#
SimSST: An R Statistical Software Package to Simulate Stop Signal Task Data^{ †}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Stop Signal Task

#### 1.2. The Horse Race Model

#### 1.3. The Key Reaction Times Distributions

#### 1.4. Motivation

#### 1.5. Study Outline

## 2. The SimSST Package

#### 2.1. The Background and Installation

- Set the simulation method (fixed SSD or tracking).
- Set the stochastic independency status of GORT and SSRT processes (Yes or No).
- Set distribution type for GORT and SSRT and also set starting SSD (one of seven distributions: ExW, ExG, SW, Gumbel, Weibull, Lognormal and Gamma).
- Simulate each process (either independently or using Copulas).
- Test if the response from the first stop trial meets $SSRT+SSD<GORT$. If so, the trial is marked as successful inhibition. Otherwise, failed inhibition.
- Test and record the status of all successive stop trials.
- (a)
- Fixed SSD Method—Repeat checking if the following stop trial status is similar to the first one without a change in the value of SSD in each stop trial.
- (b)
- Tracking Method—Check the status of the first stop trial. If a stop is successful, add a redesigned constant d (e.g., 50 ms) to the SSD to check the status of the second stop trial. Otherwise, subtract d (e.g., 50 ms) from the SSD to check that trial status. Repeat this algorithm for the successive stop trials until the end.

`line #1:> install.packages("SimSST", dependencies=TRUE)`

`line #2:> library(SimSST)`

`simssfixed()`,

`simsstrack()`and

`simssgen()`. The first two are based on an independent horse-race model, and the third one is based on a general non-independent horse-race model. Details are described in the coming sections. The reader may try to type

`?simssfixed`,

`?simsstrack`or

`?simssgen`in R for the details.

#### 2.2. The Fixed SSD Based Simulated SST Data

`simssfixed()`simulates the SST data based on the independent horse race model and the fixed SSDs and needs the following nine input parameters to simulate SST data. For given $b\ge 1$, id is a b-sized character vector for participants’ IDs. The parameters block, n, and m are b-sized numeric vectors of block numbers, total trials per block, and stop trials per block, respectively.

`SSD.b`is a b-sized vector of initial stop signal delays on each block. The

`dist.go`and

`theta.go`parameters are b-sized character vectors, specifying the type of GORT distributions, and a $b\times 3$ numeric matrix having the specification of their distributions, respectively. Similarly,

`dist.stop`and

`theta.stop`are a b-sized numeric vector specifying the type of SSRT distributions and a $b\times 3$ numeric matrix having the specification of their distributions. The function general code in R is:

`simssfixed(pid,block,n,m,SSD.b,dist.go,theta.go,dist.stop,theta.stop)`

#### 2.3. The Tracking Method Based Simulated SST Data

`simsstrack()`similarly simulates the SST data based on the independent horse race model and the tracking method in which at each stage a constant of 50 ms is added or subtracted to the following SSDs. Its parameters are the same as those of the former function

`simssfixed()`and its code in R is the following:

`simsstrack(pid,block,n,m,SSD.b,dist.go,theta.go,dist.stop,theta.stop)`

#### 2.4. The General Tracking Method Based Simulated Correlated SST Data

`simssfixed()`and

`simsstrack()`proposed in the package “SimSST” have two limitations. First, both are based on the stochastic independence of GORT and SSRT processes. Recent studies have shown a severe violation of the assumptions needed for the independent horse race paradigm [46]. Solutions to overcome this issue are considering SSDs from 200 ms and incorporation of the Spearman’s rank correlation [47] for the case of the relaxation of the assumption of stochastic independence between GORT and SSRT. (Note that for simulation purposes, the Pearson correlation is an inappropriate parameter as it may not be preserved under monotonic transformations of the original simulated data [47].) Second, there is no mechanism to connect the functions

`simssfixed()`and

`simsstrack()`. In particular, we know that replacing the constant added (subtracted) value of 50 ms to the SSD in the subsequent stop trials with 0 converts

`simsstrack()`function to the

`simssfixed()`. To solve these limitations, we may consider the generalized simulation function

`simssgen()`based on the theory of copulas with eleven parameters as follows:

`simssgen(pid,block,n,m,SSD.b,dist.go,theta.go,dist.stop,theta.stop,rho,d)`

`simssfixed()`and

`simsstrack()`and the additional parameters $rho$ and d refer to b-sized numeric vectors of the Spearman correlation of GORT and SSRT processes and the b-sized numeric vectors of measure of added (subtracted) constant to the initial SSD values, respectively. In particular, denoting the first nine parameters with “−”, we will have:

`line #1:> mySSTdata1<- simssgen(`

`pid = c("John.Smith", "John.Smith"),`

`block = c(1,2),`

`n = c(10,10),`

`m = c(4,4),`

`SSD.b = c(220,240),`

`dist.go = c("ExG", "ExG"),`

`theta.go = as.matrix(rbind(c(440,90,90),c(440,90,90))),`

`dist.stop = c("ExG", "ExG"),`

`theta.stop = as.matrix(rbind(c(120,80,70),c(120,80,70))),`

`rho=c(0,0),`

`d=c(0,0))`

`line #2:> mySSTdata1`

`line #1:> mySSTdata2<- simssgen(`

`pid = c("Jane.McDonald", "Jane.McDonald"),`

`block = c(1,2),`

`n = c(10,10),`

`m = c(4,4),`

`SSD.b = c(220,240),`

`dist.go = c("ExG", "ExG"),`

`theta.go = as.matrix(rbind(c(440,90,90),c(440,90,90))),`

`dist.stop = c("ExG", "ExG"),`

`theta.stop = as.matrix(rbind(c(120,80,70),c(120,80,70))),`

`rho=c(0,0),`

`d=c(50,50))`

`line #2:> mySSTdata2`

#### 2.5. Simulation Study: An Example

Method: | Tracking |

Stoch. Ind. Status: | Independent |

Initial SSD: | 220 ms |

d parameter: | 50 ms |

total trials: | $n=100$ k $(1\le k\le 10)$ |

stop trials: | $m=25$ k $(1\le k\le 10)$ |

GORT distribution: | ExGaussian(ExG) |

${\theta}_{go}=({\mu}_{go},{\sigma}_{go},{\tau}_{go})$ | $(440,90,90)$ |

SSRT distribution: | ExGaussian(ExG) |

${\theta}_{stop}=({\mu}_{stop},{\sigma}_{stop},{\tau}_{stop})$ | $(120,80,70).$ |

`simsstrack`in the SimSST package.

`line #1:> library(SimSST)`

`line #2:> SimSST_Data1 <- simssgen(`

`pid = c("FNLN1"),`

`block = 1,`

`n = 100,`

`m = 25,`

`SSD.b = 220,`

`dist.go = "ExG",`

`theta.go = as.matrix(rbind(c(440,90,90))),`

`dist.stop = "ExG",`

`theta.stop = as.matrix(rbind(c(120,80,70))),`

`rho=0,`

`d=50)`

`line #3:> SimSST_Data1`

Prior Setting Information: | |

${\mu}_{go}\sim Uniform[1,1000]:$ | start value: 500 |

${\sigma}_{go}\sim Uniform[1,500]:$ | start value: 150 |

${\tau}_{go}\sim Uniform[1,500]:$ | start value: 150 |

${\mu}_{stop}\sim Uniform[1,1000]:$ | start value: 300 |

${\sigma}_{stop}\sim Uniform[1,500]:$ | start value: 150 |

${\tau}_{stop}\sim Uniform[1,500]:$ | start value: 150 |

MCMC Setting Information: | |

Number of chains: | 3 |

Samples: | 50,000 |

Burn in: | 5000 |

Thining: | 5 |

Limits of Integration: | 1–10,000 |

With Trigger Failure(WTF): | No. |

## 3. Discussion

#### 3.1. Summary and Contributions

#### 3.2. Limitations and Future Work

#### 3.3. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

## Appendix A. The R Code to Convert SimSST Files to BEESTS Input Files

`line #1:> mypackages <- c("dplyr", "SimSST")`

`line #2:> lapply(mypackages, require, character.only = TRUE)`

`line #3:> Data1 <- cbind.data.frame(`

`ss_presented = recode(SimSST_Data1[,3], ‘Stop’ = "1", ‘Go’ = "0"),`

`inhibited = SimSST_Data1[,4],`

`ssd = SimSST_Data1[,8],`

`rt = SimSST_Data1[,5],`

`srrt = SimSST_Data1[,7]`

`)`

`line #4:> for(i in 1:100)`

`if(Data1$inhibited[i]==0) Data1$rt[i] <- Data1$srrt[i]`

`line #5:> BEESTS_Data1 <- (Data1[,-5])[order(Data1$ss_presented),]`

`line #6:> BEESTS_Data1`

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**Figure 1.**(

**a**) The standard stop signal task; (

**b**) Graphical representation of the complete horse race model: GORT: go reaction times, SRRT: signal respond reaction times, SSRT: stop signal reaction times, $SSD$: stop signal delay.

**Figure 2.**(

**a**–

**c**) ExG distribution hazard function change by parameters given ${\mu}_{0}=440,{\sigma}_{0}=90,{\tau}_{0}=90$; (

**d**–

**f**) SW distribution hazard function change by parameters given ${\mu}_{0}=1,{\sigma}_{0}=1,{\tau}_{0}=200$.

**Figure 3.**Sample SimSST R software package(R studio version 4.1.3) simulated SST data output with input variables as in the text.

**Figure 4.**BEETS method parameters estimations for the SimSST-based simulated SST data with true parameter values on the black horizontal line: (

**a**–

**c**) GORT ExG parameters; (

**d**–

**f**) SSRT ExG parameters.

**Table 1.**A summary of Exponentially modified Gaussian (ExG) and Shifted Wald (SW) distributions’ key features.

Distribution | $\mathbf{ExG}(\mathit{\mu},\mathit{\sigma},\mathit{\tau})$ | $\mathbf{SW}(\mathit{\mu},\mathit{\sigma},\mathit{\tau})$ |
---|---|---|

Representation | $N(\mu ,{\sigma}^{2})\oplus Exp\left(\tau \right)$ | $Wald(\mu ,{\sigma}^{2})\oplus {\delta}_{\tau}$ |

Domain | $t\in \mathbb{R}$ | $t\in {\mathbb{R}}_{\tau}^{+}$ |

Parameters | $\mu \in \mathbb{R},\sigma \in {\mathbb{R}}^{+},\tau \in {\mathbb{R}}^{+}$ | $\mu \in {\mathbb{R}}^{+},\sigma \in {\mathbb{R}}^{+},\tau \in {\mathbb{R}}_{0}^{+}$ |

$\frac{1}{\tau}exp(-\frac{t-\mu}{\tau}+\frac{{\sigma}^{2}}{2{\tau}^{2}})\mathsf{\Phi}(\frac{t-\mu}{\sigma}-\frac{\sigma}{\tau})$ | $\frac{1}{\sqrt{2\pi}\sigma {(t-\tau )}^{\frac{3}{2}}}exp(-\frac{{(t-\tau -\mu )}^{2}}{2{\sigma}^{2}{\mu}^{2}(t-\tau )})$ | |

CDF | $\mathsf{\Phi}\left(\frac{t-\mu}{\sigma}\right)-exp(-\frac{t-\mu}{\tau}+\frac{{\sigma}^{2}}{2{\tau}^{2}})\mathsf{\Phi}(\frac{t-\mu}{\sigma}-\frac{\sigma}{\tau})$ | $\mathsf{\Phi}\left(\frac{t-\tau -\mu}{\sigma \mu \sqrt{t-\tau}}\right)+exp\left(\frac{2}{\mu {\sigma}^{2}}\right)\mathsf{\Phi}\left(\frac{-(t-\tau )-\mu}{\sigma \mu \sqrt{t-\tau}}\right)$ |

Mean | $\mu +\tau $ | $\mu +\tau $ |

Variance | ${\sigma}^{2}+{\tau}^{2}$ | ${\mu}^{3}{\sigma}^{2}$ |

Skewness | $2{\sigma}^{-3}{\tau}^{3}{(1+{\sigma}^{-2}{\tau}^{2})}^{\frac{-3}{2}}$ | $3{\left(\mu {\sigma}^{2}\right)}^{\frac{1}{2}}$ |

Kurtosis | $3(1+2{\sigma}^{-2}{\tau}^{2}+3{\sigma}^{-4}{\tau}^{4}){(1+{\sigma}^{-2}{\tau}^{2})}^{-2}$ | $15\mu {\sigma}^{2}+3$ |

Hazard Function | ↗ | ↷ |

Scenario | Method | Stoch. Ind. | GORT | SSRT | Function |
---|---|---|---|---|---|

1 | Fixed SSD | Yes | ExG | ExG | simssfixed()/simssgen() |

2 | SW | simssfixed()/simssgen() | |||

3 | SW | ExG | simssfixed()/simssgen() | ||

4 | SW | simssfixed()/simssgen() | |||

5 | No | ExG | ExG | simssgen() | |

6 | SW | simssgen() | |||

7 | SW | ExG | simssgen() | ||

8 | SW | simssgen() | |||

9 | Tracking | Yes | ExG | ExG | simsstrack()/simssgen() |

10 | SW | simsstrack()/simssgen() | |||

11 | SW | ExG | simsstrack()/simssgen() | ||

12 | SW | simsstrack()/simssgen() | |||

13 | No | ExG | ExG | simssgen() | |

14 | SW | simssgen() | |||

15 | SW | ExG | simssgen() | ||

16 | SW | simssgen() |

**Table 3.**A summary of one sample t-test and one sample Wilcoxon signed-ranked test for each of individual parameters ($n=10$).

${\mathit{H}}_{0}:\mathit{\theta}={\mathit{\theta}}_{0}$ | ||||||
---|---|---|---|---|---|---|

${\mathit{H}}_{\mathbf{1}}:\mathbf{\theta}\ne {\mathbf{\theta}}_{\mathbf{0}}$ | ||||||

$\mathbf{\theta}={\mathbf{\theta}}_{\mathbf{0}}$ | ${\mathbf{\mu}}_{\mathbf{go}}=\mathbf{440}$ | ${\mathbf{\sigma}}_{\mathbf{go}}=\mathbf{90}$ | ${\mathbf{\tau}}_{\mathbf{go}}=\mathbf{90}$ | ${\mathbf{\mu}}_{\mathbf{stop}}=\mathbf{120}$ | ${\mathbf{\sigma}}_{\mathbf{stop}}=\mathbf{80}$ | ${\mathbf{\tau}}_{\mathbf{stop}}=\mathbf{70}$ |

(a) t-test | ||||||

$\widehat{\theta}(95\%CI)$ | 445(435,454) | 91(86,96) | 88(81,94) | 120(102,138) | 86(72,100) | 86(69,104) |

Sig.(2-sided) | 0.305 | 0.629 | 0.436 | 0.961 | 0.332 | 0.061 |

PE(%) | 1.1 | 1.1 | 2.2 | 0.0 | 7.5 | 22.9 |

(b) Wilcoxon test | ||||||

$\widehat{\theta}(95\%CI)$ | 443(435,464) | 90(87,98) | 89(81,94) | 120(102,138) | 87(71,110) | 89(66,110) |

Sig.(2-sided) | 0.447 | 0.953 | 0.721 | 1.000 | 0.635 | 0.076 |

PE(%) | 0.7 | 0.0 | 1.1 | 0.0 | 8.8 | 27.1 |

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Soltanifar, M.; Lee, C.H. SimSST: An R Statistical Software Package to Simulate Stop Signal Task Data. *Mathematics* **2023**, *11*, 500.
https://doi.org/10.3390/math11030500

**AMA Style**

Soltanifar M, Lee CH. SimSST: An R Statistical Software Package to Simulate Stop Signal Task Data. *Mathematics*. 2023; 11(3):500.
https://doi.org/10.3390/math11030500

**Chicago/Turabian Style**

Soltanifar, Mohsen, and Chel Hee Lee. 2023. "SimSST: An R Statistical Software Package to Simulate Stop Signal Task Data" *Mathematics* 11, no. 3: 500.
https://doi.org/10.3390/math11030500