Sequential Effects on Reaction Time Distributions: Commonalities and Differences Across Paradigms

The role of expectations

First, the role of bottom-up versus strategic influences in producing sequential effects has been considered in all of the paradigms examined here. In visual search, for example, there is a debate about the extent to which sequential effects are elicited by strategic top-down weighting of visual dimensions (e.g., color, form) which speeds target detection when the expected dimension matches the actual presented target dimension. Whereas the dimension weighting (Found & Müller, 1996) and guided search 2.0 (Wolfe, 1994) accounts assume that such a strategic top-down modulation does speed visual search with dimension repetitions, the feature priming account (Maljkovic & Nakayama, 1994) assumes that the faster search in repetition trials is driven only by automatic feature and dimension priming without any influence of strategic processes. Furthermore, some accounts postulate a mixture of automatic and strategic processing. For example, the stimulus-driven capture account (Theeuwes, 2010, 1993) assumes that stimuli are captured only based on their automatically-determined salience, although strategic top-down processes can operate at the level of stimulus processing.

While for the visual search paradigm the discussion about bottom-up and strategic influences takes place on the level of stimulus processing, for the 2CRT paradigm it is generally thought that the response-stimulus interval (RSI) determines whether responses are influenced more by automatic facilitation or by strategic expectations. For short RSIs, automatic facilitation speeds repetition responses, whereas for long RSIs the response mode shifts towards the use of strategic expectations about the upcoming stimulus which might even elicit an alternation advantage (Soetens et al., 1984, 1985).

A similar observation has been made within task-switching paradigms. Here, task-switching costs decrease with increasing cue-to-stimulus interval (CSI). Although some authors (e.g., Mayr & Kliegl, 2003; Rogers & Monsell, 1995) argue that this decrease results from an active strategic preparation process in which task-sets are pre-activated to prepare for the upcoming task, other authors (e.g., Allport & Wylie, 2000) argue that task-switching costs occur simply because previous task-sets are still activated with short CSIs, and this activation automatically interferes with the new upcoming task-set.

Automatic interference between stimulus dimensions also plays a major role in models for responses in interference tasks. According to the prominent conflict monitoring model (Blais et al., 2007; Botvinick et al., 2001), this interference elicits a conflict that is used strategically to adapt the amount of cognitive control implemented in the subsequent trial. Higher levels of cognitive control reduce the influence of irrelevant dimensions, leading to smaller congruency effects following trials in which conflict was present. An opposing view assumes, however, that sequential effects in interference tasks occur simply because of contingency learning and the automatic activation of previously stored stimulus-response bindings from episodic memory that facilitate the retrieval of the correct response (Hommel, 1998; Schmidt & De Houwer, 2011).

The contribution of memory

Second, the learning of contingencies and the retrieval of traces stored in episodic memory are also considered to be important sources of sequential effects in a number of paradigms. As mentioned in the previous paragraph, the storage and retrieval of stimulus-response bindings is one of the most prominent accounts of these effects in interference tasks (Dignath et al., 2019; Hommel, 1998).

A similar approach has also been taken for visual search. For example, Huang et al. (2004) assume that feature and dimension repetition gains are elicited through automatic retrieval of previously stored episodic memory traces. These traces facilitate performance in the case of a match with the features and dimensions in the current trial, and they harm performance if there is a mismatch.

The same rational has also been applied with respect to binding and retrieval in action control (Frings et al., 2020), leading to the only theoretical framework that tries to account for sequential effects from multiple paradigms. This framework assumes that the binding of the stimulus, the given response, and any subsequent effects of that response into an event file takes place on every trial. In the case of a match between features of the current situation with features from the previous event file, the earlier event file can be reactivated and thereby facilitate the response selection. If the current trial’s event file features mismatch those of the previous event file, however, performance is harmed.

Instead of the storage of whole events, within the 2CRT task there are also more parsimonious accounts that assume only the implicit learning of the base rate and the repetition rate of the stimuli which help to adapt expectations for the current trial (Cho et al., 2002; Goldfarb et al., 2012; Jones et al., 2013). Those accounts are especially helpful when considering not only the effects from the previous trial n – 1 on the current trial n but also when considering the effects on the current trial of multiple previous trials, n – 1, n – 2, and n – 3.

The influence of cognitive control

A third shared commonality among the accounts of sequential effects in different paradigms considers the detection of conflict and the performance adaptations carried out in response to its presence. As mentioned earlier, the detection of conflict between response-relevant and response-irrelevant stimulus dimensions and the subsequent adjustments of cognitive control to improve performance are a very prominent explanation for the occurrence of sequential effects in interference tasks (Botvinick et al., 2001, 2004). For interference tasks, even the conflict elicited by a stimulus might be stored in an event file in episodic memory and retrieved in a subsequent trial (Dignath et al., 2019).

Furthermore, other forms of conflict have also been proposed for other paradigms. For example, the conflict between a previously-activated task-set and a new, to-be activated task-set is considered to contribute to task-switching costs (Allport & Wylie, 2000). Even an alternation between stimuli from trial n – 2 to n – 1 in 2CRT tasks seems to elicit conflict and a subsequent control adaptation. This evokes an effect called alternation-based interference, which is the slowing of responses after a previous alternation compared to a previous repetition in 2CRT tasks (Jentzsch & Leuthold, 2005).

Interim summary of theoretical accounts

As can be seen from this short review, the theoretical accounts of sequential effects in different paradigms do share some assumptions concerning the role of expectations, the contribution of (implicit) memory, and the influence of cognitive control, although not all assumptions have been considered within all paradigms (e.g., conflict is rarely considered in visual search). Nevertheless, based on the reviewed theories it is still difficult to decide whether it might be fruitful in future research to focus on the commonalities among different sequential effects jointly, as has been done within the framework of binding and retrieval in action control (Frings et al., 2020), or whether it is more appropriate to investigate the sequential effects from different paradigms separately to acknowledge their distinctness.

Vincentizing

One of the simplest methods of examining pooled RT distributions—and examining effects of experimental factors upon them—is that of “Vincentizing” (Vincent, 1912). First, using standard quantile point estimation techniques (e.g., Gilchrist, 2000), researchers estimate a set of quantile points (e.g., 5th, 15th, 25th, …, 95th) within the observed RT distribution of each participant in each condition. Second, for each condition and quantile point separately, the average of these estimates across participants is taken as the estimated value of the quantile point in the group distribution for that condition (for further details see, e.g., Jiang et al., 2004; Ulrich et al., 2007; Van Zandt, 2002). It is known that the group distributions estimated in this way do not necessarily preserve the mathematical shapes of the individual-participants’ RT distributions (e.g., Sternberg, 2023; Thomas & Ross, 1980). Shape distortions may be especially problematic with some distributions that are commonly used as models for RT (e.g., ex-Gaussian, Weibull; Rouder & Speckman, 2004). Nevertheless, Vincentizing is a simple and common method of distribution averaging, and it does provide a convenient, easily visualized depiction of the between-condition differences in cumulative RT distributions, as is illustrated in Figure 3.

Percentile rank pooling

Percentile rank pooling (Miller, 2021) is a flexible non-parametric method which is especially helpful for the inspection of distributions from conditions with a low number of trials or for a comparison of distributions that differ in their scale (e.g., distributions with a different onset and variance of RTs). Its computation is based on a simple two-step logic. In a first step percentile ranks (PR) are computed for the RT of each trial separately within each participant but across multiple to-be-compared experimental conditions. The ranks are defined as

where t describes the value of the RT, L expresses the number of trials across the to-be-compared conditions with RTs less than t, E defines the number of trials with RTs equal to t, and N represents the total number of trials (Miller, 2021). In a second step, the trials from each of the conditions under investigation are selected and their computed percentile ranks ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}(t)}$ are pooled across participants separately for each of the conditions. As a result, one obtains the density or frequency (vertical axis) of each percentile rank (horizontal axis) separately for each of the conditions being compared (color). Figure 4 depicts an example of the resulting plots.

There are three characteristics of percentile rank pooling that make the method a useful option for comparing RT distributions. First, it aims to compare the full distributions of RTs in different conditions rather than just differences in a pre-specified set of quantiles or bins. Second, because percentile rank pooling does not rely on means or medians computed within each condition separately as are often used with other popular methods such as quantile averaging (Ratcliff, 1979) or bin averaging (De Jong et al., 1994), its results are more stable for conditions in which there are only a few trials per participant. Third, because percentile rank pooling considers a non-parametric transformation of RTs to percentile ranks, it simplifies the comparison of between-participant and even between-experiment conditions that might typically operate on different scales, for example, RT distributions with different means and variances. These characteristics make percentile rank pooling attractive for the present purpose of comparing repetition effects on full RT distributions across different paradigms.

Ex-Gaussian

Another way to describe RT distributions is by means of the parameters of theoretical distributions that match the shapes of RT distributions well. Hohle (1965) observed that RT distributions show similarities with an ex-Gaussian distribution. An ex-Gaussian distribution is an exponentially modified Gaussian distribution which is mathematically described as a convolution of independent exponential and normal components. Therefore, three parameters describe the distribution. τ specifies the mean of the exponential component, which reflects the length of the distribution’s upper tail and is thus responsible for its skew. The normal component is specified by μ and σ, which reflect the location and width of the distribution’s lower tail. The mean and the variance of the ex-Gaussian distribution are characterized as

and

Previous studies demonstrated that the ex-Gaussian distribution describes RT distributions very well (Luce, 1986) and can even uncover experimental effects that are hidden if one would only consider differences in mean RT. For example, Spieler et al. (2000) demonstrated that—although all interference paradigms showed facilitation of mean RT in congruent relative to incongruent trials—the effects on the parameters of the ex-Gaussian distribution can differ between paradigms. Specifically, whereas congruency influenced the normal component for all paradigms, it only influenced the exponential component for some paradigms (e.g., the Stroop task and the global-local-task).

Based on the observation that some effects are only captured in certain components of the ex-Gaussian distribution, it is tempting to interpret these components in terms of psychologically processes. Hohle (1965) suggested that the normal component captures mainly motor processes while the exponential component might reflect the time needed for the decision itself. However, more recent studies have demonstrated that such a clear distinction does not hold either in simulation-based analyses (Matzke & Wagenmakers, 2009) or experimentally (Rieger & Miller, 2020). Nevertheless, the ex-Gaussian distribution is a very useful tool to get a more detailed description of RT data (Balota & Yap, 2011; Whelan, 2008), even though one must be cautious about interpreting changes in parameter values as reflecting effects on specific underlying processes (Heathcote et al., 1991; Matzke & Wagenmakers, 2009; Rieger & Miller, 2020).

Diffusion models

Four-parameter diffusion model

In contrast to the two previous approaches, the diffusion model (Ratcliff & Murdock, 1976) is based on a specific psychological model of the decision-making process, so changes in its parameter values necessarily have specific psychological interpretations, as has been illustrated in numerous binary decision RT tasks (e.g., Goldfarb et al., 2012; Klauer et al., 2007; Schmitz & Voss, 2014; Schwarz, 1993). This model assumes that response decisions are based on a diffusion process accumulating evidence over time with a mean drift rate v, which represents task easiness or task-related ability (see Figure 1; Ratcliff et al., 2015). Once the accumulated evidence reaches one of two response criteria that are separated by a units, a response is elicited. However, the decision process can be biased towards one of the two response alternatives which is captured by the relative starting point w of the accumulation process, where w = .5 describes an unbiased accumulation process. In addition to these parameters describing the decision time, the final RT includes an additional non-decision time, t₀, which reflects the time needed for stimulus encoding and response execution (Ratcliff & McKoon, 2008). Those four parameters represent the main parameters of the diffusion model and in the following paragraphs we will refer to a diffusion model considering only these four parameters as the ‘four-parameter diffusion model’.

Figure 1

Present study

To obtain a more comprehensive view of the changes in RT distributions between repetition and alternation trials, in the present study we applied multiple analysis methods to data from four paradigms: a visual search paradigm, a 2CRT paradigm, an interference paradigm, and a task-switching paradigm. Therefore, we collected data from all four paradigms (online) utilizing similar stimuli in all paradigms to keep the task requirements as comparable as possible between paradigms. Specifically, we used a pop-out search task for the visual search paradigm, a color-majority task for the 2CRT paradigm, a color-flanker task for the interference paradigm, and both a color-majority and a form-majority task for the task-switching paradigm. In a next step we applied both descriptive distributional analyses (i.e., Vincentizing, percentile rank pooling, and ex-Gaussian distribution fitting) as well as model-based distributional analyses (i.e., seven-parameter, four-parameter, and EZ diffusion models) to these data to investigate whether the repetition versus alternation between dimensions, categories, congruency, and tasks produced similar or different changes in the various analyses’ descriptors of performance in these tasks.

Participants

We were interested in at least medium-sized effects (d_z = 0.5). When considering in G∗Power (Faul et al., 2009) a counterbalanced design between conditions and two-tailed testing with an α = .05 and a power of .95, this leads to a required sample size of n = 56 per paradigm. Participants were recruited via prolific, a platform for online studies. We restricted our participant pool to people who were between 18 and 45 years old, spoke German fluently and as a first language, had normal or corrected visual acuity with no deficits in color vision, and participated on a laptop or PC. For participation, participants received a monetary compensation of 4£. We included in the analyses only those participants with at least 70% correct responses, participants whose arcsine transformed error rates deviated at most three SDs from the sample mean of arcsine transformed error rates, and participants whose mean log RT deviated less than or equal to three SDs from the sample mean log RT. This led to final sample sizes of 56 participants within the visual search, 2CRT, and task-switching paradigms, and to 57 participants within the interference paradigm. The samples seemed comparable in the four different paradigms in terms of age and gender. In the visual search paradigm age ranged from 18 to 45 years with a mean of 29.0 years (SD = 6.5), with 24 male, 30 female, and two diverse participants. In the 2CRT paradigm age ranged from 20 to 45 years with a mean of 29.1 years (SD = 6.7), with 17 male, 37 female, and two diverse participants. In the interference paradigm age ranged from 18 to 43 years with a mean of 29.8 years (SD = 7.2), with 23 male, 33 female, and one diverse participant. In the task-switching paradigm age ranged from 19 to 45 years with a mean of 27.8 years (SD = 6.7), with 17 male, 37 female, and two diverse participants.

Design

We conducted an online study using paradigm as a between-subject factor. Within each paradigm, there were additional paradigm-dependent within-subject factors, including repetitions versus alternations of the type studied within each paradigm.

Materials and Procedure

The stimuli in all paradigms consisted of a 9 × 9 square grid of colored shapes that had the size of half of the screen height, which appeared on a white background. Each object in the array had a size of 2% of the screen height. The objects’ positions were defined by equally-spaced grid points. However, in order to avoid a symmetric pattern, we jittered each object’s position on the horizontal and vertical axis by a random offset between 0 and 2% of the screen height. The exact color and form of the objects varied between the different paradigms.

Analysis

Within each of the four tested paradigms, we first computed ANOVAs to examine effects of the within-subject factors on mean RT to obtain an initial overview of the results. We then compared the RT distributions for repetition versus alternation trials in the four paradigms, using the above-mentioned methods. Specifically, we computed the Vincentized RTs, percentile rank values ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$, and the parameters of the ex-Gaussian distribution, the EZ model, the four-parameter diffusion model, and the seven-parameter diffusion model.² We estimated the parameters separately for each participant and for repetition and alternation trials, and—because the size of the sequential effects differed depending on the stimulus type (see behavioral results)—separately for each stimulus type (e.g., target dimension for the visual search paradigm, congruency for the interference paradigm, and task for the task-switching paradigm). To obtain parameter estimates, we used maximum likelihood estimation in R (R Core Team, 2016) for the ex-Gaussian, the four-parameter diffusion model, and the seven-parameter diffusion model, and we minimized the sum of the squared differences between the predicted and observed values for the EZ model.³ More precisely, we used the optimizing function optim with the Nelder-Mead optimization algorithm. To minimize the possibility that the optimization function converged by chance on parameter estimates for a local maximum, we repeated the procedure 50 times for the ex-Gaussian distribution and the EZ model and 10 times for the four-parameter diffusion model and the seven-parameter diffusion model, selecting as the final estimates for each participant and condition the parameter estimates with the maximum likelihood.⁴ In addition, we compared whether the conclusions drawn would change depending on whether the best parameters were selected from k or only from k – 1 runs to ensure valid conclusions.

Data preparation

For all analyses, we excluded trials from the practice block, the first two trials of each experimental block, post-error trials, and trials in which the RT suggested that the participant did not perform the task adequately (e.g., due to guessing or in-attention). The latter were defined by extremely fast responses (less than 300 ms in the visual search paradigm and the interference paradigm or less than 250 ms in the 2CRT paradigm and the task-switching paradigm) or extremely slow responses (RTs greater than 3000 ms). To determine these lower RT boundaries, we checked the mean accuracy and cumulative accuracy across participants for all responses within each successive 50 ms RT interval. We took the RT boundary of the first of two successive intervals that showed performance above chance level in both the absolute and cumulative accuracy as the lower boundary for trial exclusion.⁵ This led to the exclusion of 3.8% of all trials for the visual search paradigm, 4.1% for the 2CRT paradigm, 5.1% for the interference paradigm, and 5.5% for the task-switching paradigm.

The definition of alternation and repetition trials depended on the paradigm, as indicated in the introduction. For the visual search paradigm, these were defined only in the case when targets were present in both trials n – 1 and n. A trial n in which the target dimension changed between trial n – 1 and trial n (e.g., from the color dimension in trial n – 1 to the form dimension in trial n) was coded as an alternation, whereas a trial n with the same target dimension as on trial n – 1 was coded as a repetition. In the 2CRT paradigm repetitions and alternations refer to the stimulus and response (S-R), thus repetition trials were defined by an S-R repetition (e.g., majority green, left response) between trial n – 1 and trial n, whereas alternation trials were defined by an S-R alternation. In the interference paradigm the critical trial attribute was congruency (congruent vs. incongruent). Thus, a repetition of the congruency status of trial n – 1 to trial n was coded as a repetition trial whereas an alternation of this status was coded as alternation trial. Finally, for the task-switching paradigm a repetition of the to-be-performed task (majority-color or majority-form) between trial n – 1 and trial n was defined as a repetition trial, whereas a task alternation was defined as an alternation trial.

Behavioral results

To check for the presence of sequential effects, we conducted a 2 × 2 within-subjects analysis of variance (ANOVA) with the factors of stimulus type in trial n and trial sequence (repetition vs. alternation), separately for each of the paradigms. As previously explained in connection with the distinction between repetition and alternation trials, the “stimulus type” depended on the paradigm. For the visual search paradigm, stimulus type refers to the target dimension (color vs. form target), for the 2CRT paradigm it refers to the majority color (majority green vs. majority red), for the interference paradigm it refers to the control state (congruent vs. incongruent), and finally for the task-switching paradigm it refers to the task that had to be performed (majority-color task vs. majority-form task). As the dependent variable, we included the mean RTs of trials with correct responses for each participant in each condition. Mean RTs and the corresponding values of proportion correct (PC) for all conditions are summarized in Table 1.

Visual search paradigm

For the visual search paradigm, we found a significant effect of trial sequence, F(1,55)= 66.45, p < .001, ${\displaystyle {\eta }_p^2=.547}$, with reactions being faster in repetition trials (M = 556 ms) compared to alternation trials (M = 584 ms; see Figure 2), indicating the presence of sequential effects in this paradigm. Furthermore, there was a significant effect of the target dimension in trial n, F(1,55)= 162.41, p < .001, ${\displaystyle {\eta }_p^2=.747}$, with reactions being faster for color targets (M = 501 ms) than form targets (M = 643 ms). In addition, there was a significant interaction between the target dimension in trial n and trial sequence, F(1,55)= 48.31, p < .001, ${\displaystyle {\eta }_p^2=.468}$. A post-hoc t-test revealed that the sequential effect was larger for color targets (M = 44.7 ms) compared to form targets (M = 2.4 ms; t(55)= 6.95, p < .001, d_z = 1.35), and the latter sequential effect was non-significant as the 95% confidence interval included zero (see Table 1).

Figure 2

Discussion of behavioral results

Table 1 summarizes the sequential effects on RT and PC across all four paradigms, distinguishing between stimulus types for all paradigm except 2CRT, where the stimulus types did not differ. Taken together, the results on the level of mean RTs reveal that sequential effects occur within each paradigm. However, even on the level of the mean RTs it is visible in the results of the ANOVA and in Figure 2 that there exist differences in the size of sequential effects within and across paradigms. The smallest overall sequential effects occurred for the interference task, with congruent trials showing an effect of 13.5 ms and for incongruent trials of 5.6 ms. For the visual search task, the size of the sequential effect depended strongly on the target dimension. Whereas color targets showed an effect of 44.7 ms, form targets only showed a non-significant 2.4 ms effect. In contrast, the sequential effect within the 2CRT paradigm was medium-sized at 36.6 ms. Finally, the task-switching paradigm yielded the largest sequential effects, with mean effects of 103.6 ms for the color task and 91.6 ms for the form task.

These differences in the size of the mean sequential effect across paradigms might suggest that different processes contribute to their occurrence. However, finding different effect sizes within paradigms weakens the evidence for such a conclusion. As was noted in the introduction and has been demonstrated in many previous studies (e.g., Balota & Yap, 2011; Heathcote et al., 1991; Matzke & Wagenmakers, 2009; Van Zandt, 2000; Whelan, 2008), more detailed analyses of the full distributions of RTs—rather than just the means—can provide further important clues about processing differences between conditions. Therefore, we present in the next sections a systematic description of the changes in RT distributions derived from both descriptive and model-based analyses.

Descriptive distributional analysis

Vincentizing

For an initial descriptive investigation of RT distributions, we used the method of Vincentizing. Therefore, we computed the RTs at the 5th, 10th, 15th, …, 95th quantiles separately for each participant’s responses in repetition and alternation trials for each stimulus type. In a next step, we averaged across participants to compute the resulting Vincentized RTs per quantile separately for repetition and alternation trials. Figure 3 depicts the Vincentized RTs per quantile separately for each paradigm and stimulus type. As can be seen for all paradigms, Vincentized RTs increase faster with quantile for the higher quantiles than for the lower quantiles, which is a consequence of the typical right-skew of RT distributions.

Figure 3

Additionally, for most of the graphs it can be seen that the Vincentized RTs for repetition trials are somewhat lower than for alternation trials representing the presence of sequential effects at each quantile. However, as already seen in the behavioural results, the sizes of the sequential effects differ between paradigms, as is expressed in Figure 3 by the difference between the points for repetition and alternation trials per quantile, being largest for the task-switching paradigm and smallest for the interference paradigm.

Furthermore, sequential effects seem to be strongest within different regions of the RT distributions for different paradigms. For the interference paradigm, for example, the order between repetition and alternation trials changes in the highest quantile with repetition trials showing slower RTs than alternation trials. The same pattern also occurs for the form targets of the visual search paradigm. In contrast, for the task-switching paradigm and for the color targets in the visual search paradigm, the sequential effects tend to increase with increasing quantile. And finally, for the 2CRT paradigm sequential effects seem to be relatively constant across quantiles, with only perhaps a small increase of sequential effects for the quantiles in the middle of the distribution. Thus, the Vincentized RT distributions give us a first hint that the effects of repetitions versus alternations on RT distributions do show some similarities and some differences among paradigms. However, based on these plots it is hard to draw definitive conclusions about the development of sequential effects over time, especially for the paradigms with smaller sequential effects.

Percentile rank pooling

As a second approach to investigate sequential effects on the shapes of RT distributions for each paradigm and stimulus type, we used percentile rank pooling (Miller, 2021). Therefore, in a first step, we computed the percentile rank of each RT, ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$, relative to the full set of RTs from the same participant and stimulus type (e.g., color versus form in the visual search task, congruence in the interference task). In a second step, we pooled the computed ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ values across participants and plotted histograms of these separately for repetition and alternation trials. As can be seen in Figure 4, in all paradigms there are fewer alternation than repetition trials with low percentile ranks, and the reverse is true for high percentile ranks. This provides a more detailed view of the overall sequential effects, namely, the on-average shorter RTs in repetition trials compared to alternation trials. Furthermore, from the differences in proportions of repetition and alternation trials at the low and high ends of the histograms, one can picture the strength of the sequential effect. Strong sequential effects (i.e., large differences in proportions) are present in the visual search paradigm for the color targets, in the 2CRT paradigm, and in the task-switching paradigm, but these effects are much weaker for the form target in visual search and for the interference paradigm.

Figure 4

On a first view, the percentile rank pooling analysis gives a slightly different impression of the location of the sequential effect within the RT distributions than that suggested by Vincentizing. Whereas the Vincentized distributions in Figure 3 seem to suggest that the higher RT percentiles are most affected by the sequence, the pooled percentiles in Figure 4 suggest that the sequence effects are strongest on the fastest RTs. However, this impression is elicited because the two procedures use different sub-samples to depict the distributions of sequential effects. While for Vincentizing, quantiles are computed separately for repetition and alternation trials, the percentiles of the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ are computed over the whole RT distribution. Furthermore, it is important to keep in mind that the sequential effects for different quantiles in the Vincentized distributions are only reported descriptively. However, because the variance of the RTs varies across quantiles, one needs to be cautious in interpreting those descriptive sequential effects in terms of standardized effects.

Most important for our aim of comparing the sequential effects on RT distributions across paradigms is the shape of the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ histograms, and these also seem to differ. Note that in most cases there is a monotonic increase in the frequencies of ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ values of alternation trials with increasing percentile ranks (e.g., in the visual search paradigm for color targets and in the task-switching paradigm for both tasks; see Figure 4). In contrast, in the 2CRT paradigm, it seems that the frequency of alternation-trial ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ values initially increases with increasing percentile rank but then decreases for the highest percentile ranks. Another way to describe these differing patterns is to note that the repetition effect influences the whole RT distribution in most cases but that in the 2CRT paradigm repetition has a relatively small effect on the very slowest responses. Thus, the percentile rank pooling plots also hint towards differences between the effects of repetitions versus alternations on RT distributions among paradigms. It is still difficult, of course, to draw definitive conclusions about the mechanisms responsible for these differences based only on the percentile rank pooling plots, and it remains to be seen whether the later model-based analyses will shed any light on these mechanisms.

Discussion of descriptive distributional analysis

Taken together, the patterns of sequential effects on parameters of the ex-Gaussian distributions, the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ histograms, and the Vincentized RTs revealed both similarities and differences in the nature of these effects on RT distributions. For the ex-Gaussian distribution, in all paradigms, μ was larger for alternation trials than repetition trials, reflecting a slowing of alternation responses across the whole distribution, as opposed to just a slowing of the slowest responses. This pattern can also be seen in the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ histograms, which show a similar tendency of fewer alternation than repetition responses in the lower percentiles for all paradigms, and in Vincentized RTs which show that repetition trials show in most conditions consistently lower Vincentized RTs than alternation responses.

However, there was increased slowing at the high end of the alternation distribution, as reflected in increased τ values for the ex-Gaussian distribution in the visual search and task-switching paradigms but not in the 2CRT and interference paradigms. This increase can also be seen in the Vincentized distributions, for which sequential effects increased with quantile for the task-switching paradigm and the color targets in the visual search paradigm. However, the size of the sequential effects seemed rather unaffected over quantiles for the 2CRT and interference paradigms. The change in the shapes of the RT distributions in repetition and alternation trials is also easy to see in the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ histograms. The frequencies of alternation RTs increased monotonically across percentile ranks for the visual search, the interference, and task-switching paradigms but not for the 2CRT paradigm. For the 2CRT paradigm, the frequencies for repetition and alternation trials seem to nearly converge at the high end of the distribution.

These distributional differences hint at differences between these two pairs of paradigms in the mechanisms underlying their sequential effects, and in particular at some process that produces especially large slowing in some alternation trials in the visual search and task-switching paradigms. Again, it will be interesting to see whether the later model-based analyses can shed any light on the nature of this process.

Model-based distributional analyses

Discussion of model-based distributional analyses

Like the results for the descriptive distributional analyses, the results from the model-based distributional analyses also revealed similarities and differences in the nature of sequential effects on RT distributions. For most paradigms, the drift rate v was sensitive to the sequential effects consistently across models, with a faster accumulation of evidence in repetition trials compared to alternation trials. Additionally, the time for stimulus encoding and response execution t₀ differed between repetition and alternation trials (i.e., larger t₀ for alternation trials) for the two tasks of the task-switching paradigm and the color targets in the visual search paradigm. In contrast, for the 2CRT paradigm the largest effects occurred on the relative starting point w, although the direction of the effect was not consistent across models.

Based on results from previous studies (Burnham, 2018; Schmitz & Voss, 2012) and the standard interpretation of the diffusion model parameters, it seems plausible that sequential effects map onto the drift rate v and the non-decision time t₀. Effects on the drift rate v were also observed previously in the task-switching paradigm (Schmitz & Voss, 2012) and visual search paradigm (Burnham, 2018), and these would indicate that the sequential effect arises partly due to changes in the decision making process. A higher drift rate v in repetition trials compared to alternation trials, which is the pattern we mostly observed except for the 2CRT paradigm in the four-parameter and seven-parameter diffusion models, could be interpreted as a speeded uptake of information (Ratcliff, 2014) due to a pre-activation of, for example, the target dimension in the visual search paradigm, the target category in the 2CRT paradigm, the task-set in the task-switching paradigm, or event files for all paradigms.

An effect on the non-decision time t₀, on the other hand, locates sequential effects outside the decision process and rather into processes such as stimulus encoding and response execution (Ulrich et al., 2015). A larger non-decision time for alternation trials compared to repetition trials could, for example, reflect attentional shifts on alternation trials, with participants having to shift attention to the relevant dimension or task before information accumulation could begin. This would suggest that sequential effects result from processing system adjustments made before the start of evidence accumulation in the current trial (e.g., Rogers & Monsell, 1995; Wolfe, 1994). On this account, the lack of sequential effect on t₀ in the VS form condition may have been a Type II error as this effect was also reported by Burnham (2018) for the visual search paradigm. Furthermore, the absence of this effect in the interference task suggests—like the small effect on mean RT—that there is no major reconfiguration of the processing system based on the change in congruency status, although the effect may simply have been too small to pick up.

Alternatively, changes in the non-decision time could also reflect deactivating the stimulus-response mapping from the previous trial before executing the current response as is suggested, for example, by the binding hypothesis for partial repetition costs (Hommel, 1998, 2004). This would suggest that sequential effects arise after the decision process. In our case, partial repetition costs—that is, RT costs that occur because of a repetition of the response whereas the identity of stimulus switches—could arise in alternation trials of the visual search and the task-switching paradigm. However, in the 2CRT paradigm in which the response always changes with a change in the stimulus type and in the interference paradigm for which we implemented a minimized confound design alternating in odd and even trials between different stimuli, partial repetition costs should not occur at all. This fits nicely with the observation that t₀ differs mainly for the visual search paradigm and the task-switching paradigm in the four-parameter and seven-parameter diffusion model.

For the 2CRT paradigm, effects on the relative starting point w would also be plausible as a repetition of the response category always goes along with a repetition of the response itself. Thus, an increased bias towards the correct response in repetition trials which we find in the four-parameter and seven-parameter diffusion model is equivalent to a bias towards the response of the previous trial which would reflect an automatic response activation as source of facilitation (Soetens et al., 1985). The same mechanism was introduced in the model by Goldfarb et al. (2012) and was demonstrated to be well able to explain even higher-order sequential effects. As briefly mentioned above, the direction of the effect should be consistent independent of the exact model version. However, for the EZ model we find the reversed effect.

Changes in w are not plausible for the paradigms other than 2CRT, however. In these other paradigms, the correct response either repeated in both repetition and alternation trials (e.g., in the visual search paradigm) or was orthogonal to the definition of repetition and alternation trials (as in the task-switching paradigm). Thus, there was no systematic difference between repetition and alternation trials with respect to the identity of the correct response, and hence no way for the system to be biased more toward the correct response for repetitions than for alternations, as would normally be the standard pre-activation-based interpretation of an effect on w in the diffusion model (e.g., Ratcliff & McKoon, 2008). Furthermore, for the task-switching paradigm the direction of the effect also depended on the exact diffusion model, with the EZ model again even showing an alternation advantage.

The effect on the boundary separation a also seems very implausible, because a is typically interpreted as the speed-accuracy trade-off set at the beginning of an experimental block or trial (Voss et al., 2004). However, whether a trial is a repetition or an alternation trial cannot be known before stimulus onset. In most paradigms, participants should be able to tell whether the current trial is a repetition or an alternation trial only during or even after the decision process. However, the boundary separation should already have been set before this point in time. Furthermore, the direction of the effect is also against typical expectations. In most cases, except for the four-parameter and seven-parameter diffusion model for the 2CRT paradigm, there seems to be a higher response caution in repetition trials, which is not predicted by most theoretical accounts of sequential effects (e.g., Botvinick et al., 2001; Wolfe, 1994). These observations clearly question whether the diffusion model is suitable for describing sequential effects.

Another remarkable feature of the results of all three diffusion models is that the different tasks and conditions with larger sequential effects on mean RT clearly showed significant effects on more model parameters compared to conditions with smaller sequential effects. Only the form task in the task-switching paradigm shows quite large task-switching costs (see Table 1) with few parameters sensitive to this effect. Additionally, for some conditions in which there was a significant effect on the mean RTs but no significant effect on accuracy (as indicated by the 95% confidence intervals of the delta accuracies; e.g., congruent stimuli of the interference paradigm), all three models failed to detect a difference in any parameter. In contrast, in conditions in which the accuracy differed significantly but there was no significant sequential effect on mean RTs (e.g., form targets of the visual search task), the models (except for the seven-parameter diffusion model) were well able to capture the sequential effects.

Taken together, these observations could mean that depending on task and stimulus characteristics, repetitions versus alternations differ in many different ways (i.e., corresponding to the different parameters that capture the sequential effects) or not at all. Alternatively, repetitions versus alternations could differ with respect to some process that is not really captured by the model (i.e., the model could be inappropriate for modelling sequential effects), and these many parameter changes as well as inconsistencies across and within paradigms could reflect the model’s best attempt to fit distributions that were actually generated in accordance with some other model. The examination of the sequential effects on specific parameters in specific conditions above seems to favor the latter possibility. Another test for the adequacy of a model is its actual fit to the data to which we will turn in the next section.

Model fits across paradigms

For the comparison of the model fits, we conducted a χ²-test that tested the deviation between predicted and observed number of trials per percentile for repetition and alternation trials in a combined test. Additionally, we computed a Kolmogorov-Smirnov test which considers the distance between the predicted and observed cumulative density distributions (CDF) and compares the maximum distance against a critical value, which must be done separately for the distributions obtained in repetition and alternation trials. We based both comparisons on pooled percentile ranks ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ in order to get comparable distributions that could be combined across participants to maximize the sample size of each comparison. Therefore, we generated simulated RTs for each participant and for each model using the parameter estimates from the previous analyses. In a second step, we calculated the percentile ranks separately for each paradigm and each stimulus type to compare these predicted data with the originally observed data, and these comparisons are visualized in the Appendix in Figures A1 and A2. Please note that both tests only consider the goodness of the fit but do not include a penalty for model complexity in terms of numbers of parameters or its functional complexity.

Diffusion model as a tool to explain sequential effects

In our model-based distributional analysis, we used different variants of the diffusion model because the diffusion model previously successfully passed multiple validation checks (e.g., Arnold et al., 2015; Ratcliff & McKoon, 2008; Voss et al., 2004), and thus the parameters of this model are widely interpreted in terms of cognitive processes. However, in our investigations we observed some irregularities that are difficult to reconcile with standard interpretations of its parameters in terms of processes.

First, the observations that differences between the RT distributions of repetition and alternation trials differ more strongly between stimulus types than between paradigms and even seem to depend on the size of the sequential effects in mean RTs is rather surprising. Both observations might have the same source, because in most paradigms one stimulus type showed a larger sequential effect than the other. Initial exploratory analyses revealed that these observations might be grounded in the specification of the model structure. Analyzing the variance-covariance matrix between the four main parameters of the seven-parameter diffusion model using the Fisher-information matrix revealed medium to high correlations between parameters. This suggests that the different parameter estimates considered as criteria in the t-tests are not independent, so large differences in one parameter might be associated with large differences in another parameter. Because of the high correlations among parameters, it is difficult to say which parameters are causally involved in the sequential effect and which are only incidentally involved in the effect. And, if the fit of the diffusion model does not really reveal which parameters are affected, then the model does not help to identify which cognitive processes are affected, even given the usual assumptions about which parameters are associated with which cognitive processes.

Furthermore, we observed a second irregularity—namely, that for some paradigms, parameters that most certainly should not be affected by the present sequential manipulation were sensitive to sequential effects. Two examples are the boundary separation a and the relative starting point w, which could not logically differ for repetition versus alternation trials for any paradigm except the 2CRT task. These effects are clearly inconsistent with the usual interpretations of these parameters within the diffusion model, and they thereby weaken the interpretations of effects on its parameters.

It is also noteworthy that the conclusions drawn for the present research question depended on the exact version of the diffusion model. Technically, the seven-parameter and four-parameter diffusion models are different models, so differences in the parameters that were sensitive to sequential effects might be simply explained by these different model structures. These models do share many assumptions, however, and their common parameters are generally interpreted in the same ways, making the sequential effects on different parameters somewhat puzzling. Moreover, such differences were also present for the EZ model versus the four-parameter diffusion model, which should be essentially the same model with only differences in the procedures for parameter estimation. Whereas the parameters of the EZ model are estimated via equations for which parameter search ends once a solution is found with a mean squared error being equal to 0, for the four-parameter diffusion model we used maximum-likelihood estimation. And, as had already been found in previous studies (Wagenmakers et al., 2007), search routines for maximum-likelihood estimation again had difficulties converging on realistic parameter estimates for participants with low error rates, for both the seven-parameter as well as the four-parameter diffusion model. Thus, the differences in the conclusions might result from diverging estimation routines.

However, for the paradigms under consideration low error rates are a very common observation. And still the interpretability of the parameters should be ensured, especially because we observed those violations also in the 2CRT task, a paradigm frequently used to validate the diffusion model (see, e.g., Ratcliff & McKoon, 2008). Therefore, the diffusion model accounts of sequential effects look less promising when comparing multiple paradigms than when only a single paradigm was considered (e.g., Burnham, 2018; Schmitz & Voss, 2012).

There do exist other stochastic models for RT performance such as, for example, the linear ballistic accumulator model (Brown & Heathcote, 2008), the parallel grains model (Miller & Ulrich, 2003), and the cascade model (McClelland, 1979). But just like the diffusion model, most of these models assume for simplicity that the decision maker is in the same state at the beginning of each trial, a condition that is clearly violated due to the emergence of sequential effects. Nevertheless, as the diffusion model seems to have difficulties describing sequential effects, it might be worthwhile to evaluate in future research which other stochastic models provide more plausible descriptions of sequential effects. This might help to identify processes that are sensitive to sequential effects and to include these into stochastic RT models elaborated to describe trial-to-trial variations in performance.

Nevertheless, the visual inspection of the model fits (see Appendix) reveals that the diffusion model is still able to mimic the observed changes in the RT distributions for the present paradigms. Thus, although the results of the present study reveal that the parameters of the diffusion model are hard to interpret in terms of processes influenced by sequential effects, they still might provide some descriptive information for our present research question, namely whether sequential effects also share commonalities when considering the changes in their RT distributions.

Implications concerning sequential effects

The analogous changes in the RT distributions of repetition versus alternation trials across paradigms observed in the present study strengthen the idea of sequential effects operating on the same cognitive mechanisms across paradigms. Assuming common cognitive mechanisms as is, for example, considered in the framework of binding and retrieval in action control (Frings et al., 2020), one would expect that responses in repetition and alternation trials are affected similarly irrespective of the exact paradigm. In our study these similarities are reflected in similar changes in parameter values in models such as the ex-Gaussian distribution. These analogous changes are also in line with our theoretical review in the introduction. There, we observed that theoretical accounts across paradigms do share some assumptions. These include the role of strategic expectations, the contribution of (implicit) memory, and the role of cognitive conflict. Thus, the present results provide encouragement that it would be informative to explore the commonalities across paradigms of sequential effects that could be expected based on these accounts.

There were also fairly clear differences in the distributional changes between repetition and alternation trials across paradigms. These were reflected in, for example, differences between paradigms in sequential effects on the σ and τ parameters of the ex-Gaussian distribution, as well as in the different sequential effect on the shapes of the ${\displaystyle R{T}_{\mathrm{p}\mathrm{r}}}$ and the Vincentized distributions. That repetitions and alternations affect RT distributions differently across paradigms suggests that there are task-specific sequence-sensitive processes in addition to common cognitive mechanisms. These task-specific mechanisms in sequential effects highlight that caution is required when generalizing results from sequential effects of one paradigm to another paradigm, as our results reveal that not all conclusions can be generalized across paradigms. Rather, our observations of both shared distributional changes as well as paradigm-specific distributional changes demonstrate the importance of investigating mechanisms in several paradigms, especially when testing comprehensive theoretical accounts that should be applicable across multiple paradigms.

Including several paradigms in single studies would also help to distinguish which exact mechanisms are shared between paradigms and in which mechanisms paradigms might differ. Our results suggest that sequential effects of some paradigms have more in common than other paradigms. For example, the distributional changes of the visual search paradigm for color targets as well as the task-switching paradigm seemed to show more similarities than, for example, the distributional changes within the 2CRT paradigm. It is currently unclear which processing mechanisms these paradigms share, because the similarities and differences between paradigms are seldom investigated, although this could greatly increase our understanding of fundamental cognitive processes.

Finally, our results mainly leave open the questions regarding the differing nature of sequential effects across stimulus types within each paradigm. For example, RT distributions of repetition versus alternation trials seem to behave differently for color targets and form targets of the visual search paradigm. While it is common to assume paradigm-specific cognitive mechanisms for sequential effects, stimulus-specific effects have rarely been considered. Therefore, it would be interesting for future research to investigate differences in stimulus materials as well as differences in paradigms. Ideally, this could be done using within-participant designs in which both paradigm-specific and stimulus-specific effects could be quantified.

Conclusion

The picture of sequential effects emerging from the present study is quite complex. Looking at these effects at a detailed distributional level and comparing them across multiple experimental paradigms provides a much richer picture of the effects than can be obtained by looking at mean RTs within a single paradigm. This richer picture correspondingly offers a much broader data base—and thus much greater challenges—to theoretical models designed to account for the effects. The results thus provide a further demonstration that distinctive features of RT effects are lost when only mean RTs are analyzed, and they further suggest that studying the commonalities and differences of these distinctive features across multiple paradigms can provide further insight and modelling challenges not only for the sequential effects studied here but for almost all effects that are present across multiple paradigms.