effect between prior reports (“Short” vs. “Long”).
Additionally, to visualize the variability of the sequential effect between experiments, we computed a sequential effect index as the difference in PSE between the prior short and prior long categories separately for each condition (i.e., for the TT condition, PSE in the prior short category minus the PSE in the prior long category). A 2 (task relevance: TT vs. DT) × 2 (experiments: Experiment 1 vs. Experiment 2) repeated measures ANOVA was applied on the sequential effect index to assess the main effects and the interaction effect. To assess the decisional carry-over effect between experiments, we calculated a decisional carry-over effect index as the difference in PSE between prior short and prior long reports separately for each experiment. A two-tailed independent sample t-test was performed on the decisional carry-over effect index to assess the effect between experiments (Experiment 1 vs. Experiment 2).