#  A Bayesian Mixture Modelling of Stop Signal Reaction Time Distributions

Mohsen Soltanifara, Michael Escobarb, Annie Dupuisc, and Russell Schachard
CONTACT Mohsen Soltanifar. Email<EMAIL_ADDRESS>a,b,c
Biostatistics Division, Dalla Lana School of Public Health, University of
Toronto, Toronto, M5T 3M7, ON, Canada ; a,c,d Department of Psychiatry, The
Hospital for Sick Children, 555 University Avenue, Toronto, M5G 1X8, ON,
Canada

###### Abstract

The distribution of single Stop Signal Reaction Times (SSRT) in the stop
signal task (SST) as a measurement of the latency of the unobservable stopping
process has been modeled with a nonparametric method by Hans Colonius (1990)
and with a Bayesian parametric method by Eric-Jan Wagenmakers and colleagues
(2012). These methods assume equal impact of the preceding trial type
(go/stop) in the SST trials on the SSRT distributional estimation without
addressing the case of the violated assumption. This study presents the
required model by considering two state mixture model for the SSRT
distribution. It then compares the Bayesian parametric single SSRT and mixture
SSRT distributions in the usual stochastic order at the individual and the
population level under ex-Gaussian distributional format. It shows that
compared to a single SSRT distribution, the mixture SSRT distribution is more
diverse, more positively skewed, more leptokurtic, and larger in stochastic
order. The size of the disparities in the results also depends on the choice
of weights in the mixture SSRT distribution. This study confirms that mixture
SSRT indices as a constant or distribution are significantly larger than their
single SSRT counterparts in the related order. This offers a vital improvement
in the SSRT estimations.

###### keywords:

Stop Signal Reaction Times, Mixture Distribution, Stochastic Order, Bayesian
Parametric Approach

##  1 Introduction

Inhibition refers to the ability to actively suppress, interrupt or delay an
action [1]. Inhibition itself is a key dimension of executive control, which
on its own is required for an organism to adjust behaviour according to
changing conditions; this could be assessing inappropriateness of the current
course of thought and action, changing goals, and changing world, [2, 3, 4].
Response inhibition is the ability to stop responses that are no longer
appropriate [3]. Examples of response inhibition in daily life includes
braking quickly when driving into an intersection while another vehicle is
running through a red light [4]. In order to empirically study response
inhibition in a laboratory setting, two paradigms have been suggested: The
Go/No-go task and the stop signal task (SST), and the later is widely used,
[1, 5]. The stop signal paradigm includes two response tasks: go task and the
stop task(e.g. stop 25% of the time). In go trials, the go reaction time
(GORT) is the response to the stimulus such as “X” and “O” presented on the
computer screen. In stop trials, the stop signal reaction time (SSRT) is the
unobserved latency of the stopping response upon observing the stop signal
(e.g. an auditory tone such as “beep”) after stop signal delay (SSD or
$T_{d}$) extra time [6, 7]. Often, the adjustment of stop signal delays
$T_{d}$ is done by the more reliable tracking method in which depending to
previous trial’s success or failure the $T_{d}$ is increased or decreased by
50 ms with the aim of achieving 50% overall successful inhibition at the end
of the paradigm. In the go trials and stop signal response trials the observed
reaction times and the unobserved latency of the stopping response(i.e. GORT,
SRRT, and SSRT, respectively) are measured in milliseconds. In young adults
trying to stop continuous actions, such as typing, the SSRT is close to 200 ms
[8].

Figure 1: Graphical representation of the complete horse race model[[3],
modification] : GORT: go reaction times, SRRT: Signal respond reaction times,
SSRT: Stop Signal reaction times; Td: stop signal delay(SSD)

Several models have been proposed for the evaluation and description of
response inhibition in the stop signal paradigm, including the deadline model,
the independent horse race model, the interactive horse race model and the
Hanes-Carpenter model [9, 10, 11, 12]. In this study, the independent horse
race model is considered. It provides a theoretical framework in which
researchers can measure the Stop Signal Reaction Times (SSRT), and its
associated factors [13]. There are two types of the horse race model: the
independent model with constant SSRT index, and the complete independent model
with non-constant SSRT distribution; in this study, we focus on the second
model(See Figure 1). SSRT measurements have been a key tool used by
psychopathologists to make inferences about a patient’s ability to inhibit
thought and action (i) on the spectrum of clinical groups (e.g. ADHD, OCD,
autism, schizophrenia ); and, (ii) across various tasks and experimental
conditions [13]. SSRT measurement’s precise estimation affects such inferences
profoundly.

There are several methods of estimation of SSRT depending to two contexts in
the SST literature: (i) as a constant index, or (ii) as a non-constant random
variable. Within each context these methods mentioned below refined the
earlier proposed methods given their associated contexts.

There are four estimation methods of SSRT as a constant index including the
mean crude method, the Logan 1994 integration method[3], the weighted
method[14] and the time series based state space method[15]. Given a subject
with go reaction time GORT random variable in the go trials with quantile
function $Q_{GORT}$, $n$ stop signal delays $T_{d}$ , and the probability of
successful inhibition(SI) the first two point indices of constant SSRT in the
entire SST cluster are defined as:

$\displaystyle SSRT_{Crude}^{c}$ $\displaystyle=$
$\displaystyle\overline{GORT}-\overline{T_{d}}$ (1) $\displaystyle
SSRT_{Logan1994}^{c}$ $\displaystyle=$ $\displaystyle
Q_{GORT}(1-P(SI|\overline{T_{d}}))-\overline{T_{d}}.$ (2)

Given higher reliability and less bias in the second index versus the first
index, the second index has been recommended as the plausible index of
constant SSRT [16]. The third and fourth indices are essentially improvements
of the second index under their associated contexts. For the third point index
of constant SSRT, partitioning the entire SST cluster in to two clusters of
type-A SST cluster (trials following a go trial) and type-B SST cluster
(trials following a stop trial) and calculating trial type related Logan 1994
SSRT as $SSRT_{A}$ and $SSRT_{B}$ with corresponding weights
$W_{A}=\\#\text{Type A Stop}/\\#\text{Total Stop},$ $W_{B}=1-W_{A},$ the third
point index of constant SSRT is defined as:

$SSRT_{Weighted}^{c}=W_{A}.SSRT_{A}+W_{B}.SSRT_{B}.$ (3)

Finally, transforming raw tri-variate SST time series data to tri-variate
state-space time series data using missing data EM algorithm with log normally
distributed outputs $GORT^{ss.ln}$ and $T_{d}^{ss.ln},$ the fourth point index
of constant SSRT is defined as:

$SSRT_{SS.Logan1994}^{c}=Q_{GORT^{ss.ln}}(1-P(SI|\overline{T_{d}^{ss.ln}}))-\overline{T_{d}^{ss.ln}}.$
(4)

Several researchers have shown that merely focusing on measures of central
tendency in reaction times RT including SSRT gives insufficient information
regarding the nature of the data. For instance, different clinical groups may
have the same mean reaction times but the shape of their distributions differ
either in their tails as seen in an ADHD group compared to the controls [17]
or their domain of variance in a schizophrenia group versus control [18].
These observations lead the researchers to study the entire SSRT distribution
(Figure 1).

There are two main methods to estimate SSRT as a random variable including the
Colonius’s nonparametric method [19] and the Bayesian parametric method [20,
21]. The first method retrieves the cumulative distribution function of SSRT
given go reaction times GORT in go trials with density $f_{GORT}$, signal
respond reaction times SRRT in the failed stop trials with density $f_{SRRT}$,
n stop signal delays $T_{d},$ and probability of successful inhibition(SI) as:

$F_{SSRT}(t)=1-(1-P(SI|T_{d})).(\frac{f_{SRRT}(t+T_{d}|T_{d})}{f_{GORT}(t+T_{d})}),\
\ 0<t,T_{d}<\infty.$ (5)

Although the first method theoretically gives the entire non-parametric
distribution of SSRT, in practice it cannot be implemented for empirical data
as it requires an unrealistically high number of trials for accurate
estimations; and, in the case of such estimations for the case of simulated
data it has underestimated the mean of SSRT and overestimated the variance of
SSRT [4, 22]. These observations lead researchers to propose the second method
of estimation of SSRT in the Bayesian context under given parametric
distributional assumptions for the involved GORT, SRRT and SSRT in the SST
data, [20]. The Bayesian Parametric Approach (BPA) presents a novel parametric
approach to estimate the entire distribution of SSRT, which applies to real
data with a low trial number [21]. Depending on the individual or hierarchical
data, the BPA estimates parameters of the SSRT distribution separately for
each participant (called individual BPA or IBPA), or estimates the parameters
of SSRT for each participant with information from the entire population, as
well as parameters of SSRT for the entire population(called hierarchical BPA
or HBPA) [23, 24]. The BPA is based on the idea of censoring, in which the
censored observation on the right side of the censoring point $(T_{d}+SSRT)$
in the signal inhibit stop trials are omitted, but their number is known. It
also assumes the fundamental assumption of the independent horse race model,
that is, the GORT and SSRT are independent distributions [21]. Assuming that
the distribution of GORT and SSRT follows a parametric form, such as Ex-
Gaussian, Ex-Wald, Shifted Wald, Shifted Lognormal or Shifted Weibull with
parameter $\theta=(\theta_{1},\theta_{2},\theta_{3}),$ the goal of the BPA is
to estimate simultaneously the parameters $\theta_{go},\theta_{stop}.$ First,
given $G$ go trials with go RT times $t=(t_{g})_{g=1}^{G}$, the log likelihood
is given by:

$Log(L(\theta_{go}|t))=\sum_{g=1}^{G}log(f_{GO}(t_{g}|\theta_{go}))$ (6)

Second, given S stop trials, including R signal respond stop trials with go
SRT times $t=(t_{r})_{r=1}^{R}$, and $I$ signal inhibit trials with delays
$T_{d}=(t_{d_{s}})_{s=1}^{S},(S=R+I),$ the log-likelihood of is given by:

$\displaystyle Log(L(\theta_{go},\theta_{stop}|t,T_{d}))$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{R}log(f_{go}(t_{r}|\theta_{go}))+\sum_{r=1}^{R}log(1-F_{stop}(t_{r}-T_{d_{r}}|\theta_{stop}))$
(7) $\displaystyle+$
$\displaystyle\sum_{i=1}^{I}log(\int_{0}^{\infty}(1-F_{go}(t|\theta_{go}))\times
f_{stop}(t-T_{d_{S-I+i}})dt).$

Finally, although standard MLE methods can be applied to estimates posterior
estimates $\theta_{go},\theta_{stop},$ given BPA intends to handle both
individual and hierarchical, it applies Markov Chain Monte Carlo (MCMC)
sampling to estimate them [20, 21]. The software used for the computation
includes WINBUGS [20] and Bayesian Ex-Gaussian Estimation of Stop-Signal RT
distributions (BEESTS) [21]. Several studies have used the BPA approach in
estimating parameters of SSRT distribution, for the case of Ex-Gaussian
distribution assumption with $\theta=(\mu,\sigma,\tau).$ For example, it has
been shown that more practice in stop trials corresponds to lower estimated
$\mu$ and higher estimated $\tau$, for the SSRT distribution, [25]. Next, the
BPA approach has shown that tyrosine consumption corresponds with lower
estimated $\mu$ for the SSRT distribution [26]. Finally, the mixture BPA
approach has been used to show the existence of trigger failures111Trigger
Failure(TF) refers to the situation in which the participant fails to
correctly dignose and interpret the stop signal leading to his inability to
attempt to inhibit the ongoing go process, [27]. on stop signal performance in
healthy control participants in two studies of inhibition deficiency in
schizophrenia [28].

However, as mentioned by Logan [4] there is little known about the
inhibition’s aftereffects and the type of questions of interest. One related
unanswered question is whether there exists any aftereffects of the non-
inhibited(e.g. go) trials and inhibited (e.g. stop)trials on inhibition and in
case of affirmative answer how to measure SSRT as a random variable. Here, in
both the nonparametric and Bayesian parametric methods mentioned above, there
is an implicit assumption about the aftereffects of go trials and stop trials
in SST data; that is, the impact of the preceding trial type, either go or
stop, on the current stop trial SSRT is assumed to be the same. Most of the
SST literature has taken this assumption for granted, and to the best of
authors’ knowledge, there are few studies to date that dealt with this
question and the estimation of the SSRT distribution when this assumption is
violated. Some studies have shown that after a go trial the participants have
a lower go reaction time GORT versus after the stop trial [29]. This implies
that the GORT distribution after each type of trial(go/stop) will differ,
impacting the participant’s ability to stop after each trial type [30]. There
are only two studies in the SST literature that partially answered this
question when SSRT is considered as constant index [14, 15]. Here, it was
shown when considering SST data in a longitudinal context
$SSRT_{Weighted}^{c}>SSRT_{Logan1994}^{c};$ and when considering SST data in a
missing time series context $SSRT_{(SS.Logan1994)}^{c}>SSRT_{Logan1994}^{c}.$
Both studies’ results were valid for the empirical SST data and the simulated
SST data. Given that constant SSRT index can be considered as a degenerate
random variable these results partially shed light on the proposed question
for the case of a non-degenerate SSRT random variable. However, it is still
unknown in which order context these comparisons over random variables can be
conducted; additionally, in which mechanism the pairwise comparisons of the
involved paired sets of random variables is conducted.

This study offers an estimation of the SSRT distribution given the violated
assumption of equal impact of the preceding trial type(go/stop) on the current
stop trial SSRT distribution using the notion of two state mixtures [31] and
proposes parametric mixture Bayesian modelling on the entire SST data set. The
outline of the study is as follows: First, as in [14] for each participant the
overall empirical SST data is partitioned into type A cluster SST data and
type B cluster SST data. Using IBPA method the fitted SSRT Ex-Gaussian
parameters are calculated for the cluster type SSRT distributions and the
single SSRT distribution. The study’s empirical data provides an example of
the violated assumption. Second, a mixture SSRT random variable is introduced
as a natural generalization of its constant index $SSRT_{Weighted}^{c}$
counterpart in equation (3) and its Bayesian parametric form under the
ExGaussian distributional assumption with uninformative uniform priors for its
components parameters is considered. Then, considering the mean of posterior
parameters as their point estimations the key descriptive and shape statistics
of the mixture SSRT $(SSRT_{Mixture})$ are compared with those of type A SSRT
$(SSRT_{A})$, type B SSRT $(SSRT_{B}),$ and the single SSRT $(SSRT_{Single})$.
Third, we compare the involved pairs of distributions in usual stochastic
order $(<_{st})$ at the individual and population level. The comparisons at
the population level use our proposed Two Stage Bayesian Parametric Approach
(TSBPA) and our proposed Paired Samples Parametric Distribution Test (PSPDT).
Finally, the earlier comparisons are repeated and discussed in terms of the
involved weights in the definition of proposed mixture SSRT
$(SSRT_{Mixture}).$

##  2 Materials & Methods

###  2.1 The Data and Study Design

This study’s data and design are previously described at the [32]. That study
included 16,099 participants aged 6 to 19 years old and was conducted at the
Ontario Science Center in Toronto, Canada between June 2009 and September
2010. Each participant completed 4 blocks of 24 trials with a random 25% stop
signal trials in each block. There was 96 trials in total(24 stop signals; and
72 go trials). Every go trial began with a 500 ms fixation point followed by a
stimulus: An O or X presented for 1000 ms in the center of a computer screen.
With an initial stop signal delay $T_{d}$ of 250 ms after the go stimulus,
each stop trial included a audio stop signal cue presented through headphones
to the participant in the context of the tracking method.

###  2.2 The Sample and Variables

####  2.2.1 Cluster Type SST data

A random sample of 44 participants was selected for further analysis. The
entire stop signal task data for each participant was partitioned to type-A
stop signal task data (all trials preceded by go trials) and type-B stop
signal task data (all trials preceded by stop trials) as shown in the Table 1,
[14]. These participants each had minimum 10 type B stop trials.

Table 1: Partition of stop task (SST) data to Type A SST data and Type B SST data given previous trial type (go/stop), [14] Data |  | Previous Trial
---|---|---
|  |  | Go | Stop
Current Trial |  | Go | $Go_{A}$ | $Go_{B}$
| Stop | $Stop_{A}$ | $Stop_{B}$

For each participant four types of SST data clusters were considered: Type -A
cluster, Type-B cluster, Type-S Single cluster (entire single SST data), and
type-M Mixture cluster (composition of type-A and type-B clusters) . Using
IBPA, the parameters of the corresponding Ex-Gaussian SSRTs’ parameters
$\theta=(\mu,\sigma,\tau)$ were calculated as described in the next
subsections.

####  2.2.2 Ex-Gaussian Random Variable

Heathcote (1996), [33] formulated the Ex-Gaussian (ExG) distribution with
parameters $(\mu,\sigma,\tau)$ with density given by:

$f_{ExG}(t|\mu,\sigma,\tau)=\frac{1}{\tau}exp(\frac{\mu-t}{\tau}+\frac{\sigma^{2}}{2\tau^{2}})\times\Phi(\frac{\mu-t}{\sigma}-\frac{\sigma}{\tau}):\\\
\sigma,\tau>0,-\infty<t<\infty$ (8)

where $\Phi$ is the standard normal cumulative distribution function. The
first four non-central moments are given by:

$\displaystyle E(ExG)$ $\displaystyle=$ $\displaystyle\mu+\tau,$
$\displaystyle E(ExG^{2})$ $\displaystyle=$
$\displaystyle\mu^{2}+2\mu\tau+\sigma^{2}+2\tau^{2},$ $\displaystyle
E(ExG^{3})$ $\displaystyle=$
$\displaystyle\mu^{3}+3\mu\sigma^{2}+6\mu\tau^{2}+3\mu^{2}\tau+3\sigma^{2}\tau+6\tau^{3},$
$\displaystyle E(ExG^{4})$ $\displaystyle=$
$\displaystyle\mu^{4}+4\mu^{3}\tau+6\mu^{2}\sigma^{2}+12\mu^{2}\tau^{2}+24\mu\tau^{3}+12\mu\sigma^{2}\tau+3\sigma^{4}+12\sigma^{2}\tau^{2}+24\tau^{4}.$

Finally, this random variable is right-skewed and leptokurtic with the
following variance, skewness and kurtosis shape statistics:

$\displaystyle Var(ExG)$ $\displaystyle=$ $\displaystyle\sigma^{2}+\tau^{2}$
$\displaystyle\gamma_{ExG}$ $\displaystyle=$ $\displaystyle
2(1+\sigma^{2}\tau^{-2})^{\frac{-3}{2}}$ $\displaystyle\kappa_{ExG}$
$\displaystyle=$ $\displaystyle
3\bigg{(}\frac{1+2\sigma^{-2}\tau^{2}+3\sigma^{-4}\tau^{4}}{(1+\sigma^{-2}\tau^{2})^{2}}\bigg{)}.$
(10)

####  2.2.3 Mixture SSRT Random Variable

Given single SSRT by $SSRT_{S}$, type-A SSRT by $SSRT_{A}$ type-B SSRT by
$SSRT_{B}$, and $W_{A}\sim Bernoulli(W_{A}^{c})$ where the type A trial type
weight $W_{A}^{c}$ is given by $W_{A}^{c}=\\#\text{Type A Stop}/\\#\text{Total
Stop},$ $W_{B}^{c}=1-W_{A}^{c},$ the Single SSRT and Mixture SSRT random
variables were defined as follows222Note that with notation SSRT for random
variable SSRT and $SSRT^{c}$ for constant SSRT estimated with frequentist
methods, we have: $E(SSRT)=SSRT^{c}$. Consequently, definitions in equation
(11) are natural generalizations of constant SSRT estimations with frequentist
methods [14] to general non-constant random variables.333Here onward, $W_{A}$
given the context is either a Bernoulli random variable or a constant number
$W_{A}^{c}$ defined as above.:

$\displaystyle SSRT_{Single}$ $\displaystyle=$ $\displaystyle SSRT_{S},$
$\displaystyle SSRT_{Mixture}$ $\displaystyle=$ $\displaystyle W_{A}\times
SSRT_{A}+W_{B}\times SSSRT_{B}.$ (11)

In the Bayesian context and using IBPA and under Ex-Gaussian parametric
assumption we have $SSRT_{S}\sim
ExG(\theta_{S}=(\mu_{S},\sigma_{S},\tau_{S})),$ $SSRT_{A}\sim
ExG(\theta_{A}=(\mu_{A},\sigma_{A},\tau_{A}))$ and $SSRT_{B}\sim
ExG(\theta_{B}=(\mu_{B},\sigma_{B},\tau_{B})),$ where the parameter point
estimations are the means of the associated posterior distributions in IBPA.
The Bayesian Mixture Ex-Gaussian SSRT model can be formulated as follows.
Figure 2 presents the model using plate notation. Here, we have:

$K=2:$ Number of cluster types,

$N=96:$ Number of trials in SST data,

$\theta_{i}=(\mu_{i},\sigma_{i},\tau_{i}):$ Parameters of Ex-Gaussian SSRT
distribution of the ith cluster $(i=1:A;i=2:B),$

$\mu_{i}\sim U[\alpha_{1},\beta_{1}],(i=1:A;i=2:B):$ Here:
$\alpha_{1}=10,\beta_{1}=2,000,$

$\sigma_{i}\sim U[\alpha_{2},\beta_{2}],(i=1:A;i=2:B):$ Here:
$\alpha_{2}=10,\beta_{2}=2,000,$

$\tau_{i}\sim U[\alpha_{3},\beta_{3}],(i=1:A;i=2:B):$ Here:
$\alpha_{3}=10,\beta_{3}=2,000,$

$\phi=(\phi_{1},\phi_{2}):$ Prior Probability of clusters
$(\phi_{1}=W_{A},\phi_{2}=W_{B}),$

$z_{i}\sim Bernoulli(W):\ \ W=W_{A},$

$x_{i}:$ ith SST trial,

$x_{i}|_{stop}\sim ExG(\mu_{z_{i}},\sigma_{z_{i}},\tau_{z_{i}}).$

Figure 2: Bayesian Mixture Ex-Gaussian SSRT model using plate notation. Filled
in shapes indicate known values.

The first four moments of the Mixture SSRT are as follows:

$E(SSRT_{Mixture}^{k})=W_{A}E(SSRT_{A}^{k})+W_{B}E(SSRT_{B}^{k}):\ \ 1\leq
k\leq 4.$ (12)

Consequently, the variance, the skewness and the kurtosis of the Mixture SSRT
are computed by:

$\displaystyle Var(SSRT_{Mixture})$ $\displaystyle=$ $\displaystyle
W_{A}E(SSRT_{A}^{2})+(1-W_{A})E(SSRT_{B}^{2})$
$\displaystyle-(W_{A}E(SSRT_{A})+(1-W_{A})E(SSRT_{B}))^{2},$
$\displaystyle\gamma_{SSRT_{Mixture}}$ $\displaystyle=$
$\displaystyle\frac{1}{Var^{\frac{3}{2}}(SSRT_{Mixture})}\bigg{(}E(SSRT_{Mixture}^{3})$
$\displaystyle-3E(SSRT_{Mixture})E(SSRT_{Mixture}^{2})+2E^{3}(SSRT_{Mixture})\bigg{)},$
$\displaystyle\kappa_{SSRT_{Mixture}}$ $\displaystyle=$
$\displaystyle\frac{1}{Var^{2}(SSRT_{Mixture})}\bigg{(}E(SSRT_{Mixture}^{4})-4E(SSRT_{Mixture})E(SSRT_{Mixture}^{3})$
$\displaystyle+6E^{2}(SSRT_{Mixture})E(SSRT_{Mixture}^{2})-3E^{4}(SSRT_{Mixture})\bigg{)}.$

######  Remark 1.

Using new equation (11) for SSRT, Colonious’s proposed nonparametric method
for retrieving the entire SSRT CDF for given type-A weight $W_{A},$ type A
delay $T_{dA}$, type-A signal respond density $f_{SRRTA}$, type-A GORT density
$f_{GORTA}$, type-A probability of successful inhibition $P(SI|T_{dA})$; and,
the corresponding type B information yields the following mixture form:

$\displaystyle F_{SSRT}(t)$ $\displaystyle=$ $\displaystyle
1-W_{A}\bigg{(}(1-P(SI|T_{dA})).(\frac{f_{SRRT}(t+T_{dA}|T_{dA})}{f_{GORT}(t+T_{dA})})\bigg{)}$
(14) $\displaystyle-
W_{B}\bigg{(}(1-P(SI|T_{dB})).(\frac{f_{SRRT}(t+T_{dB}|T_{dB})}{f_{GORT}(t+T_{dB})})\bigg{)},$
$\displaystyle\hskip 142.26378pt0<t,T_{dA},T_{dB}<\infty.$

######  Remark 2.

The mixture modelling for SSRT proposed here can be applied with other non-Ex-
Gaussian parametric RT distributions such as Ex-Wald, Wald, [34] Gamma,
Weibull and Lognormal [35, 36] with the required modifications in estimations.

###  2.3 Statistical Analysis

For each participant IBPA under Ex-Gaussian parametric distribution was run
three times: one for its associated cluster type-A, cluster type-B and single
type-S SST data (a total of 132 times). We then calculated the mean posterior
estimates of $\theta_{S}=(\mu_{S},\sigma_{S},\tau_{S}),$
$\theta_{A}=(\mu_{A},\sigma_{A},\tau_{A}),$ and
$\theta_{B}=(\mu_{B},\sigma_{B},\tau_{B}).$ Then, the parameters, the
descriptive statistics and the shape statistics for type-A SSRT $(SSRT_{A}),$
type-B SSRT $(SSRT_{B}),$ type-S single SSRT $(SSRT_{Single})$ and type-M
Mixture SSRT $(SSRT_{Mixture})$ were calculated. The next steps of the
analysis depended to the context and procedure described in the following.

####  2.3.1 Comparisons Context: Real Numbers and Random Variables

Two sets of comparisons were conducted: (i) within a real numbers contexts;
and, (ii) within a real valued random variables context. For the first set of
comparisons, paired t-test (PROC TTEST, ‘SAS/STAT’ software version 9.4, [37])
were conducted to compare fitted parameters of the Ex-Gaussian distribution,
the descriptive summary statistics and the shape statistics in the usual real
numbers order $(<)$ across cluster types. For the second set of comparisons,
the two samples Kolmogorov Smirnov(KS) tests ( ks.test package stats, ‘R’
software version R.3.4.3, [38] ) under assumption of 96 points for the
involved random variables CDFs were conducted to compare the SSRT random
variables in usual stochastic order $(<_{st})$ across cluster types. Such test
at the individual is:

$\displaystyle H_{0}$ $\displaystyle:$ $\displaystyle
SSRT_{Single}(\overrightarrow{\theta_{S}})=_{st}SSRT_{Mixture}(\overrightarrow{\theta_{M}}),$
$\displaystyle H_{1}$ $\displaystyle:$ $\displaystyle
SSRT_{Single}(\overrightarrow{\theta_{S}})\neq_{st}SSRT_{Mixture}.(\overrightarrow{\theta_{M}})$
(15)

####  2.3.2 Comparisons Procedure: Random Variables

Given two sets of stop signal reaction times distributions
$\\{SSRT_{Single}(\theta_{S_{k}})\\}_{k=1}^{44}$ and
$\\{SSRT_{Mixture}(\theta_{M_{k}})\\}_{k=1}^{44}$, our problem of interest was
an overall comparison between these two groups of distributions in usual
stochastic order $<_{st}$ [39]. Our proposed problem was dealt with two steps
as follows:

Step(1): Two Stage Bayesian Parametric Approach(TSBPA444This proposed analysis
is neither completely hierarchical Bayesian analysis nor completely
conventional meta analysis. It has components of both methods. On one hand, it
has two separates one stage Bayesian analyses. On the other hand, it
calculates overall population level estimates in the second analysis with
consideration of non-zero correlations.)

Referring to equation (11) we define overall SSRT distributions per single S
cluster type and mixture M cluster type as the following:

$\displaystyle SSRT_{O.Single}(\overrightarrow{\theta_{S}})$ $\displaystyle=$
$\displaystyle SSRT_{Single}(\overline{\theta_{S}}),$ $\displaystyle
SSRT_{O.Mixture}(\overrightarrow{\theta_{M}})$ $\displaystyle=$ $\displaystyle
SSRT_{Mixture}(\overline{\theta_{M}}).$ (16)

where
$\overrightarrow{\theta_{S}}=\overline{\theta_{S}}=\overline{\theta_{T}}=(\overline{\mu_{T}},\overline{\sigma_{T}},\overline{\tau_{T}})$
and
$\overrightarrow{\theta_{M}}=\overline{\theta_{M}}=(\overline{W_{A}},\overline{\theta_{A}},\overline{\theta_{B}})$
with $\overline{W_{A}}=\sum_{k=1}^{44}W_{A_{k}}/44$ and
$\overline{\theta_{A}}=(\overline{\mu_{A}},\overline{\sigma_{A}},\overline{\tau_{A}}),\overline{\theta_{B}}=(\overline{\mu_{B}},\overline{\sigma_{B}},\overline{\tau_{B}})$
being computed by a Two Stage Bayesian Parametric Approach(TSBPA) method
described as follows.

In the TSBPA (See Figure 3), the data, the priors and the posterior
estimations are considered as below [40, 41, 42]. We conduct the first stage
with 3 chains, 5,000 burn in out of 20,000 simulations in BEESTS 2.0 software.
Then we consider the mean of posterior estimates
$\mu_{stop},\sigma_{stop},\tau_{stop}$ as their point estimates
$E(\mu_{stop}|x)\rightarrow\mu_{stop},E(\sigma_{stop}|x)\rightarrow\sigma_{stop},E(\tau_{stop}|x)\rightarrow\tau_{stop}$
in the second stage of meta-analysis. We conduct this stage with 3 chains with
5,000 burn in out of 100,000 simulations in WINBUGS1.4 software [43]. Finally,
we consider the mean of posterior estimates
$\mu_{\mu_{stop}},\mu_{\sigma_{stop}},\mu_{\tau_{stop}}$ in the second stage
as estimates of
$\overrightarrow{\theta_{S}}=\overline{\theta_{S}}=\overline{\theta_{T}}=(\overline{\mu_{T}},\overline{\sigma_{T}},\overline{\tau_{T}})$
respectively for the case of overall data S. We repeat this process for the
case of type A SST data and type B SST data for estimation of
$\overline{\theta_{A}}=(\overline{\mu_{A}},\overline{\sigma_{A}},\overline{\tau_{A}}),\overline{\theta_{B}}=(\overline{\mu_{B}},\overline{\sigma_{B}},\overline{\tau_{B}})$,
respectively.

[b]  Stage(1) Data Individual Priors $GORT\sim
ExG(\mu_{go},\sigma_{go},\tau_{go})$ $SRRT\sim
ExG(\mu_{go},\sigma_{go},\tau_{go},\mu_{stop},\sigma_{stop},\tau_{stop},SSD)I_{[1,1000]}^{+}$
$\mu_{go},\sigma_{go},\tau_{go}\sim U[10,2000]$ $SSRT\sim
ExG(\mu_{go},\sigma_{go},\tau_{go},\mu_{stop},\sigma_{stop},\tau_{stop},SSD)I_{[1,1000]}^{+}$
$\mu_{stop},\sigma_{stop},\tau_{stop}\sim U[10,2000]$

[b]  Stage(2) Data Priors $(\mu_{stop},\sigma_{stop},\tau_{stop})^{\prime}\sim
MVN(M_{3\times 1},\sum_{3\times 3})$
$\rho_{\mu,\sigma},\rho_{\mu,\tau},\rho_{\sigma,\tau}\sim U[-0.99,+0.99]$
$\mu_{stop}\sim N(\mu_{\mu_{stop}},\sigma_{\mu_{stop}}^{2})$
$\mu_{\mu_{stop}},\beta_{20},\beta_{30}\sim N(0,1000)$
$\sigma_{stop}|\mu_{stop}\sim
N(\mu_{\sigma_{stop}},\sigma_{\sigma_{stop}}^{2})$
$\sigma_{\mu_{stop}},\sigma_{\sigma_{stop}},\sigma_{\tau_{stop}}\sim
N(0,10)I_{[0,1000]}^{+}$ $\tau_{stop}|(\mu_{stop},\sigma_{stop})\sim
N(\mu_{\tau_{stop}},\sigma_{\tau_{stop}}^{2})$
$\mu_{\sigma_{stop}i}=\beta_{20}+\beta_{21}.\mu_{stop}i$
$\mu_{\tau_{stop}i}=\beta_{30}+\beta_{31}.\mu_{stop}i+\beta_{32}.\sigma_{stop}i$

Figure 3: Two-Stage Bayesian Parametric Approach (TSBPA) with ExGaussian
distributional assumption framework. Filled in shapes indicate known values.

Step(2):Paired Samples Parametric Distribution Test(PSPDT555This proposed test
can be considered as a distributional counterpart of the paired z-test in the
real numbers. )

Using overall estimates in Step (1) we then conduct the following paired
samples parametric distribution test hypothesis testing for $K=44$ at the 5%
significance level:

$\displaystyle H_{0}$ $\displaystyle:$ $\displaystyle
SSRT_{O.Single}(\overrightarrow{\theta_{S}})=_{st}SSRT_{O.Mixture}(\overrightarrow{\theta_{M}}),$
$\displaystyle H_{1}$ $\displaystyle:$ $\displaystyle
SSRT_{O.Single}(\overrightarrow{\theta_{S}})\neq_{st}SSRT_{O.Mixture}(\overrightarrow{\theta_{M}}).$
(17)

where the Two-Sample Kolmogorov-Smirnov Statistics $D_{(n,m_{k})}$ for the kth
$(1\leq k\leq K)$ comparisons of the simulated distributions in (17), the
following average two-samples KS statistics were considered as the test
statistics for the comparison of distributions in the test of (17):

$\overline{D_{n,m}}=\frac{1}{K}\sum_{k=1}^{K}D_{n,m_{k}}$ (18)

We reject the null hypothesis $H_{0}$ in favour of alternative hypothesis
$H_{1}$ at given $\alpha$-level (e.g. 0.05) whenever:

$\overline{D_{n,m}}>c(\alpha)\sqrt{\frac{1}{m}+\frac{1}{n}}:c(\alpha)=\sqrt{\frac{-1}{2}ln(\frac{1}{2})},\alpha=0.05,n=m=96.$
(19)

The Two-Sample K-S test analysis was conducted with R3.4.3 software as before.
The hypothesis testing in (17) were repeated for other comparisons between
cluster type SSRT indices including $SSRT_{A}$ vs $SSRT_{Single}$; $SSRT_{B}$
vs $SSRT_{Single}$;and $SSRT_{B}$ vs $SSRT_{A}$.

######  Remark 3.

The test (17) for the degenerate case of $K=1$ reduces to the usual two sample
K-S test at the individual level (15).

##  3 Results

The results are divided into three subsections. In subsection 3.1, we explored
the posterior mean Ex-Gaussian parameter estimations of cluster type-A,
cluster type-B, single and mixture SSRT distribution and compared them and the
shape statistics including skewness and kurtosis across cluster type indices.
Next, in subsection 3.2, we compared single SSRT and mixture SSRT
distributions in stochastic order at two levels: (i) the individual level;and,
(ii) the population level. For the individual level we applied IBPA, and for
the population level, we used TSBPA. Finally, in subsection 3.3, we compare
the comparisons results for the descriptive statistics and the entire SSRT
distribution in terms of the cluster weights ($W_{A}$).

###  3.1 Descriptive Analysis of Ex-Gaussian Parameters and Shape Statistics

This section includes two sets of descriptive results: First, the results for
cluster-type related parameters, mean and standard deviation of the Ex-
Gaussian SSRT; Second, the results for cluster-type related shape statistics
of skewness and kurtosis of the involved random variables.

Table 2. presents the descriptive results for the type-A, type-B, single, and
mixture fitted SSRT Ex-Gaussian random variable using IBPA666See Supplementary
Material (Table 1) for parameter estimates across the three cluster types.. As
it is shown, while there is no significant difference between mean posterior
of parameters $\theta=(\mu,\sigma,\tau)$ as well as mean and standard
deviation between cluster type SSRTs, the mentioned list of quantities of both
cluster type SSRTs are significantly larger than the one of single SSRT.
Hence, we conclude at this stage that the mean of mixture SSRT is
significantly larger than the one of single SSRT. This result is consistent
with the frequentist approach [14]. However, it is observed that the variance
has significantly increased and consequently the precision has significantly
decreased. We remind the reader that the mean and standard deviation of type-A
SSRT and type-B SSRT are significantly different than those of single SSRT.
Thus, the study SST data provides an example of the case of violated
assumption of equal impact of the preceding trial type (go/stop) on the
current stop trial SSRT . Otherwise, all these descriptive statistics would
have been equal across type-A, type-B and type-S single SST clusters.

Table 2: Descriptive results for parameters, mean and standard deviation of
fitted IBPA Ex-Gaussian distribution to SSRT given cluster type(n=44).

[b]  (a) Descriptive Results Parameter(mean(95$\%$CI))
Statistics(mean(95$\%$CI)) Cluster Type $\mu$ $\sigma$ $\tau$ Mean St.d Type S
93.1 116.2 103.6 196.8 157.8 (83.6,102.7) (104.1,128.2) (87.7,119.6)
(173.5,220.1) (139.4,176.2) Type A 124.1 164.5 140.9 265.0 217.7 (107.4,140.7)
(150.4,178.3) (126.9,155.0) (235.8,294.2) (199.1,236.2) Type B 123.9 168.0
129.7 253.6 213.2 107.2,140.6) (155.8,180.3) (115.3,143.9) (222.9,284.2)
(195.7,231.0) (b) Two Sample ttest Parameter(mean(95$\%$ CI))
Statistics(mean(95$\%$ CI)) Comparison $\mu$ $\sigma$ $\tau$ Mean St.d Type B
vs. Type A -0.2 3.6 -11.3 -11.4 -4.4 (-22.1,21.7) (-18.1,25.3) (32.6,10.1)
(-53.5,30.7) (-33.5,24.7) Type B vs. Type S 30.8*** 51.7*** 26.0*** 56.8***
55.5*** (17.9,43.6) (37.5,66.0) (10.9,41.1) (32.2,81.4) (35.4, 75.7) Type A
vs. Type S 30.9*** 48.2*** 37.3*** 68.2*** 59.9*** (18.5,43.3) (36.5,59.8)
(24.8,49.8) (48.3,88.1) (44.4,75.4) Type M vs. Type S - - - 63.7*** 71.4*** -
- - (57.2,70.1) (62.5,80.3)

  * •

Notes: $\overline{W_{A}}=0.59,$
∗p-value$<0.05$;∗∗p-value$<0.005$;∗∗∗p-value$<0.0005$.

Figure 4 shows the difference between skewness and kurtosis of fitted IBPA Ex-
Gaussian SSRT random variables by cluster type. As shown in Figure 4 (a) while
each of components of Mixture SSRT has smaller skewness versus the Single SSRT
upon combination into Mixture SSRT, the resultant Mixture SSRT has
significantly larger skewness compared to the Single SSRT. Similar results
hold for the case of kurtosis as shown in Figure 4 (b).

Figure 4:  Plot of statistics difference of fitted IBPA Ex-Gaussian SSRT
random variable by cluster type (n=44): (a) Skewness; (b) Kurtosis.

Given summary statistics and shape statistics comparison results between
single SSRT and mixture SSRT, one naturally considers comparison of their
associated distributions. In the next section, we deal with this topic.

###  3.2 Bayesian Mixture SSRT Estimation and Comparisons

This section deals with individual and overall level estimations of Single
SSRT and Mixture SSRT and their comparisons in usual stochastic order. Table 3
presents the results of two-sample KS hypothesis test at the individual level
given by (15) by direction and p-values for the sample of 44 subjects based on
IBPA. Similar hypothesis testing are conducted replacing $\neq$ with $<$ and
$>$ in an alternative test. With the exception of one participant( case 34),
the result show that the single SSRT is smaller than mixture SSRT in
stochastic order. This is consistent with the direction of constant index SSRT
results [14].

Table 3: Two-Sample Kolmogorov–Smirnov test results for the Single SSRT
distribution versus Mixture posterior SSRT distribution, (n = 44).

[b]  Alternative Hypothesis Unequal Greater Less $\\#$ Statistics p-value
Statistics p-value Statistics p-value 1 0.2708 0.0017 0.0417 0.8465 0.2708
0.0009 2 0.2604 0.0029 0.0208 0.9592 0.2604 0.0015 3 0.3333 0.0001 0.0729
0.6002 0.3333 0.0001 4 0.3438 0.0001 0.0417 0.8465 0.3438 0.0001 5 0.2396
0.0079 0.0312 0.9105 0.2396 0.0040 6 0.2812 0.0009 0.0417 0.8465 0.2812 0.0005
7 0.3021 0.0003 0.0729 0.6002 0.3021 0.0002 8 0.2188 0.0200 0.0312 0.9105
0.2188 0.0101 9 0.3646 0.0001 0.0521 0.7707 0.3646 0.0001 10 0.2396 0.0079
0.0312 0.9105 0.2396 0.0040 11 0.3229 0.0001 0.0417 0.8465 0.3229 0.0001 12
0.3229 0.0001 0.0000 1.0000 0.3229 0.0001 13 0.3021 0.0003 0.0208 0.9592
0.3021 0.0002 14 0.2396 0.0079 0.0104 0.9896 0.2396 0.0040 15 0.3229 0.0001
0.0625 0.6873 0.3229 0.0001 16 0.2500 0.0048 0.0312 0.9105 0.2500 0.0025 17
0.3229 0.0001 0.0417 0.8465 0.3229 0.0001 18 0.3542 0.0001 0.0000 1.0000
0.3542 0.0001 19 0.2604 0.0029 0.0417 0.8465 0.2604 0.0015 20 0.2604 0.0029
0.0312 0.9105 0.2604 0.0015 21 0.3854 0.0001 0.1042 0.3529 0.3854 0.0001 22
0.3438 0.0001 0.0417 0.8465 0.3438 0.0001 23 0.3646 0.0001 0.1146 0.2835
0.3646 0.0001 24 0.3021 0.0003 0.0104 0.9896 0.3021 0.0002 25 0.4896 0.0001
0.0833 0.5134 0.4896 0.0001 26 0.2604 0.0029 0.0104 0.9896 0.2604 0.0015 27
0.3958 0.0001 0.0729 0.6002 0.3958 0.0001 28 0.2708 0.0017 0.0208 0.9592
0.2708 0.0009 29 0.2396 0.0079 0.0104 0.9896 0.2396 0.0040 30 0.3750 0.0001
0.0729 0.6002 0.3750 0.0001 31 0.4062 0.0001 0.0312 0.9105 0.4062 0.0001 32
0.2500 0.0048 0.0208 0.9592 0.2500 0.0025 33 0.2500 0.0048 0.0208 0.9592
0.2500 0.0025 34 0.1562 0.1923 0.0000 1.0000 0.1562 0.0960* 35 0.2500 0.0048
0.0104 0.9896 0.2500 0.0025 36 0.3021 0.0003 0.0312 0.9105 0.3021 0.0002 37
0.1979 0.0463 0.0104 0.9896 0.1979 0.0233 38 0.3125 0.0002 0.0312 0.9105
0.3125 0.0001 39 0.2917 0.0005 0.0625 0.6873 0.2917 0.0003 40 0.3750 0.0001
0.0521 0.7707 0.3750 0.0001 41 0.2188 0.0200 0.0417 0.8465 0.2188 0.0101 42
0.3021 0.0003 0.0208 0.9592 0.3021 0.0002 43 0.2188 0.0200 0.0104 0.9896
0.2188 0.0101 44 0.4062 0.0001 0.0312 0.9105 0.4062 0.0001

  * •

Notes: IBPA: $\\#$Chains = 3; Simulations = 20,000; Burn-in = 5,000 (for both
single and mixture parameters); The sample size for K-S test for each
distribution was n=m=96; ${}^{*}:$ Exceptional case.

For an overall conclusion using a paired samples parametric distribution test,
we test the hypothesis (17) from TSBPA given uninformative priors. As before,
similar hypothesis testing are conducted with replacing $\neq$ with $<$ and
$>$ in the alternative test. The choice of TSBPA rather than HBPA was out of
consideration for pairwise non-zero correlations in the second stage of the
analysis. In this regard, one key missing characteristics in the HBPA is the
relaxing assumption of zero correlation of mean posterior parameters at the
individual level. This assumption is violated given cluster-S SSRT mean
posterior parameters Pearson correlations
$\rho_{\mu\sigma}=0.20,\rho_{\mu\tau}=0.64,\rho_{\sigma\tau}=0.66;$ cluster-A
SSRT mean posterior parameters Pearson correlations
$\rho_{\mu\sigma}=0.52,\rho_{\mu\tau}=0.81,\rho_{\sigma\tau}=0.74;$ and,
cluster-B SSRT mean posterior parameters Pearson correlations
$\rho_{\mu\sigma}=0.69,\rho_{\mu\tau}=0.95,\rho_{\sigma\tau}=0.80$. Table 4
presents the results of the paired samples parametric distribution test using
TSBPA:

Table 4: Two-Sample Kolmogorov–Smirnov test results for the Single SSRT
distribution versus the Mixture SSRT distribution, (n = 44).

[b]  Alternative Hypothesis Unequal Greater Less Comparison Statistics p-value
Statistics p-value Statistics p-value $SSRT_{Single}$ vs. $SSRT_{Mixture}$
0.2100 $<$0.0302 0.0736 $>$0.9999 0.2095 $<$ 0.0312 $SSRT_{Single}$ vs.
$SSRT_{B}$ 0.2017 $<$0.0422 0.0816 $>$0.9999 0.1984 $<$0.0562 $SSRT_{Single}$
vs. $SSRT_{A}$ 0.2273 $<$0.0152 0.0658 $>$ 0.9999 0.2256 $<$0.0152 $SSRT_{B}$
vs. $SSRT_{A}$ 0.1326 0.3702 0.1075 0.6502 0.0653 $>$ 0.9999

  * •

Note: The sample sizes for K-S test for both distributions were $n=m=96$.
$\overrightarrow{\theta_{S}}=(\overline{\mu_{T}},\overline{\sigma_{T}},\overline{\tau_{T}})=(78.4,93.9,73.1),$
$\overrightarrow{\theta_{A}}=(\overline{\mu_{A}},\overline{\sigma_{A}},\overline{\tau_{A}})=(94.0,134.5,104.8),$
$\overrightarrow{\theta_{B}}=(\overline{\mu_{B}},\overline{\sigma_{B}},\overline{\tau_{B}})=(90.9,142.3,99.0)$
and
$\overrightarrow{\theta_{M}}=(\overline{W_{A}},\overrightarrow{\theta_{A}},\overrightarrow{\theta_{B}}),$
with $\overline{W_{A}}=0.59.$

As we observe from Table 4, the results are conclusive. The single SSRT is
smaller than cluster type-A SSRT, cluster type-B SSRT; and, Mixture SSRT
(p-value $<$0.05). Also, given consideration of mean of SSRT distribution as
its point index estimation the result regarding comparison of single SSRT
versus Mixture SSRT is consistent with the direction of the frequentist
results as:

$SSRT_{Single}^{c}=E(SSRT_{Single})=151.5<195.2=E(SSRT_{Mixture})=SSRT_{Weighted}^{c}.$

Note that there is no significant difference between cluster type A and
cluster type B SSRT (p-value=0.370). We remind the reader that one may guess
that cluster type-A SSRT is smaller than the cluster type-B SSRT in stochastic
order given the test p-value 0.650 versus 0.999, but this claim’s verification
needs much larger sample size than n=44.

Figure 5. shows the plot of the overall density and cumulative distribution
function of cluster type SSRTs with overall TSBPA parameter estimates given in
Table 4. As it is observed in Figure 5(b), while there is no much distinction
between cumulative distributions of cluster type-A SSRT, cluster type-B SSRT
and Single SSRT, the cumulative distribution of single SSRT is clearly on the
left side of that of Mixture SSRT.

Figure 5:  The density and cumulative distribution function(CDF) of overall
sample cluster type SSRT, single SSRT and mixture SSRT with Ex-Gaussian
parametric distribution: (a) density; (b) cumulative distribution function.

In this and the previous section, we considered the cluster type weight
$(W_{A})$ in its fixed individual values. In the next section, we study its
role in the comparison results as a key variable on its own.

###  3.3 The Role of Cluster Type Weights in the Comparisons

This section deals with the comparison of the descriptive statistics of mean,
variance and the entire distribution of SSRT indices in terms of individual
optimal weights. The following proposition determines the optimal weights
values [15]:

######  Proposition 3.1.

The weight $W_{A}=0.75(W_{B}=0.25)$ is the optimal weight given independence
of assignment of stop or go process to the given trial in the tracking SST
data.

First, to study the impact of individual weights on the disparities of the
mean estimations across indices, we consider average differences of the new
index $SSRT_{Mixture}$ mean versus the established index $SSRT_{Single}$ mean
in terms of the individual weights $(W_{A})$. Considering $W_{A}$ as the main
variable it follows that:

$\displaystyle\overline{\Delta E(SSRT)}(W_{A})$ $\displaystyle=$
$\displaystyle\overline{E(SSRT_{Mixture})}-\overline{E(SSRT_{Single})}$ (20)
$\displaystyle=$
$\displaystyle\overline{E(W_{A}.SSRT_{A}+(1-W_{A}).SSRT_{B})}-\overline{E(SSRT_{Single})}$
$\displaystyle=$ $\displaystyle
W_{A}.\overline{E(SSRT_{A})}+(1-W_{A}).\overline{E(SSRT_{B})}-\overline{E(SSRT_{Single})}$
$\displaystyle=$
$\displaystyle(\overline{E(SSRT_{A})}-\overline{E(SSRT_{B})}).W_{A}+(\overline{E(SSRT_{B})}-\overline{E(SSRT_{Single})}$
$\displaystyle=$
$\displaystyle((\overline{\mu_{A}}+\overline{\tau_{A}})-(\overline{\mu_{B}}+\overline{\tau_{B}})).W_{A}+((\overline{\mu_{B}}+\overline{\tau_{B}})-(\overline{\mu_{S}}+\overline{\tau_{S}}))$
$\displaystyle=$ $\displaystyle 11.4\ \ W_{A}+56.8,\ \ \hskip 85.35826pt(0\leq
W_{A}\leq 1).$

Figure 6(a) presents the average disparities of mean mixture SSRT and mean
single SSRT versus individual weights $W_{A}$ for the extrapolated range of
$[0,1].$ As shown, the average difference between two index means is linear in
terms of the individual weight $W_{A}$ with the minimum value of 56.8 ms (for
minimum sample weight of 0.00) and maximum value of 68.2 ms (for maximum
sample weight of 1.00). Also, their corresponding averaged disparities equals
to 63.5 ms (at the sample overall weight of 0.59 ). Finally, the disparities
of two index means maximizes to 65.3 ms using the optimal weight of 0.75.

Second, to examine the impact of individual weights on the disparities of the
variance estimations across indices, similar to the case in section 3.2 we
consider average differences of the new index $SSRT_{Mixture}$ variance versus
the established index $SSRT_{Single}$ variance in terms of the individual
weights $(W_{A}).$ Considering $W_{A}$ as the main variable it follows that:

$\displaystyle\overline{\Delta Var(SSRT)}(W_{A})$ $\displaystyle=$
$\displaystyle\overline{Var(SSRT_{Mixture})}-\overline{Var(SSRT_{Single})}$
$\displaystyle=$
$\displaystyle\overline{Var(W_{A}.SSRT_{A}+(1-W_{A}).SSRT_{B})}-\overline{Var(SSRT_{Single})}$
$\displaystyle=$
$\displaystyle-\overline{(E(SSRT_{A})-E(SSRT_{B}))^{2}}.W_{A}^{2}$
$\displaystyle+\bigg{(}\overline{(E(SSRT_{A})-E(SSRT_{B}))^{2}}$
$\displaystyle+\overline{(Var(SSRT_{A})-Var(SSRT_{B}))}\bigg{)}.W_{A}$
$\displaystyle+\overline{Var(SSRT_{B})-Var(SSRT_{Single})}$ $\displaystyle=$
$\displaystyle-\overline{(\mu_{A}+\tau_{A}-\mu_{B}-\tau_{B})^{2}}.W_{A}^{2}$
$\displaystyle+(\overline{(\mu_{A}+\tau_{A}-\mu_{B}-\tau_{B})^{2}}+\overline{(\sigma_{A}^{2}+\tau_{A}^{2}-\sigma_{B}^{2}-\tau_{B}^{2})}).W_{A}$
$\displaystyle\overline{\sigma_{B}^{2}+\tau_{B}^{2}-\sigma_{S}^{2}-\tau_{S}^{2}}$
$\displaystyle=$ $\displaystyle-18885.9\ \ W_{A}^{2}+21106.8\ \
W_{A}+20330.3,\hskip 28.45274pt(0\leq W_{A}\leq 1).$

In particular, the average variance differences attains its maximum value at:

$W_{A}=\frac{\overline{(\mu_{A}+\tau_{A}-\mu_{B}-\tau_{B})^{2}}+\overline{(\sigma_{A}^{2}+\tau_{A}^{2}-\sigma_{B}^{2}-\tau_{B}^{2})}}{2\overline{(\mu_{A}+\tau_{A}-\mu_{B}-\tau_{B})^{2}}}\approx
0.56.$

Figure 6 (b) presents the average disparities of mixture SSRT variance and
single SSRT variance versus individual weights $W_{A}$ for the extrapolated
range of $[0,1].$ As shown, the average difference between two indices
variances follows a quadratic increasing- decreasing pattern in terms of the
weights $W_{A}$ with maximum values attained closed to $W_{A}\approx 0.56.$
Next, the disparities for the optimal weight $W_{A}=0.75$ is smaller than that
of population weight $W_{A}=0.59.$ However, across all weights spectrum the
average SSRT variance differences are positive showing that the new mixture
SSRT index has higher variance than the current single SSRT index ; and,
consequently its precision is smaller.

Figure 6: Plot of difference of SSRT index statistics ($SSRT_{Mixture}$ vs
$SSRT_{Single}$) by weight $W_{A}(n=44)$: (a) Means; (b) Variances

Finally, to explore the impact of cluster type weights $(W_{A})$ on the
overall SSRT distributions comparison results for the hypothesis testing (17)
we considered the averaged two-sample KS test statistics S as a function S of
weights $S=S(W_{A})$ and calculated the corresponding p-values. Figure 7
presents the results in terms of the weights. As shown in figure 7(b), for
almost all ranges of the weights $W_{A}$, the single SSRT is significantly
smaller than mixture SSRT in stochastic order. Next, the disparity is the
weakest when $W_{A}$=0 with corresponding p-value=0.0562 and it is the
strongest when $W_{A}$=1 with corresponding p-value=0.0152. Finally, the
disparity at the optimal weight $W_{A}$=0.75 is stronger than that of
population weight $W_{A}$=0.59 with corresponding p-values of 0.0262 and
0.0312, respectively.

Figure 7: Plot of overall test results of single SSRT versus mixture SSRT :
(a) Two sample KS test statistics; (b) cut off point of the test p-value

##  4 Discussion

###  4.1 Present Work

This study presented a mixture Bayesian parametric approach for a more
illuminating estimation of SSRT distribution by considering two subtype SST
cluster information suggesting a new estimation of the SSRT distribution.
Also, it introduced two novel statistical methodologies accompanied by their
empirical applications: TSBPA and PSPDT. It was hypothesized that considering
cluster type information in the calculations of the new mixture SSRT
distribution would impact the estimation of SSRT distribution. This yields to
a distributional counterpart to the case of constant index SSRT [14, 15].

The results confirmed the hypothesis through three observations: (i) the
descriptive and shape statistics, (ii) the distributional comparisons at the
individual level and the population level, and (iii) with validity of the
results in the first two observations across the entire spectrum of the
weights. Similar to the constant index SSRT [14], in the majority of cases the
mixture SSRT is different from the single SSRT in shape statistics and the
stochastic order. However, in two special distinct cases, they are the same:
(i) type A cluster SST is empty $(W_{A}=0)$ and (ii) type B cluster SST is
empty $(W_{A}=1).$

This study confirmed that SSRT depends on non-horse race related factors in
each round of SST experimental trial such as memory aftereffects and
proportion of cluster type stop trials. It has shed light into preparation
aspect of performance of choice stop signal reaction times by treating the
aftereffects of the previous trial type as memory in the two-state mixture
model [31]. In addition, given that skewness of the RT distributions increases
with memory involvement (versus a perceptual decision)[44], the increase in
reported skewness in the mixture SSRT versus single SSRT confirms that the
proposed mixture model successfully captures the memory involvement in the
decision process[44]. Also, as in the context of the horse race independent
model an increase in kurtosis of SSRT is proportional to more extreme values
of the right tail of SSRT distribution and hence higher probability of failed
inhibition in the stop trials(and vice versa) , the increase in reported
kurtosis in the mixture SSRT versus single SSRT gives evidence that the
proposed mixture model optimally uses the information given by pre pushed
failed inhibitions in the stop trials in the estimation of SSRT distribution.

This study’s findings for the SSRT distribution were consistent with the
constant index SSRT when considering the impact of sub-cluster types in the
estimations [14, 15]. In details, there were consistent results between the
usual comparison of the single SSRT and weighted SSRT (as constant indices)
and the stochastic comparison of the single SSRT and mixture SSRT (as non-
constant random variables). Indeed, we found that:

$\displaystyle SSRT_{Single}^{c}$ $\displaystyle<$ $\displaystyle
SSRT_{Weighted}^{c},$ (22) $\displaystyle SSRT_{Single}$ $\displaystyle<_{st}$
$\displaystyle SSRT_{Mixture}.$ (23)

On one hand, if we look at two sides of the equation(22) as degenerate random
variables then we are led to the equation (23). On the other hand, if we take
expectation from both sides of the equation (23), then we are led to the
equation (22).

The study’s novel statistical methodological contributions involved the Two-
Stage Bayesian Parametric Approach (TSBPA), and the Paired Samples Parametric
Distribution Test(PSPDT). TSBPA’s advantage was that it considers the
underlying non-zero correlation between estimated mean posterior parameters at
the first stage in the final calculations in the second stage. This feature is
neglected in HBPA. PSPDT offered a novel method to compare a paired sets of
parametric random variables using the two-samples KS test. An application of
both proposed methods was provided in this study.

There are limitations in the current study. First, the sample size was
relatively small $(n=44).$ To show more precise comparisons, a larger sample
size needed. Second, the TSBPA assumes a multivariate normal distribution form
for the mean posterior parameters at the second stage, which may not hold.
Third, in TSBPA, when comparing overall Mixture SSRT and overall Single SSRT
there is no specific restriction on the simulation sample sizes in equation
(18). Here, while we set the sample sizes to $n=m=96$ (the SST data trials
size), there could be other choices. Finally, given the structure of the
equations for the shape statistics (skewness and kurtosis) in terms of the
cluster weights $(W_{A}),$ unlike the descriptive statistics in section 3.3,
there was no simple closed form for the averaged differences of the new index
$SSRT_{Mixture}$ skewness (or kurtosis) versus the established index
$SSRT_{Single}$ skewness (or kurtosis) in terms of the individual weights
$(W_{A}).$ Similar to the descriptive statistics, existence of such a simple
closed formula would shed more light on the average disparities of skewness
and kurtosis of the two indices across a spectrum of the individual weights.

###  4.2 Future Work

The proposed approach in modelling the SSRT distribution in this study should
be replicated in the future research in several different directions. This may
shed light on further unknown corners. New work includes (i)considering of the
larger number of SST trials,(ii) examining the order of trials,(iii)expanding
these methods to other clinical populations,(iv) considering trigger failures
in the modelling,(v) interpreting the shape statistics, and (vi) estimating
signal respond reaction times (SRRT).

First, research has recommended that reliable estimates of SSRT for adults
requires 200 SST trials with 50 stops [16]. Hence, the current work’s approach
needs to be replicated for SST data with 400 trials including sub-cluster
types of 200 trials with 50 stops for confirmation and generalization
purposes.

Second, supplementary research to this study must address the presumption of
equal impact in the order of trials for the same cluster type weights $W_{A}.$
For example, for the case of $W_{A}=1,$ one may consider two schemes within
the study of 96 SST trials: In the first scheme, trials numbered $2k(1\leq
k\leq 25)$ are stop trials. In the second scheme, trials numbered $98-2k\
(1\leq k\leq 25)$ are the stop trials. Currently, there is no known study that
investigates if in the same participant, these schemes lead to the same
$SSRT_{Mixture}$ or not.

Third, the work subsequent to this study should apply the proposed
$SSRT_{Mixture}$ to study the inhibitory deficiency in different clinical
groups such as ADHD, OCD, autism and schizophrenia in terms of descriptive
statistics, shape statistics plus the differential disparities across these
clinical groups.

Fourth, there are trigger failures that impact the estimations [27]. Given the
probability of trigger failures(TF) of $P_{T}(TF);$ $P_{A}(TF)$ and
$P_{B}(TF)$ for the overall SST data, cluster-A SST data and cluster-B SST
data respectively, there remains an open question on their relationships and
the impact of the cluster type trigger failures in the estimations of the
$SSRT_{Mixture}$ and on the above results. The results of such consideration
generalizes this study’s findings in terms of trigger failures and control
them in order to eliminate a potential confounding variable, trigger failure
status.

Fifth, this study merely reported and compared the shape statistics for
skewness and kurtosis across the cluster type SSRTs, single SSRT and the
mixture SSRT distributions. There is a need to investigate into psychiatric
and psychopathological interpretations of these shape statistics given the Ex-
Gaussian parametric distribution assumption.

Finally, this and the earlier study [14] addressed the estimation of stop
signal reaction times (SSRT) in the case of the violated assumption of similar
aftereffects of the prior trial type. It is plausible to conduct a counterpart
investigation on to the estimation of the signal respond reaction times (SRRT)
constant index and distribution.

###  4.3 Conclusion

There has been a great deal of interest in the aftereffects of inhibition on
the estimation of SSRT in the SST literature from early 1990s. This study
addressed the problem in part and presented a two-sate mixture model of SSRT
distribution by considering the prior trial type with results consistent with
results of the constant SSRT index in the literature [14]. Given the
consistency of results across constant index and non-constant random variable
contexts, the researchers are recommended considering mixture SSRT
distribution $(SSRT_{Mixture})$ as the most informative estimation of the
latency of stopping.

## Disclosure Statement

This work has previously been presented as a research poster in Statistical
Society of Canada 2018 annual meeting in Montreal, Canada. Mohsen Soltanifar,
Michael Escobar and Annie Dupuis have no financial interests to disclose.
Russell Schachar has equity in ehave and has been the on the Scientific
Advisory Board(SAB) in Lilly and Highland Therapautic Inc, Toronto, Canada.

## Funding

This work has been funded by University of Toronto’s Queen Elizabeth II (QEII)
graduate scholarship in science and technology 2017-2018 and, by Canadian
Institute of Health Research (CIHR) operating grant (PIs: Russell Schachar:
MOP-93696; Paul Arnold: MOP-106573).

## Note on Contributors

Mohsen Soltanifar (Ph.D) is a sessional course instructor jointly working at
the University of Toronto and the Hospital of Sick Children, Toronto. His
research area includes quantitative psychology with statistical focus on
multilevel, frequentist and bayesian mixture modelling and time series.

Michael Escobar (Ph.D) is a professor of biostatistics at the University of
Toronto. His research interests include nonparametric bayesian modelling,
mixture modelling and applied statistical methods in psychiatric research.

Annie Dupuis (Ph.D) is an adjunct professor of biostatistics at the University
of Toronto and an independent biostatistical consultant in Toronto, Canada .
Her research areas include attention deficit hyperactivity disorder, autism
spectrum disorders, and cystic fibrosis.

Russell Schachar (M.D) is a professor of psychiatry at the University of
Toronto, and a senior scientist leading a cognitive neurosciences laboratory
in the Research Institute at the Hospital for Sick Children, Toronto. He is
the TD Bank chair in Child and Adolescent psychiatry. His research focuses on
attention deficit hyperactivity disorder.

## ORCID

Mohsen Soltanifar: https://orcid.org/0000-0002-5989-0082
Michael Escobar: https://orcid.org/0000-0001-9055-4709
Annie Dupuis: https://orcid.org/0000-0002-8704-078X
Russell Schachar: https://orcid.org/0000-0002-2015-4395

## Abbreviations

$\begin{array}[]{ll}\text{ADHD}&\text{Attention Deficit Hyperactivity
Disorder}\\\ \text{BEESTS}&\text{Bayesian Ex-Gaussian Estimation of Stop
Signal RT distributions}\\\ \text{BPA}&\text{Bayesian Parametric Approach}\\\
\text{ExG}&\text{Ex-Gaussian distribution}\\\ \text{GORT}&\text{Reaction Time
in a go trial}\\\ \text{GORTA}&\text{Reaction Time in a type A go trial}\\\
\text{GORTB}&\text{Reaction Time in a type B go trial}\\\
\text{HBPA}&\text{Hierarchical Bayesian Parametric Approach}\\\
\text{IBPA}&\text{Individual Bayesian Parametric Approach}\\\
\text{MCMC}&\text{Marco Chain Monte Carlo}\\\ \text{OCD}&\text{Obsessive
Compulsive Disorder}\\\ \text{PSPDT}&\text{Paired Samples Parametric
Distribution Test}\\\ \text{SSD}&\text{Stop Signal Delay}\\\
\text{SRRT}&\text{Reaction Time in a failed stop trial}\\\
\text{SRRTA}&\text{Reaction Time in a failed type A stop trial}\\\
\text{SRRTB}&\text{Reaction Time in a failed type B stop trial}\\\
\text{SSRT}&\text{Stop Signal Reaction Times in a stop trial}\\\
\text{SSRTA}&\text{Stop Signal Reaction Times in a type A stop trial}\\\
\text{SSRTB}&\text{Stop Signal Reaction Times in a type B stop trial}\\\
\text{SST}&\text{Stop Signal Task }\\\ \text{TF}&\text{Trigger Failure}\\\
\text{TSBPA}&\text{Two Stage Bayesian Parametric Approach}\end{array}$

## Supplementary Materials

Additional supporting information may be found in the online version of this
article at the publisher’s website on the data of Mean posterior Ex-Gaussian
parameters estimations across trial types by IBPA (n = 44).

## References

  * [1] Johnston S, Dimoske A, Smith J, Bang R.J, and et al. The Development of Stop Signal and Go/No-go Response Inhibition in children Aged 7-12 Years: Performance and Event Related Potential Indices. International Journal of Psychology, 2007; 63: 25-38.
  * [2] Schachar R.J, Logan G.D, Robaey P, Chen S, and et al. Restraint and Cancellation: Multiple Inhibition Deficits in Attention Deficit Hyperactivity Disorder. Journal of Abnormal Child Psychology, 2007; 35: 229-238.
  * [3] Matzke D, Verbrugen F, and Logan G.D. The Stop Signal Paradigm. Steven Handbook of Experimental Psychology and Cognitive Neuroscience(4th ed), Vol 5 Methodology. John Wiley $\&$ Sons, Inc, 2018.
  * [4] Logan G.D. On the Ability to Inhibit Thought and Action: A User’s Guide to the Stop Signal Paradigm. In D. Dagenbach, $\&$ T.H. Carr (Eds), Inhibitory Process in Attention, Memory and Language. San Diego : Academic Press, 1994.
  * [5] Trommer B.L, Hoeppner J.B, Larber R, and Armstrong K.J. The Go-Nogo Paradigm in Attention Deficit Disorder. Annals of Neurology, 1988; 24: 610-614.
  * [6] Schachar R.J, Tannock R, and Logan G.D. Inhibitory Control, Impulsiveness and Attention Deficit Hyperactivity Disorder. Clinical Psychology Review, 1993; 13: 721-739.
  * [7] Schachar R.J, Tannock R, and Logan G.D. Deficient Inhibitory Control in Attention Deficit Hyperactivity Disorder. Journal of Abnormal Child Psychology, 1995; 23: 411-437.
  * [8] Logan G.D. On the Ability to Inhibit Complex Movements: A Stop Signal Study of Type Writing. Journal of Experimental Psychology: Human Perceptions and Performance, 1982; 8(6): 778-792.
  * [9] Ollman R.T, and Billington M.J. The Deadline Model for Simple Reaction Times. Cognitive Psychology, 1972; 3: 311-336.
  * [10] Logan G.D. Attention, Automaticity, and the Ability to Stop a Speeded Choice Response. In J. Long $\&$ A. D. Baddeley (Eds). Attention and Performance IX. Hillsdale, NJ, Erlbaum, 1981.
  * [11] Boucher L, Palmeri T.J, Logan G.D, and Schall J.D. Inhibitory Control in Mind and Brain: An Interactive Race Model of Countermanding Saccades. Psychology Review, 2007; 114(2): 376-397.
  * [12] Hanes D.P, and Carpenter R H.S. Countermanding Saccades in Humans. Vision Research, 1999; 39: 2777-2791.
  * [13] Logan G.D, and Cowan W.B. On the Ability to Inhibit Thought and Action: A Theory of an Act of Control. Psychological Review, 1984; 91: 295-327.
  * [14] Soltanifar M, Dupuis A, Schachar R, and Escobar M. A frequentist mixture modelling of stop signal reaction times. Biostatistics $\&$ Epidemiology, 2019; 3(1), 90-108.
  * [15] Soltanifar M, Knight K, Dupuis A, Schachar R, and Escobar M. A Time Series-based Point Estimation of Stop Signal Reaction Times. In JSM Proceedings, SSC Section, Alexandria, VA: American Statistical Association, 2019: 1515-1528.
  * [16] Verbruggen F, Aron A.R, Band G, Beste C, and et al. Capturing the Ability to Inhibit Actions and Impulsive Behaviors: A Consensus Guide to the Stop Signal Task. Elife, 2019; 9.
  * [17] Steensen C.L, Elbaz Z.K, and Douglas V.I. Mean Response Times, Variability , and Skew in the Responding of ADHD Children: A Response Time Distributional Approach. Acta Psychologica, 2000; 104: 167-190.
  * [18] Belin T.R, and Rubin D.B. The Analysis of Repeated Measures Data on Schizophrenic Reaction Times Using Mixture Models. Statistics in Medicine, 1995; 14: 747-768.
  * [19] Colonius H. A Note on the Stop Signal Paradigm, or How to Observe the Unobservable. Psychological Review, 1990; 97: 309-312.
  * [20] Matzke D, Dolan C.V, Logan G.D, Brown S.D, and et al. Bayesian Parametric Estimation of Stop signal Reaction Time Distributions. Journal of Experimental Psychology: General, 2013; 142(4):1047-1073.
  * [21] Matzke D, Love J, Wiecki T.V, Brown S.D, and et al. Release the BEESTS: Bayesian Estimation of Ex-Gaussian Stop Signal Reaction Time Distributions. Frontiers in Psychology, 2013; 4: Article 918.
  * [22] Band G.P.H, van dr Molen M.W, and Logan G.D. Horse Race Model Simulations of the Stop Signal Procedure. Acta Psychologica, 2003; 112: 105-142.
  * [23] Gelman A, and Hill J. Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge, England: Cambridge University Press, 2007.
  * [24] Rouder J.N, Lu J, Speckman P, Sun D.C, and et al. A Hierarchical Model for Estimating Response Time Distributions, Psychologic Bulletin $\&$ Review, 2005; 12: 195-223.
  * [25] Chevalier N, Chatham C.H, and Munakata Y. The Practice of Going Helps Children to Stop: The Importance of Context Monitoring in Inhibitory Control. Journal of Experimental Psychology: General, 2014; 143: 959-965.
  * [26] Colzato L.S, Jonkess B.J, Sellaro R, van der Wildenburg W.P.H, and et al. Eating to Stop: Tyrosine Supplementation Enhances Inhibiting Control but not Response Execution. Neuropsychologica, 2014; 62: 398-402.
  * [27] Matzke D, Love J, and Heatcote A. A Bayesian Approach for Estimating the Probability of Trigger Failures in the Stop Signal Paradigm. Behavior Research, 2017; 49: 267-281.
  * [28] Dupuis A, Indralingam M, Chevrier A, Crosbie J, and et al. Adjustment in the Stop Signal Task: development in children and adolescents, 2018. doi: 10.111/cdev.13062.
  * [29] Kramer A.F, Humphrey D.G, Lanish J, Logan G.D, and Strayer D.L. Aging and Inhibition. Conference on Cognition and Aging, Atlanta, GA, 1992.
  * [30] Verbruggen F, and Logan G.D. Models of Response Inhibition in the Stop Signal and Stop Change Paradigms. Neuroscience and Biobehavioral Reviews, 2009; 38: 647-661.
  * [31] Luce R.D. Response Times: Their Role in Inferring Elementary Mental Organization. Oxford University Press, 1986; 274-275.
  * [32] Crosbie J, Arnold P.D, Peterson A, Swanson J, and et al. Response Inhibition and ADHD Traits: Correlates and Heritability in a Community Sample. Journal of Abnormal Child Psychology, 2013; 41: 497-597.
  * [33] Heatcote A, RTSYS: A DOS Application for the Analysis of Reaction Times Data. Behavior Research Methods, Instruments and Computers, 1996; 28(3), 427-445.
  * [34] Schwarz W. The Ex-Wald Distribution as a Descriptive Model of Response Times, Behavioural Research Methods, Instruments and Computers, 2001; 33(4): 457-469.
  * [35] Prado Martin M.d, and Fermin Dr. A Theory of Reaction Times Distributions, 2008 (unpublished). Retrieved on April 29, 2017(http://cogprints.org/6310).
  * [36] Rouder J.N. Are Un-shifted Distributional Models Appropriate for Response Time ? Psychometrica, 2005; 70: 377-381.
  * [37] SAS/STAT Software version 9.4. SAS/STAT Software version 9.4 of the SAS System for Windows. Cary, NC: SAS Institute, 2012.
  * [38] R. Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2018.
  * [39] Shaked M, and Shantikumar J.G. Stochastic Orders and their Applications. Academic Press, 1994; 323-348.
  * [40] Bajkiewicz S, Thompson J.R, Sutton A.J, Cooper N.J, and et al. Multivariate Meta Analysis of Mixed Outcomes : a Bayesian Approach . Statistics in Medicine, 2013; 32: 3962-3943.
  * [41] Sylwia Bajkiewicz, John R. Thompson, Alex J. Sutton, Nicola J. Cooper, and et al. Use of Bayesian Meta Analysis to Estimate the HAQ for Mapping Onto the EQ-5D Questionnaire in Rheumatoid Arthritis. Statistics in Medicine, 17: 109-115, 2014.
  * [42] Bajkiewicz S, Thompson J.R, Riley R.D, and Abrams K.R. Bayesian Meta Analytical Methods to Incorporate Multiple Surrogate Endpoints in Drug Development Process. Statistics in Medicine, 2016; 35:1063-1089.
  * [43] Lunn D.J, Thomas A, Best N, and Spiegelhalter D. WinBUGS- a Bayesian Modelling Framework: Concepts, Structure, and Extensibility. Statistics and Computing, 2000; 10: 325-337.
  * [44] Hockley W.E. Analysis of Response time Distributions in the Study of Cognitive Processes. Journal of Experimental Psychology: Learning, Memory and Cognition, 1984; 10(4): 598-615.