Introduction

Among executive functions, the ability to inhibit and change actions in order to regulate impulses and attain long-term goals has a central place. A core experimental method used to measure the concept of response inhibition is the Stop-signal task, originally developed by Logan and Cowan [1]. In this task, on a majority of trials (Go-trials) a Go-signal is presented, to which participants have to respond as quickly as possible, producing the Go-reaction time (Go-RT). On a minority of trials (Stop-trials), the Go-signal is followed at some delay (the Stop-signal delay, SSD) by a Stop-signal, which instructs the participant to withhold the Go-response. A fundamental result is that withholding the Go-response becomes more difficult with longer SSDs [1].

An influential model of the Stopping-process in the Stop-signal task is the independent horse-race model [1]. The horse-race model assumes that there are two independent processes at play during a Stop-trial: the Go-process, which starts with presentation of the Go-signal and concludes with a Go-response, and the Stop-process, starting with the presentation of the Stop-signal and concluding with the withholding of the Go-response. Both processes race to conclusion and whichever finishes first determines the response. Since a longer delay between Go- and Stop-signal gives the Stop-process less time to finish, trials with longer SSDs make it harder to correctly withhold the Go-response. In a common Stop-signal experiment, a participant will manage to withhold the Go-response only on some Stop-trials, but not on others, producing an erroneous response. The Go- and Stop-process are thought to be largely independent (for a review, see [2]).

The horse-race model allows computation of the time the stopping process takes (the Stop-signal reaction time, SSRT). The SSRT can be computed by determining the SSD at which the likelihood of producing an erroneous Go-response (referred to as p(error)) is 0.5, and then subtracting that SSD from the mean Go-RT during simple Go-trials. The SSD for which p(error) is 0.5 is referred to as the critical SSD. To estimate the critical SSD, the staircase tracking procedure is often used. With this method, a correctly withheld Go-response increases the SSD by a certain duration, while an erroneously executed Go-response decreases the SSD by a certain amount. In practice, this method usually converges on an SSD at which p(error) is at 0.5, such that this SSD can be subtracted from the mean Go-RT (this form of computing the SSRT is known as the mean method). Should the method converge at a p different from 0.5, the Go-RTs are rank-ordered, and the SSD is subtracted from the nth Go-RT, where n is the total number of Go-RTs multiplied by p (the so-called integration method, [2]). An illustration of the horse race model’s different quantities can be found in the supplementary material (S1 Fig).

One disadvantage of the Stop-signal task is its potential length, especially if different conditions are included, because for every Stop-trial several Go-trials have to be included to make the Go-response the prepotent response, and because it takes a relatively large number of Stop-trials to reliably estimate the SSRT [3]. As a possible solution to this problem, Livesey and Livesey [4] proposed a different method to select an upcoming Stop-trial’s SSD and estimate the critical SSD used for computing the SSRT, based on an adaptive Bayesian estimation algorithm known in the psychophysical literature as the PSI method [5].

In the following, our description of the method closely follows the original work of Kontsevich and Tyler [5], which should be consulted for further detail. We present the full algorithm here so that changes in its application to the Stop-signal task discussed later in this article will be easier to follow.

In short, the relationship between SSD and the likelihood of a response in a Stop-trial follows a psychometric function, such as the Weibull cumulative density function chosen by Livesey and Livesey [4]: (1) (2) The Threshold parameter governs at which SSD p(error) is 0.5, the Slope determines the rise of the function, and the ErrorRate determines the minimum and maximum the function can take on.

A participant’s performance in the task is described by a vector λ_participant = (Threshold_participant, Slope_participant, ErrorRate_participant) in this three-dimensional parameter space. Our knowledge of a participant’s performance λ_participant at the start of the task is formalized as a probability distribution p₀(λ) over the parameter space, which assigns each possible performance vector λ a probability p₀ of matching the participant’s performance λ_participant. When having no prior knowledge, at the outset of the task we can set p₀(λ) as the uniform distribution.

Step 1: At the outset of Stop-trial 1, p₀ represents our prior knowledge of the participant’s performance. For each SSD and response that is possible in trial 1, we can compute the posterior probability of each parameter vector λ, according to Bayes’ theorem: (3) where p(response|λ,SSD) is the psychometric function in Eqs 1 and 2. Thus, for each SSD we might present in the upcoming trial 1, and for each response the participant might give in return, we compute the posterior distribution over the parameter space.

Step 2: For each of these possible SSD-response combinations we compute the entropy H₀ of the resulting posterior distribution p₀(λ|SSD,response) as: (4) The entropy function stems from information theory and quantifies the amount of information necessary to fully specify the state of a system [6]. In the current context, the entropy of each posterior distribution quantifies how much information is still needed to completely specify which vector λ in parameter space corresponds to our participant’s performance vector λ_participant. When deciding which SSD to offer on Stop-trial 1, we choose that SSD which yields the lowest entropy, and thus the most information about the participant’s performance.

Step 3: Since the entropy function in Eq 4 is also dependent on the participant’s response, we compute the current likelihood of the two possible responses (success, error) for each SSD, using the prior probability distribution p₀(λ) and the psychometric functions in Eqs 1 and 2: (5) (6) With these, we can compute the expected entropy E[H₀(SSD)] as (7)

Step 4: In Stop-trial 1, we present the SSD yielding the lowest expected entropy and record the participant’s response.

Step 5: After having finished trial 1, our updated prior knowledge p_1(λ) is the posterior distribution p₀(λ|SSD,response) from Eq 3 corresponding to the SSD we eventually offered in trial 1 and the response the participant gave. We now continue with the following trials t = 2, 3 … n, and on each trial, in order to determine the SSD to offer, steps 1 to 4 are now repeated, except that where we used our previous prior distribution p₀(λ), we now use our updated prior distribution p_t−1(λ).

Step 6: At the end of the task, after n Stop-trials have been concluded, we derive the parameter vector most likely to match our participant’s parameter vector λ_participant by computing the expected value of λ: (8)

In a number of Stop-signal task simulations and behavioral experiments, Livesey and Livesey [4] showed that the PSI method can quickly and accurately estimate the SSRT. In terms of correlation with the actually simulated SSRT, as well as mean absolute deviation, the PSI method was comparable to the traditional staircase. However, the staircase was shown to provide a systematically biased estimate of the SSRT, overestimating short SSRTs and underestimating long SSRTs, even at more than 30 Stop-trials, while the PSI method arrived at a relatively unbiased estimate already after 10 Stop-trials. The PSI method thus promises as a viable alternative to the staircase method when quick and unbiased estimation of the SSRT is required.

The purpose of this study is to investigate the influence of Go-response slowing on the PSI method. In Livesey and Livesey [4], the Go-RTs in both the simulations and the behavioral tasks are constant over the course of the experiment. However, it is known that the Stop-signal task can invite a gradual slowing of the Go-RTs over the course of the experiment [7, 8]. There is a strategic element to this slowing, since participants will sometimes try to slow their Go-responses in order to still catch the Stop-signal. Since this increases the participant’s chance of correctly responding, the staircase algorithm will respond to this by further increasing the SSD, motivating the participant to further slow the Go-response. In this way, the staircase even promotes this strategic slowing. It is an open question how the PSI method can deal with this slowing effect in the Stop-signal task. In order to test this, we employ a version of the so-called Stop-change task, a variant of the Stop-signal task we have previously found to elicit strong Go-response slowing. Previous research has shown the processes involved in stopping and changing to be partly overlapping [9–12].

In the following behavioral study we implement a slightly modified version of the PSI method, the so-called PSI marginal method [13, 14]. In the original PSI method, as used by Livesey and Livesey [4], the error rate determining the minimum and maximum p(error) the psychometric function can take on is usually fixed. Prins [13] showed that this can lead to a biased estimation of the threshold parameter, which is central to estimating the SSRT in the Stop-signal task. As a remedy, Prins proposed also adding the error rate-parameter to the parameter space, thus estimating three free parameters instead of two. As a further refinement, he proposed minimizing the entropy of a posterior distribution that is marginalized over irrelevant parameters (e.g. in the Stop-signal task, minimizing entropy of a posterior that includes the threshold parameter but is marginalized over slope and error rate, since these are not relevant to SSRT estimation).

In terms of the formulation above, this means changing Eq 4 so as not to compute the entropy H₀(SSD,response) of the full posteriors, but instead to compute: (9) where p₀(Threshold|SSD,response) is the marginalized posterior distribution: (10) Since we implement these recommendations, we will refer to the method we tested as the PSI marginal method from now on, and will refer to the method proposed by Livesey and Livesey as the original PSI method.

Behavioral experiment 1: the PSI marginal method

Results

Experiment 1 was mostly explorative in nature to elucidate the behavior of the PSI marginal method in a task involving Go-response slowing. See Fig 1 for an example participant with gradual Go-response slowing in the PSI marginal method’s block. Since the maximum SSD offered was 1200 ms, the Go-RTs of this participant exceeded this SSD after about half the experimental run. In terms of the two methods’ response to Go-RT slowing, the behavior illustrated in Fig 1 causes a problem for the PSI method. As can be seen in Fig 2, when Go-RTs exceed the maximum threshold considered by the algorithm (at Change-trial 9), the critical SSD (equal to the threshold parameter) receiving the highest probability reaches the upper end of the parameter space, in this case 1200 ms, and the method is not able to estimate the participant’s true critical SSD. In effect, the PSI marginal method is not able to keep up with the slowed Go-RTs. Notice that the SSRT is ultimately computed with the expected parameter combination, not the maximum probability parameters, but nevertheless reaching the boundary of the parameter space invalidates the method’s SSRT estimates. Furthermore, when the mean Go-RT has slowed down considerably, the critical SSD estimated by the method is, after a sufficient number of trials, based on this slowed Go-RT. However, when computing the SSRT, the mean Go-RT of the entire block is usually used. Thus, the critical SSD subtracted from the mean Go-RT might only be based on a slowed subset of Go-RTs toward the end of the experimental block. Although we only tested the PSI marginal method, both of these problems equally exist for the original PSI method.

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Fig 1. Go-RT slowing in Experiment 1.

A participant’s Go-RTs for one condition. The line indicates the increasing number of Change-trials in that condition, the circles are Go-RTs.

https://doi.org/10.1371/journal.pone.0210065.g001

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Fig 2. Likelihood estimates during Experiment 1.

The PSI marginal method’s posterior distributions for the example participant. Probabilities are averaged over the error rate dimension, colors indicate the probability of a parameter combination, the point of maximum probability is indicated in black. Colors range from minimum to maximum probability per distribution. Around Change-trial 9, the most likely threshold parameter reaches the boundaries of the method’s parameter space.

https://doi.org/10.1371/journal.pone.0210065.g002

A further consequence arising from response slowing in the Stop-signal task is the overall accuracy of the participants. Although the staircase method is designed to converge on an accuracy of around 0.5 and can to some degree “follow” the participant’s slowed Go-responses, if responses are being slowed down too quickly, they can escape the staircase and allow the participant to attain an accuracy substantially higher than 0.5. This is especially true if the experiment contains multiple conditions and thus, the amount of response slowing between the different conditions’ interleaved trials is larger. However, an accuracy of around 0.5 is necessary for an accurate computation of the SSRT. Although the PSI marginal method is not explicitly designed to converge at 0.5, it also usually settles around this value. In Fig 3, the proportion of successful change trials is plotted, averaged over bins of 8 Change-trials, for the two different methods and with participants split into two groups, based on whether they showed substantial Go-response slowing. From the figure it becomes clear that only the staircase method for non-slowing participants was able to maintain an accuracy of about 0.5 throughout the experiment. Slowing participants were able to achieve a substantially higher accuracy, for both the staircase and PSI marginal method. Non-slowing participants for the PSI marginal method, on the other hand, stayed below 0.5 performance for the length of the experiment.

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Fig 3. Accuracy throughout Experiment 1.

The proportion of successful Change-trials (averaged over bins of 8 trials) of the PSI marginal and Staircase methods. Participants were split into two groups based on whether they showed substantial Go-response slowing.

https://doi.org/10.1371/journal.pone.0210065.g003

What is striking, furthermore, is that in the PSI marginal method’s block, slowed subjects initially achieve a higher accuracy (around 0.5) than non-slowing subjects (0.2). This is owed to the extensive parameter range for critical SSDs (50 ms– 1200 ms) we used in order to allow some response slowing. Before gaining knowledge about a participant’s behavior, the method tends to choose SSDs from the center of its parameter range, which in this case is between 600 and 650 ms, since these are most informative. These SSDs are already too large for non-slowing subjects, leading to their poor performance, while slowing subjects are better able to deal with these trials.

We have included further visualization of the relation between SSRT estimates of the staircase- and PSI method in the supplementary material (S3 Fig).

Since it is clear the PSI marginal method has considerable problems with participants slowing their Go-responses in the Stop-signal task, we propose the PSI adjusted method, which keeps a running estimate of the current mean Go-RT during the experiment. This adjusted method does not estimate the critical SSD, but the difference between estimated mean Go-RT and the critical SSD, thus in effect estimating SSRT directly. In this way, the method’s estimated SSRT is always based on the current mean Go-RT, and also does not change substantially over the course of the experiment, thus not reaching the boundaries of the considered parameters. Additionally, even dramatic Go-response slowing should not be able to cause an accuracy substantially larger than 0.5.

The PSI adjusted method

In the following the PSI adjusted method, which can also deal with slowed Go-RTs, will be explained in more detail. A participant’s performance is now characterized by a three-dimensional vector ω = (SSRT_participant,Slope_participant,ErrorRate_participant). Thus, we estimate SSRT directly, instead of the threshold parameter. The relation between SSRT and threshold is: (11) Since the psychometric function is defined in terms of the threshold parameter, we use Eq 11 to convert a given SSRT value to its corresponding threshold value. In order to do that, we need an estimate of the participant’s mean Go-RT. Since during response slowing, the Go-RT changes throughout the task, the first step in each Stop-trial is to estimate the current mean Go-RT. This is done via linear regression: over a sliding window including the last several Go-trials, the Go-RTs are regressed on their respective trial numbers. The resulting linear regression equation is used to predict the Go-RT of the current trial, which serves as an estimate of meanGo-RT in Eq 11.

The logic behind this is that a participant’s SSRT is a characteristic not changing with response slowing. The slowing only affects the threshold parameter, which indicates the SSD at which p(error) = 0.5. This threshold trails behind the continuously slowing Go-RT, while the distance between the Go-RT and the threshold (the SSRT) remains constant. By estimating the changing Go-RT and adjusting the threshold parameter for it, we can estimate the unchanging SSRT.

After having estimated the current Go-RT at the outset of the Stop-trial, we also need to adjust our SSD range. Until now we have not touched upon the range of SSDs to choose from, but since we now adjust for slowing Go-RTs, we have to ensure that the SSDs that can be offered in our Stop-trial match the range of threshold values that are possible under our current estimate of the Go-RT and the range of SSRT parameters in our parameter space. On each Stop-trial, we let SSDs range from where min(SSRT) and max(SSRT) are the minimum and maximum SSRTs of our parameter space. In practice, we also truncate negative SSDs to 0. After these adjustments at the start of a Stop-trial, the method follows the PSI marginal method outlined earlier, except that all distributions are now defined over ω parameter space.

In order to compare the properties of our PSI adjusted method during the Stop-signal task to the PSI marginal method as well as the classic staircase method, we ran three sets of simulations. The first set is very similar to those presented by Livesey & Livesey [4] and investigates the qualities of the SSRTs estimated by the different methods under the condition of stationary Go-RTs. The second and third set of simulations investigate the effects of Go-RT slowing on the different methods.

Simulations 1: Constant reaction times

Results

As Fig 4A shows, the correlations between all four methods’ SSRT estimates and the true SSRTs develop similarly over the course of the experiment, reaching the 0.9 mark around 18 trials and further increasing from there. No substantial differences between the methods are obvious.

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Fig 4. Different methods’ performance with stationary GO-RTs.

Performance of the PSI marginal method, PSI adjusted method, and the staircase method with stationary Go-RTs. (A) Correlations of estimated SSRTs with simulated SSRTs. (B) Mean absolute deviation between estimated and simulated SSRTs. (C) Linear slope between estimated and simulated SSRTs. Filled areas indicate the standard error of the mean over the 150 simulated experiments.

https://doi.org/10.1371/journal.pone.0210065.g004

Fig 4B shows the mean absolute deviation of SSRT estimates for the different methods. Again, the time course over the experiment is generally similar for the different methods, with only a small underperformance of the PSI methods between trials ten to 25, corresponding to a mean absolute deviation of around 2.5 ms after 12 trials.

The most interesting result is depicted in Fig 4C, where the linear slope between a method’s SSRT estimate and the true SSRT is plotted over the course of the experiment. This can be seen as an index of biased SSRT estimation, with slopes smaller than one meaning lower SSRTs are being overestimated or higher SSRTs being underestimated, or both. An unbiased estimation would result in slopes of one. As can be seen, there is a clear ordering of performance for the different methods, with the PSI methods performing best and equally well, the staircase integration method performing worse and the staircase mean method performing worst. The differences in performance are larger in the lower trial numbers. Since the staircase- and PSI methods settle on different ultimate slope values, it will take progressively longer for the staircase methods to reach the same slope values as the PSI methods the closer that value is to one. For example, it takes the staircase integration method 4 trials longer than the PSI methods to reach a slope of 0.9, but 11 trials longer to reach a slope of 0.95, and 21 trials longer to reach a slope of 0.97.

Simulations 2: Increasing reaction times

Results

As Fig 5A shows, when Go-responses are slowed over the course of the experiment, the correlations for the staircase and PSI adjusted method are fairly similar to when responses are not being slowed. The PSI marginal method’s performance, however, deteriorates from around trial 12 onwards, due to it not being able to react to the slowing Go-responses (and changing critical SSDs). Not visible in the graphic, the correlation coefficients for the PSI marginal method decrease until 0.2 at trial 100.

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Fig 5. Different methods’ performance with slowing GO-RTs.

Performance of the PSI marginal method, PSI adjusted method, and the staircase method with increasing Go-RTs. (A) Correlations (B) Mean absolute deviation (C) Linear slope. Filled areas indicate the standard error of the mean.

https://doi.org/10.1371/journal.pone.0210065.g005

In Fig 5B, it can be seen that the staircase-integration method, the staircase-mean method and the PSI marginal method all at some point deteriorate in performance due to not being able to keep up with the slowing Go-RTs. The staircase-integration method performs best until around trial 30, while the staircase-mean method stops improving at a mean absolute deviation of around 35 ms. While initially performing worse than the staircase-integration method, the PSI adjusted method is able to keep up with slowing Go-RTs and steadily decrease absolute deviation. The deviations for the staircase- and PSI marginal method eventually increase to around 80 and 275 ms, respectively.

In terms of the slope between estimated and actual SSRTs, the staircase-mean method stays consistently below one. The PSI marginal method deteriorates in performance around trial 12. Not shown in the graphic, the slope eventually decreases down to almost 0 at trial 100. The staircase integration method quickly surpasses a slope of one, again not able to keep up with response slowing. The PSI adjusted method quickly rises towards a slope of one and stays there for the remainder of the trials. As can be seen in the supplementary materials (S4 Fig), the stronger the response slowing, the stronger and earlier the adverse effects on SSRT estimation occur.

Generally, while in the case of constant RTs (simulations 1) the PSI methods simply attain a given degree of precision in SSRT estimates in fewer trials than the staircase methods, with increasing Go-RT slowing it becomes more a question of being able to deal with the slowing at all.

Simulations 3: “Pure” slowing

Results

The results can be seen in Figs 6–8. Correlations, mean absolute deviations and linear slopes were computed as for simulations 1 and 2. To compare the performance of the two methods to the case of no Go-response slowing, we also show the results of simulation 1 again (in black), in which Go-RTs were constant.

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Fig 6. PSI marginal method’s performance for different degrees of slowing inside parameter space.

Panel A shows correlation coefficients between simulated and estimated SSRTs, panel B shows mean absolute deviation between simulated and estimated SSRTs, panel C shows mean linear slope relating simulated and estimated SSRTs. Colors indicate different degrees of response slowing. Until trial 100 (red line), all simulated participants are inside the method’s respective parameter space.

https://doi.org/10.1371/journal.pone.0210065.g006

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Fig 7. PSI adjusted method’s performance for different degrees of slowing inside parameter space.

As in Fig 6, but now the performance of the PSI adjusted method is depicted.

https://doi.org/10.1371/journal.pone.0210065.g007

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Fig 8. Marginal entropy of the PSI marginal method over the course of the task.

The marginal entropies of the threshold, slope and error rate parameters are shown, over the course of the task, for the PSI marginal method. Line pattern indicates which parameter’s entropy is shown. The upper panel shows the results for 5 ms of slowing per Stop-trial, the middle panel for 10 ms, and the lower panel for 15 ms.

https://doi.org/10.1371/journal.pone.0210065.g008

As can be seen in Fig 6, the effects of Go-RT slowing on the PSI marginal method’s performance are already apparent before the boundaries of the parameter space are reached (trial 1 to 100): the stronger the response slowing, the lower the correlation between simulated and estimated SSRTs (Fig 6A), ranging from around 0.9 between trials 25 and 75 for 5 ms slowing, to around 0.75 for 15 ms of slowing. When the boundaries of the parameter space are reached (and crossed; after the red line), correlation coefficients start to decrease more strongly.

The same effect of response slowing is apparent for mean absolute deviation (B) between simulated and estimated SSRTs: the stronger Go-RT slowing, the larger the deviation becomes, even before reaching the parameter space’s boundaries. At trial 100, when the end of parameter space is reached, the mean absolute deviation for 5 ms of slowing lies at around 100 ms; for 10 ms, it lies at around 430 ms; for 15 ms, it lies just below 700 ms. Not shown here, the results also indicate that the increased absolute deviation during the first 100 trials is mainly due to incorrect estimation of the Go-RT and only to a lesser extent the result of erroneous estimation of the critical SSD.

In the linear slopes between simulated and estimated SSRTs (Fig 6C), the pattern is not so clear while participants remain inside parameter space: while slopes seem to slightly oscillate around 1, especially for stronger response slowing, the marked deterioration of performance for the PSI marginal method only begins after the threshold parameter space was left, with slopes sharply declining.

As visible in Fig 7, the PSI adjusted method is able deal with response slowing both when participants remain inside the parameter space as well as when they leave parameter space. In mean absolute deviation as well as linear slopes, some gradation is visible, with stronger response slowing leading the decreased performance (insets), but the size of these effects is not comparable to the PSI marginal method’s deterioration of performance.

In Fig 8, we show the entropy of the PSI marginal method’s posterior distributions over the course of the simulated experiments, separately for 5, 10 and 15 ms response slowing per Stop-trial. Furthermore, for each degree of response slowing, we separately show the entropy for threshold parameter (summed over slope and error rate), slope (summed over threshold and error rate) as well as error rate (summed over threshold and slope parameters). The parameter dominating the overall entropy, which can be envisioned as the sum of all three entropies, is the threshold parameter. It is clear that after a short drop in entropy for the threshold parameter (until trial 25 for 5 ms), the value increases gradually until trial 100. Starting at trial 100, when the boundaries of the parameter space are hit, overall entropy decreases again, which is largely driven by the threshold parameter. This shows that while response slowing leads to an increase in entropy when the parameter space has not been left yet, especially for the threshold parameter, the event of hitting the boundaries of parameter space and leaving it leads to decreased entropy, most likely due to the probabilities of the posterior distributions concentrating at the outer edges of parameter space.

Behavioral experiment 2: The PSI adjusted method

As the results of Experiment 1 showed, the presence of strong response slowing caused different problems for the PSI marginal method. After finding a good performance of the PSI adjusted method in the simulations, we wanted to put the method to the test in human participants and see whether these problems persisted. As in Experiment 1, a group of participants completed two blocks of the Stop-change task, one with SSDs controlled by the staircase method, and one controlled by the PSI adjusted method described above.

Results

We present the results of Experiment 2 in a similar manner to Experiment 1. Figs 9 and 10 are taken from a participant showing Go-response slowing similar to the example participant from Experiment 1 (Figs 1 and 2). Even though there is considerable slowing, the SSRT estimate of the PSI adjusted method is fairly stable across change trials, allowing the method to optimally sample SSDs despite Go-responses being slowed.

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Fig 9. Go-RT slowing in Experiment 2.

As Fig 1, a participant’s Go-RTs for one condition in the PSI adjusted method’s block. The line indicates the increasing number of Change-trials, the circles are Go-RTs.

https://doi.org/10.1371/journal.pone.0210065.g009

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Fig 10. Likelihood estimates during Experiment 2.

As Fig 2, the PSI adjusted method’s posterior distribution for the example participant. Colors indicate probability of the parameter combinations, the error rate dimension was averaged over. Despite considerable Go-response slowing, the parameters receiving the highest probability do not reach the upper boundaries of the parameter space.

https://doi.org/10.1371/journal.pone.0210065.g010

Fig 11 compares the PSI marginal and PSI adjusted method with respect to their parameter estimates over the course of the experiment. For the PSI marginal method (Experiment 1), there is a considerable number of participants, specifically those slowing their Go-responses, for which the threshold parameter receiving the highest probability in the posterior distribution reaches the method’s upper boundary, thus limiting the method’s ability to estimate the true SSRT. Furthermore, a smaller number of participants reach the lower boundary of the PSI marginal method’s parameter space. These are participants showing particularly fast Go-responses.

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Fig 11. Parameter estimates for Experiment 1 and 2.

Highest probability parameters for the PSI marginal method (Experiment 1) and PSI adjusted method (Experiment 2) over the course of the experiment. The participants were split intro groups depending on whether the showed go-response slowing. Every line represents one of the stop-change task’s conditions of one participant; thus, there are four lines per participant. Black lines indicate a parameter reaching the upper end of the parameter space, red lines indicate parameters reaching the lower end. Parameter estimates were slightly smoothed using a cubic spline to facilitate visual inspection. Note that while maximum probability parameters are shown, SSRTs are eventually computed based on the expected parameters.

https://doi.org/10.1371/journal.pone.0210065.g011

While for the PSI marginal method strong response slowing causes a parameter drift toward the upper boundary, in the PSI adjusted method the maximum probability parameters are visibly more stable over time, both for slowing and non-slowing participants. There is, however, a small number of participants in the slowing group whose parameter estimates in the PSI adjusted method drift toward the lower end of the parameter space. There thus appears to be some residual influence of response slowing on parameter estimation. Importantly, this problem is far less pronounced than in the PSI marginal method used in Experiment 1. We come back to the reasons behind this effect in the discussion.

Another problem we noted for the PSI marginal method and staircase in Experiment 1 was that for slowing participants, accuracy levels substantially deviated from 0.5. As Fig 12 shows, the PSI adjusted method is successful at maintaining accuracy at roughly 0.5 throughout the experiment, for both slowing and non-slowing participants. As in Experiment 1, the staircase method is only able to maintain this intended accuracy level for non-slowing participants.

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Fig 12. Accuracy throughout Experiment 2.

As Fig 3, the proportion of successful Change-trials for the PSI adjusted and Staircase methods, separately for slowed and non-slowed participants.

https://doi.org/10.1371/journal.pone.0210065.g012

As for experiment 1, we included further results on the relation between SSRT estimates of the staircase- and PSI method in the supplementary material (S5 Fig).

A further question relevant to the PSI adjusted method is how many trials are optimal when estimating Go-RTs through linear regression. While the up-to 40 trial long window used by us to predict current mean Go-RT appears to have worked fine, we present additional analysis of Go-RTs in supplementary S6 Fig, in which mean and standard deviation of the residuals of the RT prediction are shown, as a function of different window sizes. While the variability of residuals reaches a stable minimum with window sizes larger than 10 to 15 trials for both slowing and non-slowing subjects, the mean residual stays between -1 and 1 ms roughly between window sizes of 1 and 30 trials. Thus, according to our behavioral data, window sizes over 10 to 15 trials, but not substantially larger than 30 trials, are recommended. It should be noted that the optimal window size depends on the timescale and form of changes in RTs, which in turn likely depend on the specific task, such that other numbers of trials might be better suited to predict Go-RT in different tasks.

Discussion

The original PSI method for the Stop-signal task [4] was proposed to allow optimal sampling of SSDs in Stop-trials, thus allowing shorter experimental lengths. In this article, we showed that the PSI marginal method, and by implication the original PSI method, cannot adequately deal with the kind of response slowing often found in Stop-signal tasks. We proposed a PSI adjusted method that continuously estimates Go-RTs and adjusts for possible slowing, thereby allowing optimal sampling of SSDs despite slowing. In three sets of Stop-signal task simulations, we compared the performance of this adjusted method to the PSI marginal method as well as the classic staircase method. Furthermore, we tested how the PSI adjusted method fares when employed with human participants showing strong Go-response slowing.

From the simulation results, it is clear that when there is no response slowing, the PSI methods offer improved performance in estimating SSRTs, especially for small trial numbers and when unbiased estimation of the spectrum of SSRTs is important. In the face of response slowing, as is common in Stop-signal tasks, it is crucial to correct the PSI method for the changing response speed. When this is achieved, as in our proposed PSI adjusted method, this method can offer improved performance over the staircase method even under considerable response slowing.

Another interesting result most clearly visible in Fig 4 is that the traditional staircase combined with the mean method is considerably worse for estimating SSRTs at low trial numbers than when combined with the integration method. This performance decrement is visible when inspecting the slope between simulated and estimated SSRTs, showing that either smaller SSRTs are overestimated or larger SSRTs are underestimated. The cause of this behavior is that when using the mean method, it is assumed that the SSDs offered up to this point correspond to a performance of 0.5. However, since the staircase takes a certain time to reach that performance level, the SSDs subtracted from the mean Go-RT will introduce a certain bias, which varies depending on the true SSRT of the participant. With the integration method, on the other hand, this bias is corrected for by using not the mean Go-RT, but a percentile of the RT distribution that corresponds to the participant’s behavior.

A further benefit of the PSI methods not discussed so far is that for every point during the experiment, they provide a likelihood distribution over the parameters underlying the participant’s performance, and thus a likelihood distribution over the SSRT estimate at that point. This allows computing a confidence interval of sorts around the updated SSRT estimate after every Stop-trial (similar to the illustration of SSRT estimation in Fig 7). This opens up interesting new experimental opportunities, such as only running a Stop-signal task until a certain level of confidence in SSRT has been reached, or dynamically controlling stimulus presentation based on the SSRTs that are likely or unlikely for a participant.

As shown in the results section of Experiment 2, there appears to be a residual effect of Go-response slowing on SSRT estimation even for the PSI adjusted method. This is likely the result of less than optimal estimation of Go-RTs for this method, particularly an underestimation of Go-RTs. The mechanism behind this is the following: when participants have longer Go-RTs than estimated by the method, they are capable of still stopping for SSDs that would be too long under their actual SSRT, thus leading to a reduced estimated SSRT. Since the PSI adjusted method presented here predicts Go-RTs by linearly regressing previous RTs on trial number, it is clear that any kind of non-linear change in response times cannot be accommodated. Furthermore, the PSI adjusted method only predicts mean Go-RTs, thereby ignoring other properties of the Go-RT distribution, such as variance and skewness, which would be taken into account under the ex-Gaussian parameterization of the RT distribution. As a future improvement to the method, more flexible and behaviorally realistic models of response slowing could be used to better capture changes in RTs throughout the experiment. It stands to reason that this would further improve the statistical properties of the PSI adjusted method.

S3 and S5 Figs show the relationship between the SSRT estimates of the staircase blocks and the PSI method’s blocks, separately for Experiment 1 and 2. While the number of data points is not very large, it is clear that the correlations between estimates from the staircase method and the PSI methods are far from perfect. This is likely caused by response slowing, since the correlation coefficients are larger in the non-slowing groups, both for Experiment 1 and 2. In these groups, the adjusted PSI method produces SSRT estimates more in line with the staircase method than the PSI marginal method does. It is an open question to which degree the correlation-diminishing effects of response slowing are stemming from the PSI methods’ SSRT estimates or the staircase method’s estimates. Both the PSI methods and the staircase method might contribute to this deviation of estimates: the PSI marginal method does not take into account response slowing, and the adjusted PSI method’s estimate of Go-RTs might be suboptimal, while the staircase method might just as well not be able to deal effectively with Go-RT slowing. Future research might elucidate the source of these incongruencies in SSRT estimates.

While the method of optimal RT estimation is of technical relevance, there is also the more theoretical question of whether Go-response slowing invalidates the Stop-signal and related tasks. Indeed, it has often been pointed out that the independent horse race model underlying the analysis of behavior in this task assumes independence between the Go-process driving the Go-response and the Stop-process responsible for inhibiting this response. Since Go-response slowing in this task is generally related to the presence and prevalence of Stop-trials [15], this might call into question the validity of Stop-signal tasks that provoke considerable response slowing. We think these tasks are nevertheless important precisely because of this slowing.

While the mathematical independent horse race model understands the Go- and Stop-process as independent, it has been noted that pure independence can be paradoxical when addressing the instantiation of response stopping in the brain [16]. After all, the Stop-process must exert some form of influence over the Go-process in order to prevent its execution. Interestingly, Boucher et al. [17] showed that effects explained by the independent horse race model can be captured in an interactive horse race model where Go- and Stop-process do interact in a delayed and pronounced manner. We think that when discarding Go-response slowing simply as a nuisance when estimating SSRT, one is discarding a property inherent in the very neural inhibition system that is being studied. In this point we agree with other authors who have made Go-response slowing in Stop-signal tasks an explicit subject of experimental investigation [15, 18–20]. In this area of research, response slowing in the context of Stop-signals has often been taken as an indicator of proactive inhibition, and some results suggest that the neural processes involved in reactive response inhibition are also partly involved in proactive inhibition and can be indexed by response slowing [19, 21]. Indeed, quantifying not just reactive but also proactive inhibition in the same task has been useful in analyzing neural data and providing a more complete picture of the inhibition system [21, 22].

This is not to say, of course, that response slowing cannot be an obstacle to correct estimation of SSRT, as has been shown previously [3]. If precise estimation of only SSRT is paramount, experimental methods of preventing slowing can be crucial, such as giving participants feedback on their RTs. However, we think that in order to investigate the entire response inhibition system, Go-RT slowing must also be accounted for, which necessitates methods of SSRT estimation in the presence of slowed Go-responses.

A criticism that could be levelled against our behavioral experiments is that we used a non-traditional Stop-task, namely the Stop-change task, and included different conditions, which simulation studies usually do not. However, as described earlier, the processes involved in changing a response to another one have been found to partly overlap those of outright stopping a response [9–12]. Furthermore, while simulation studies often do not acknowledge different experimental conditions for lack of a clear way of simulating them, many behavioral experiments do include multiple conditions. Since this can increase the amount of response slowing per Stop-trial, it is important to consider the influence this has on estimating SSRTs.

We have presented behavioral as well as simulation data showing that the PSI marginal method and even the traditional staircase method in the Stop-signal task have problems accommodating excessive Go-response slowing. Consequently, we proposed an extension of the PSI method that accommodates this often observed behavior. Our results show this PSI adjusted method to be an advantageous alternative to previous methods of SSD choice especially when the number of Stop-trials that can be presented is limited.

Supporting information