Title: Bayesian Multilevel Single Case Models Using 'Stan'. a New Tool to Study Single Cases in Neuropsychology

Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 Title: Bayesian Multilevel Single Case Models using 'Stan'. A new tool to study single cases in Neuropsychology

Authorship: Michele Scandola1,4,*; Daniele Romano 2,3,4

Affiliations: 1 University of Verona, Human Sciences Department 2 University of Milano-Bicocca, Psychology Department 3 NeuroMi, Milan Center for Neuroscience 4 BASIC-NPSY Research Group

* Corresponding author Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 Abstract (for broad readership) Research in Neuropsychology is based on single cases. However, the range of statistical tools specifically designed for single cases is limited.

The current gold standard is the Crawford's t-test which has proved to be reliable, but is restricted to simple experimental designs (i.e. two variables with up to two levels each) and it is limited in terms of making inferences with regard to support for alternative hypotheses.

Bayesian Multilevel Single Case models (BMSC) overcome these limitations and allow for inferences relating to both null and alternative hypotheses in complex experimental designs which use the Bayesian framework.

As part of the study, we first validated the method by comparing the BMSC and Crawford’s t-test in a simulation that enabled us to verify the reliability of the instruments precisely. The BMSC proved to be ten times more reliable than the Crawford’s test. We then showed how BMSC is useful in complex designs by means of an example using real data.

Notably, the BMSC provides individual results not only for the control group and for single cases but also for the differences between the control group and the single cases. It therefore provides a comprehensive vision of the whole experimental design, without the need for it to fit multiple models. The real data that we used for the purposes of testing demonstrated showed that with a few lines of code, it is possible to analyse a data set that includes a single case and a control group in depth. The BMSC follows the recent trend which involves a shift inattention from p-values to other inferential indices and estimates. Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 Abstract (for peers) Research in Neuropsychology is based on single cases. However, the range of statistical tools specifically designed for single cases is still limited.

The current gold standard is the Crawford's t-test, but it is crucial to note that this is limited to simple designs and it is not possible to make inferences relating to support for the null hypothesis using this method.

The Bayesian Multilevel Single Case models (BMSC) provide a novel tool that grants one the flexibility of linear mixed model designs. BMSC is also able to support both null and alternative hypotheses in complex experimental designs using the Bayesian framework.

We compared the BMSC and Crawford’s t-test in a simulation study involving a case of no-dissociation and a case of simple dissociation between a single case patient and a series of control groups of different sizes

(N=5, 15, or 30). We then showed how BMSC is useful in complex designs by means of an example using real data.

The BMSC proved to be ten times more reliable than the Crawford’s test. Notably, the BMSC model provides a comprehensive vision of the whole experimental design, interpolating a single model. It follows the recent trend which involves a shift in attention from p-values to other inferential indices and estimates. Peer-reviewed, published version:

The aim of a neuropsychologist is to understand anatomo-clinical relations and since Broca's pioneering investigations (Broca, 1861) a great deal of research has been devoted toto how the brain works based on the behaviour of patients with a brain lesion (Caramazza, 1986).

Neuropsychology has historically invariably been based on the study of single cases (Broca, 1861; Finger,

1994; SCOVILLE & MILNER, 1957), a method that still contributes a great deal to the discipline (Thiebaut de Schotten et al., 2015). The first neuropsychological studies were based on detailed descriptions of the condition of the patient in vivo, and the post-mortem analysis of patient’s brain. Paul Broca’s classic work on the patient known as Tan (so called because the only syllables he could utter were “tan tan”) is typical of this procedure (Broca, 1861). Subsequently, there were a number of single case studies, or case reports, which indicated the importance and potential of this type of research, for example the study of the famous patient

HM by Corkin (Corkin, 2013) which revealed certain mechanisms relating to memory, the case of SM

(Feinstein, Adolphs, & Damasio, 2011) on mechanisms relating to fear, the investigation of agnosia with patient DF (Goodale et al., 1994) and the research into neglect carried out by Bisiach & Luzzatti (1978), just to cite a few of the most significant studies which still influence current literature on the subject.

Nowadays there is a longstanding debate in the field of Neuropsychology on the merits and usefulness of single case studies in current research, with scientists frequently taking one of the two opposite positions, either favouring the study of single cases or the study of groups of patients with similar conditions

(Caramazza & Coltheart, 2006). In single case studies, scientists tend to study the peculiarities that distinguish patients with certain conditions in depth, in this way describing in detail every aspect of the effects of the lesion. This approach is used in particular when the patient presents an extremely rare – or even unique – clinical condition and collecting data from a group of patients with the same condition is therefore not feasible. The importance of single cases in contemporary Neuropsychology is confirmed by the fact that now there are specialist journals which specifically accept single case studies (e.g. Neuropsychologia and the

Journal of Neuropsychology). These have a section dedicated to single cases (e.g. Cortex) or are even Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 entirely dedicated to single case studies (e.g. Neurocase). Notably, the number of publications per year of studies of single cases is still increasing (see Figure 1).

Figure 1. The number of articles in the 1969 – 2019 period found in Pubmed by using the Search query:

"single case" OR "case study" OR "case report".

A single case study is usually described in detail, accurately reporting all the peculiarities of a patient’s condition, reporting both what is normal and what is abnormal, from a quantitative and a qualitative point of view.

In addition to anatomo-clinical descriptions of single case patients, comparisons have also been with control samples of healthy people, or other typologies of patients that have in common certain aspects relating to the single case patient. This has been done in order to determine whether the behaviour of the patient is different from the norm or from another clinical population. This method is currently the most frequently used and it is based on two main statistical approaches which make it possible to compare the performance of the patient under investigation and the controls. Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 According to the first approach, a patient might be administered a test (or a series of tests) involving normative data. The aim of these is to provide a standard measure of cognitive functions. For example, the token test (De Renzi & Faglioni, 1978) measures the ability of the subject to comprehend verbal language by means of giving the person short standardised commands that they are then required to execute. Another is the digit span test (Orsini et al., 1987) which measures the short-term memory of a person by asking them to remember fixed series of numbers of increasing length. Psychometric validations were carried out on these tests in order to assess the distribution of the performance of healthy people with a view to subsequently identifying an abnormal performance. Several, slightly different methods are used to identify a cut-off score which separates an abnormal performance from a score associated with a healthy person (usually the 95th centile). The most common method involves converting the single case performance into a z-score according to the mean and variance of the population, and a rare event is defined as abnormal (the cut-off is typically set at 5% for bidirectional tests, corresponding to a Z-score of ±1.96).

While this method is rock solid, it can only be used for those tests that have been validated on a large control sample. However, validation studies are long and worth the effort only if the task is also of interest for clinical purposes. Additionally, a definition of a cut-off which is based only on the healthy population may have drawbacks. While the cut-off is certainly able to distinguish 95% of the healthy population (as its specificity is known), we do not know the percentage at which an abnormal performance is detectable, that is, we do not know its sensitivitye since the patients’ data are not taken into account to determine the cut-off.

A better cut-off, maximising both specificity and sensitivity, is given by the item-response theory by means of receiver-operating characteristics analisys (Macmillan, 2002). However, this approach is not always feasible since it requires large samples both for the control group and the patients group.

In 1998, John Crawford proposed a second method which derived from work done by Rohlf and Sokal

(Rohlf & Sokal, 1995) and this is the current gold standard. Crawford is credited with being the first to propose statistical tests which were able to compare single cases to control groups quantitatively, while maintaining high qualitative standards in both first-type and second-type errors (Crawford, Howell, &

Garthwaite, 1998; John R. Crawford & Garthwaite, 2005; John R Crawford et al., 2010). In the Crawford’s t- Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 test, instead of comparing the performance of a large population with a normal distribution, a comparison is made with a relatively small control group that has done exactly the same task as the single case patient. A

Student-t distribution is used to compare the patient’s performance with the control group, treating the single case as a group of N=1. Crawford’s team then developed further statistical tests to be applied to different experimental designs, but these have the same basis as the first test.

The method developed by Crawford has several advantages: i) it can be used for non-standard testing; ii) the updated version makes it possible for up to two conditions (or tasks) within the same design to be investigated ( Crawford & Garthwaite, 2007) and iii) it can be used within a Bayesian framework in which a credible interval for the p-value is calculated ( Crawford et al., 2010). Crawford demonstrated that with small sample sizes (i.e. N<10), the method was capable of containing the alpha error probability to its nominal value of 5%. In contrast, the Z-score conversion is unreliable with small sample sizes with type 1 errors that can be as much as 20% ( Crawford & Garthwaite, 2005).

The ease with which Crawford's method can be applied has enormous advantages for the application of single case studies in reliable, modern experimental designs. However, there are still some limitations.

Firstly, until now complex designs with more than two levels have not been supported. Secondly, although it can be done within a Bayesian framework, the current output provided does not allow a straghtforward way to understand if the null hypothesis is supported. Furthermore, multiple single cases cannot be analysed in small groups and lastly, only variables at the interval level of measurement can be analysed.

The present study forms part of a larger project the aim of which is to start from Crawford’s point of view for single case analysis and subsequently develop a method. The characteristics of this method are ideally as follows:

1. it allows complex experimental designs to be taken into account, with the application of multilevel

methods in the analysis of single cases (Gelman & Hill, 2006; Pinheiro & Bates, 2000; Stroup,

2012), with fixed (or population-level) and random (or group-level) effects;

2. its results are given in terms of Bayes Factors (Kass & Raftery, 1995; Raftery, 1995) in order to

observe whether the effect seen in a single case is different from the control group (alternative Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 hypothesis), and when it refers to the same statistical population as the control group (null

hypothesis);

3. it computes results for the control group, for the single case and for the differences between the

control group and the single case within the same method;

4. it can take into account different families of statistical populations, not only the Gaussian family;

5. it is able to group together a small number of single cases (if these are not sufficient in number to

become a statistically representative group) thereby obtaining weighted mean effects.

In this paper, a Bayesian model for linear dependent variables is presented. This model has the same flexibility as multilevel methods and provides Bayes Factors for both the control group and the single case as well as the differences between them. The objective was to create a novel tool that grants the flexibility of linear mixed model designs with the possibility of testing both the null and the alternative hypotheses of complex designs within the Bayesian framework.

The method was validated by a simulation study, and we demonstrate the potential of this method by means of an application on real data. At this stage, the BMSC can be used in the R environment and the related functions are collected in the bmscstan package.

The Statistical Method

As a first step, we developed bmscstan for linear cases. The likelihood distribution for the control group is normally distributed as follows:

2 ( yc∨b) N (X c β+Zb ,σ c )

where yc is the dependent, linear variable, Xc is the contrast matrix for the fixed effects, β is the unknown vector of the fixed effects, Z is the contrast matrix for the random effects, b the unknown vector for the random effects and σ2 is the scale parameter.

The likelihood distribution for a single case is:

2 ys N [ X s ( β+δ ) , σ S] Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 where ys is the dependent variable for the single case, Xs is the contrast matrix for the fixed effects, β is the unknown vector of the fixed effects of the control group and δ is the unknown vector of the differences between the fixed effects of the control group and the single case.

The two contrast matrices Xc and Xs do not have to have the same number of rows, but they must have the same number of columns and the same contrasts (e.g. if a factor has sum-to-zero contrasts in X c, in Xs, you should use the same type of contrast). From a practical point of view, the experimental design must be the same for both the controls and the single case, but the number of observations does not need to be the same.

Prior distributions

The default distributions of the unknown vectors of the fixed effects β and of the differences between the control group and the single case δ is:

β N (μ=0 ,σ 2=10)

δ N (μ=0,σ 2=10)

The user can choose between two other options:

Cauchy distribution option Student's t-distribution option β Cauchy (μ=0,σ2=√2/2) β t (μ=0,σ2=10,ν=3)

δ Cauchy (μ=0,σ2=√2/2) δ t (μ=0,σ2=10,ν=3) The Cauchy distribution option is more conservative compared to the default option, while the Student's t- distribution option is as conservative as the default option but more robust to outliers.

2 2 2 σ c and σ S have very vague priors: N (μ=0,σ =1000).

If there are any random effects these are shown on vector b which is distributed along with a multivariate normal distribution with mean zero and an unknown covariance matrix. b N (μ=0,Σ) where Σ is given by: Peer-reviewed, published version:

Ω LKJ ( ξ ) with ξ>0 and D(σ) being a vector of standard deviations.

Parameter and Bayes Factor estimation

The estimation of parameters and the computation of Bayes Factors, both by means of the bmscstan package, are computed using its principal function: BMSC.

This function requires a formula written in lme4-like syntax (Bates, Maechler, Bolker, & Walker, 2015) which describes the supposed relation between the dependent and the independent variables, the data frame of the control group, and the data frame of the single case. Moreover, further optional variables can be set

(further details in the help page of the package).

The bmscstan package uses Stan on the backend to estimate the parameters (Carpenter et al., 2017).

Therefore, all the sampling methods available in Stan can be used to estimate the parameters in the bmscstan package.

The summary function from the bmscstan package gives as the output for each parameter the mean, the standard error, the standard deviation and the 2.5%, 25%, 50%, 75%, 97.5% quantiles of the posterior distribution (with the 2.5% and 97.5% quantiles being the Bayesian Credible Interval). Furthermore, the output will also give the Gelman and Rubin's convergence diagnostic (Gelman & Rubin, 1992), the Effective

Sample Size (Geyer & Geyer, 1992) and the Bayes Factor BF10 (the amount of evidence relating to the alternative hypothesis against the null hypothesis) by means of the Savage-Dickey density ratio

(Wagenmakers, Lodewyckx, Kuriyal, & Grasman, 2010).

The output of the summary function is divided into four blocks (for an example see the section entitled “A working example”).

1. In the first block, the formula and the priors chosen are shown; Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 2 2. in the second block, the fixed effects of the control group (the β vector and σ c) is shown, with the

Gelman and Rubin's diagnostic index, the Effective Sample Size and the BF10;

3. in the third block, the fixed effects of the single case is presented, calculated from the sum of the β

and δ vectors; the Gelman and Rubin's diagnostic index and Effective Sample Size are not presented

since these are a combination of the distributions of the second and fourth block;

4. in the fourth block, the fixed effects of the difference between the Control Group and the Single

2 Case is presented (the δ vector and σ S).

Installation

The bmscstan package can be installed in its latest version by means of devtools::install_github("michelescandola/bmscstan").

This package depends on the rstan (Stan Development Team, 2020), bayesplot (Gabry, Simpson, Vehtari,

Betancourt, & Gelman, 2019), logspline (Kooperberg, 2019), LaplacesDemon (Statisticat & LLC., 2020) and ggplot2 (Wickham, 2016) packages.

Validation

In order to validate our method, we decided to test it in a classic test-retest condition. Therefore, we simulated data from normal distributions with two conditions (1 and 2) for both the control group and the single case, with 30 trials for each condition and each individual.

We simulated 1000 samples of a single case and various different control groups of different dimensions: 5,

15 or 30 individuals, under the null (no differences between the single case and the control group) and the alternative hypothesis (the presence of a difference between the single case and the control group). This means there was a total of 6000 simulations.

For each simulation, we compared the results obtained from testing the single case patient against the control group, both with the Crawford’s t-test and the BMSC. We grouped the simulations by different sized control Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 groups (either 5, 15, or 30) and two conditions:1) impairment (alternative hypothesis) and 2) non-impairment

(null hypothesis). Peer-reviewed, published version:

Figure 1: Crawford's tests under the null hypothesis in Conditions 1 and 2. H1 = significant result; ns = not statistically significant result. Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 Results showed that under the null hypothesis, the Crawford's tests showed a first-type error < 0.05 (see

Figure 1). Here we see that the Crawford’s test has excellent properties in containing the first-type error, limiting the observation of false positives. Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 Figure 2: BMSC tests under the null hypothesis. Results were grouped as follows:H1 if BF10 > 3, H0 if BF10

< 1/3, nc = not conclusive, when BF10 was comprised between 1/3 and 3.

In Figure 2, it can be seen that the BMSC method is able to contain the first-type error, and that in the majority of cases it is even able to show that the null hypothesis is true, something which is impossible with the classic Crawford’s test. Peer-reviewed, published version:

Figure 3: Crawford's tests under the alternative hypothesis. Results were grouped as follows:

H1 = significant result, i.e., p-value <.05; ns = not statistically significant result, i.e., p-value>.05.

In Figure 3, the Crawford test is very powerful, as can be seen in Condition 2, and it maintains a good first- type error in Condition 1. Peer-reviewed, published version:

Figure 4: BMSC test under the alternative hypothesis. Results were grouped as follows:H1 if BF10 > 3, H0 if

BF10 < 1/3, nc = not conclusive, when BF10 was comprised between 1/3 and 3.

BMSC is very powerful in Condition 2 and maintains a good first-type error in Condition 1.

A working example In order to explore how to use this package, we present a step-by-step tutorial on how to analyse data from a real single case (available within the package).

The data sets come from an experiment in a study entitled "The Body Sidedness Effect" (Ottoboni, Tessari,

Cubelli, & Umiltà, 2005; Tessari, Ottoboni, Baroni, Symes, & Nicoletti, 2012). The participants were an individual patient with a unilateral brachial plexus lesion and 16 healthy controls.

The Body Sidedness Effect is an experimental paradigm which is useful in that it implicitly investigates the

Body Structural Description (Medina & Coslett, 2010). It is based on the seminal Simon task (Simon, 1969) in which blue or red circles are shown on the left or right side of a computer screen. Participants are requested to identify the colour of the circles. They are required to use their right hand for the blue circles and their left hand for the red circles. Even if the position of the circles on the computer screen is task- irrelevant, the participants’ reaction times were longer when the circle appeared on the opposite side with Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 respect to the hand they were using to respond (e.g. a blue circle on the left side of the screen). In an experiment carried out relating to the Body Sidedness Effect, the circles were presented at the centre of the screen, superimposed to a task-irrelevant image of a right or left hand or a right or left the foot. Similarly to the Simon Effect, if the laterality of the background image was incongruent to the hand that was supposed to be used to answer, the reaction times were longer, indicating evidence of an effect relating to a preserved

Body Structural Description (Ottoboni et al., 2005; Tessari et al., 2012).

The first step is to load the library bmscstan and investigate the data set.

> library( bmscstan ) > data( BSE ) > > str(data.ctrl)

'data.frame': 4049 obs. of 5 variables: $ RT : int 785 641 938 841 486 425 408 394 611 387 ... $ Body.District: Factor w/ 2 levels "FOOT","HAND": 1 1 1 1 1 1 1 1 1 1 ... $ Congruency : Factor w/ 2 levels "Congruent","Incongruent": 2 2 2 2 2 2 1 1 1 1 ... $ Side : Factor w/ 2 levels "Left","Right": 1 1 1 1 2 1 1 1 2 2 ... $ ID : Factor w/ 16 levels "HN_017","HN_019",..: 1 1 1 1 1 1 1 1 1 1 ... > > contrasts( data.ctrl$Side ) <- contr.treatment( n = 2 ) > contrasts( data.ctrl$Congruency ) <- contr.treatment( n = 2 ) > contrasts( data.ctrl$Body.District ) <- contr.treatment( n = 2 ) > > str(data.pt) 'data.frame': 467 obs. of 4 variables: $ RT : int 562 424 411 491 439 593 504 483 467 413 ... $ Body.District: Factor w/ 2 levels "FOOT","HAND": 1 1 1 1 1 1 1 1 1 1 ... $ Congruency : Factor w/ 2 levels "Congruent","Incongruent": 1 2 2 1 1 2 2 1 1 2 ... $ Side : Factor w/ 2 levels "Left","Right": 1 2 1 2 1 1 2 1 2 2 ... > > contrasts( data.pt$Side ) <- contr.treatment( n = 2 ) > contrasts( data.pt$Congruency ) <- contr.treatment( n = 2 ) > contrasts( data.pt$Body.District ) <- contr.treatment( n = 2 )

In our data set, the dependent variable is "RT" (reaction times). The independent variables are Congruency

(levels: congruent/incongruent), Body.District (levels: FOOT/HAND) and Side (levels: Left/Right). In all cases, these independent variables have a sum-to-zero matrix contrast.

In both data sets, dependent and independent variables must have the same name. It is possible, for example, that the factor Body.District has a treatment contrast, and Congruency a sum-to-zero contrast, but the typology of contrast has to be the same for both data sets.

The ID factor is only present in the control group data set because it represents the individual participants. Peer-reviewed, published version:

> qqnorm(data.ctrl$RT) > qqline(data.ctrl$RT) > > qqnorm(data.pt$RT) > qqline(data.pt$RT) >

A B) )

Figure 5: Q-Q plot for control and single case data. A) Control group. B) Single case data. The data were not normally distributed, as it is possible to observe from the Q-Q plots. Therefore, we removed outliers.

> out <- boxplot.stats( data.ctrl$RT )$out > data.ctrl <- droplevels( data.ctrl[ ! data.ctrl$RT %in% out , ] ) > out <- boxplot.stats( data.pt$RT )$out > data.pt <- droplevels( data.pt[ ! data.pt$RT %in% out , ] )

Because data from the two Body Districts were collected in two different blocks, we wanted to group our random-effects not only by participant but also by Body District. In classic lme4 syntax, we wanted to fit the following model:

RT Body .District ×Congruency × Side+ (Congruency × Side|ID /Body .District ¿

However, bmscstan::BMSC is not able to manage a syntax for a random part such as this

( Congruency × Side|ID /Body . District ¿. Further details about how to specify the random part of the analysis are reported in the help section of the package (help(bmscstan::randomeffects)). Peer-reviewed, published version:

( Congruency × Side|ID ¿+( Congruency × Side|ID : Body . District ¿.

Therefore, we only needed to create a factor that represented the interaction between ID and Body.District.

> data.ctrl$BD_ID <- interaction( data.ctrl$Body.District , data.ctrl$ID )

It was then possible to fit the model using the BMSC function and could be tested the quality of the estimates by means of Posterior Predictive P-values (Gelman, 2013).

> mdl <- BMSC(formula = RT ~ Body.District * Congruency * Side + > (Congruency * Side | ID) + (Congruency * Side | BD_ID), > data_ctrl = data.ctrl, > data_sc = data.pt, > cores = 4, > seed = 2020) > > pp_check( mdl )

Figure 6: Posterior Predictive P-value check for the Control group and the Single case data.

The posterior predictive check appeared to be in agreement with the data observed.

To read the results output, we needed to use the summary function. Peer-reviewed, published version:

RT ~ Body.District * Congruency * Side + (Congruency * Side | ID) + (Congruency * Side | BD_ID)

[1] "Prior: normal"

In the first part of the output of the summary function, the fitted object gives the information concerning the formula you use, and the prior distributions selected for the coefficients.

Fixed Effects for the Control Group

mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat BF10 (Intercept) 199.88 0.0670 7.79 184.55 194.52 199.90 205.20 215.072 13519 1 1.58e+49 Body.District2 17.20 0.0618 5.80 6.22 13.14 17.11 21.15 28.931 8823 1 54.8 Congruency2 16.10 0.0844 6.75 4.08 11.33 15.72 20.53 30.272 6382 1 21.2 Side2 19.11 0.0912 7.50 5.01 13.88 18.95 24.30 33.919 6763 1 31.9 Body.District2:Congruency2 -10.03 0.0422 4.99 -19.84 -13.38 -9.97 -6.65 -0.375 13994 1 3.81 Body.District2:Side2 -4.87 0.0448 4.85 -14.37 -8.15 -4.92 -1.54 4.553 11759 1 0.796 Congruency2:Side2 -7.22 0.0805 7.76 -22.65 -12.40 -7.13 -1.90 7.653 9310 1 1.17 Body.District2:Congruency2:Side2 6.42 0.0614 6.25 -5.99 2.26 6.53 10.69 18.550 10362 1 1.13

mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat sigmaC 75.7 0.00662 0.882 74 75.1 75.7 76.3 77.5 17715 1

In the second part, the fixed effects for the control group are shown, with the descriptive statistics for the posterior distributions, the diagnostic statistics relating to the quality of the posterior distributions (n_eff that is the Effective Sample Size, Geyer & Geyer, 1992) and the Rhat that is the Gelman and Rubin's diagnostic index (Gelman & Rubin, 1992), and the Bayes Factor which is computed by means of the Savage-Dickey

Density Ratio (BF10).

Here we can observe several BF10 > 3: the first BF10 concerns the intercept of the linear model, meaning that the average reaction times are different from zero. Thus, there are the BF10s concerning the Body.District2,

Congruency2 and Side2 main effects, and the Congruency2:Side2 interaction, indicating that there could be a specific effect that needs further investigation.

Other coefficients had inconclusive BF10s: Body.District2:Congruency2:Side2, Congruency2:Side2 and

Body.District2:Side2 results. At this point, Occam’s razor principle was applied, and even if the BF10s did not clearly support H0, we consider that, among a series of possible explanatory models with the same adequacy of prediction, the best model was the most parsimonious and therefore the one with fewer coefficients.

Fixed Effects for the Patient Peer-reviewed, published version:

mean se_mean sd 2.5% 25% 50% 75% 97.5% BF10 (Intercept) 394.8 0.0715 6.39 382.1 390.5 394.9 399.27 407.19 1.65e+70 Body.District2 40.5 0.0762 6.82 27.2 35.8 40.5 45.21 53.83 455825 Congruency2 47.1 0.0804 7.19 33.5 42.2 47.0 52.01 61.42 2859122 Side2 50.7 0.0790 7.07 36.9 45.8 50.7 55.54 64.68 2068424 Body.District2:Congruency2 -26.9 0.0928 8.30 -43.1 -32.6 -26.9 -21.17 -10.42 145 Body.District2:Side2 -21.5 0.0908 8.12 -37.4 -27.0 -21.5 -16.06 -5.65 25.4 Congruency2:Side2 -14.2 0.0989 8.84 -31.7 -20.2 -14.3 -8.28 2.91 3.3 Body.District2:Congruency2:Side2 -10.4 0.1096 9.80 -29.8 -16.9 -10.4 -3.81 8.79 1.71

The third part shows the fixed effects of the individual patient. We observed the presence of a Body.District2 effect (BF10 = 455,825), Congruency2 (BF10 = 2,859,122), Side2 (BF10 = 2,068,424),

Body.District2:Congruency2 (BF10 = 145), Body.District2:Side2 (BF10 = 25.4) and a Body.District2:Side2

70 interaction (BF10 = 3.3), and also the intercept (BF10 = 1.65e ). It was possible to see that the effects were different from the effects seen in the control group.

The diagnostic statistics are not shown here since these effects were computed based on the fixed effects of the control group and the fixed effects of the difference between the control group and the individual patient, the diagnostic indexes of whom are reported.

Fixed Effects for the difference between the Patient and the Control Group

mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat BF10 (Intercept) 194.97 0.0630 7.83 179.54 189.6 195.07 200.37 210.071 15448 1 Inf Body.District2 23.29 0.0628 6.95 9.66 18.6 23.39 28.03 36.945 12250 1 134 Congruency2 31.04 0.0698 7.41 16.30 26.2 31.01 35.99 45.651 11283 1 1275 Side2 31.60 0.0754 7.75 16.11 26.4 31.67 37.07 46.469 10561 1 800 Body.District2:Congruency2 -16.82 0.0625 7.69 -31.99 -22.0 -16.84 -11.69 -1.750 15118 1 8.47 Body.District2:Side2 -16.61 0.0598 7.59 -31.49 -21.8 -16.68 -11.37 -1.474 16132 1 7.31 Congruency2:Side2 -7.02 0.0698 7.99 -22.67 -12.4 -6.99 -1.68 8.566 13132 1 1.15 Body.District2:Congruency2:Side2 -16.85 0.0624 8.51 -33.74 -22.4 -16.91 -11.10 -0.122 18626 1 6.37

mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat sigmaP 64.3 0.0203 2.65 59.3 62.5 64.2 66 69.7 17052 1

In the fourth part of the output, there are the fixed effects for the differences between the control group and the individual patient.

Here we see that the intercept (BF10 = Inf) and all effects, with the exception of Congruency2:Side2 (BF10 =

1.15), were different from the fixed effects of the control group. A more detailed analysis on

Body.District2:Congruency2:Side2 (BF10 = 6.37) below.

The posterior distributions of the effect were then plotted.

> plot( mdl ) Peer-reviewed, published version:

Figure 7: Plot and visual comparison of the coefficents obtained by means of the BMSC.

A more detailed analysis of the results of both the Control group and Single Case can be seen in the vignette of the package with the command vignette(“bmscstan-use"). For the purposes of this article, only the Body.District2:Congruency2:Side2 interaction seen in the difference between the controls and the single case is considered.

The bmscstan::pairwise.BMSC function is extremely useful at this point. This function computes the

BF10 for all the possible comparisons relating to the effect we were interested in. The subjects are the BMSC model and the coefficient that needs to be explored, and you need to state whether you are interested in the performance of the control group, in the performance of the single case, or just in the difference between the control group and the single case by setting the subject as “who” equal to "control", “single case” or “delta”.

> ph <- pairwise.BMSC( mdl1 , "Body.District2:Congruency2:Sides" , who = "delta" )

> print( ph , digits = 3 ) Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 The output of this function is divided into two parts. In the first part, we see the marginal distributions of each level of the effect, with the mean, standard deviation, standard error, and the 2.5%, 25%, 50%, 75%,

97.5% quantiles of the distribution, and its BF10 against zero.

Pairwise Bayesian Multilevel Single Case contrasts of coefficients divided by Body.District2:Congruency2:Side2

Marginal distributions

mean se_mean sd 2.5% 25% 50% 75% 97.5% BF10 (not zero)

FOOT Congruent Left 195 0.0875 7.83 180 190 195 200 210 Inf

FOOT Incongruent Right 251 0.1516 13.56 224 241 251 260 276 7.69e+34

FOOT Incongruent Left 226 0.1137 10.17 206 219 226 233 246 2.65e+23

FOOT Congruent Right 227 0.1166 10.43 206 219 226 234 247 8.94e+28

HAND Incongruent Left 232 0.1365 12.21 208 224 233 241 256 5.83e+18

HAND Congruent Right 233 0.1428 12.77 208 225 233 242 258 3.25e+23

HAND Congruent Left 218 0.1085 9.70 199 212 219 225 237 1.02e+27

HAND Incongruent Right 224 0.1778 15.90 192 213 224 234 255 5.39e+12 In the second part, the full list of all the possible pairwise comparisons appears. You should only

look only at those that make sense for your study. In this case, we report only some of them, that is,

those whose BF10 is > 3 and, of interest for this study,

 FOOT Congruent Left - FOOT Incongruent Left: BF10=621.74

 FOOT Incongruent Right- FOOT Congruent Right: BF10=18.45

 FOOT Congruent Left - FOOT Congruent Right: BF10=737.96

 FOOT Incongruent Right- FOOT Incongruent Left: BF10=17.16

 HAND Congruent Right- HAND Congruent Left: BF10=3.24

 FOOT Incongruent Right- HAND Incongruent Right: BF10=26.00

 FOOT Congruent Left- HAND Congruent Left: BF10=174.55 Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 You might also want to plot these marginal distributions. This is easily done by means of the bmscstan::plot.pairwise.BMSC function.

> plot( ph )

Figure 8: Plot of the intervals relating to the marginal distributions of a three-way interaction.

Discussion

In the present study, we describe a new instrument which can be used to compare a single case to a control group in terms of their performance. This method represents a much needed tool in that studies in the field of neuropsychology are frequently based on individual patients with particular conditions and unique lesions and it is often difficult to form groups of patients with similar conditions.

Neuropsychology as a discipline has its origins in the study of single cases (e.g. Broca, 1861). This was developed further by other scholars (see SCOVILLE & MILNER, 1957) and significant advances are still Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 being made (see D’Imperio et al., 2017; Moro et al., 2012, 2015; Moro, Pernigo, Urgesi, Zapparoli, &

Aglioti, 2008; Moro, Zampini, & Aglioti, 2004; Thiebaut de Schotten et al., 2015).

The BMSC method performs well in terms of first-type and second-type errors and it gives researchers in the field of Neuropsychology the freedom to design complex experimental studies, allowing them to analyse all of the data within a unique, comprehensive model. It also gives results for the control group, the single case and the differences between the control group and the single case. Moreover, it enables the researcher to test both the null and alternative hypotheses. Furthermore, a constellation of satellite functions make it possible to check whether the model is representative of the data, to plot the results with different options, and to automatically perform pairwise analysis when necessary.

The aim of the modified version of the t-test developed by Crawford and colleagues ( Crawford et al., 1998) was to distinguish between a defective performance from the performance of a small control group (e.g.

N<10). This test has the great advantage of keeping the probability of type 1 errors to its nominal value in cases of small sample sizes for the controls, even in simulations with groups of less than 5 people ( Crawford

& Garthwaite, 2005). This was confirmed in the simulation study presented here. Nonetheless, in Crawford’s tests there are some limitations, especially regarding the experimental design and the unidirectional conclusions (as for any test under the Null-Hypothesis Statistical Testing framework). They can be used to determine simple or double dissociations in a patient by comparing a single datapoint to a control group but multiple trials cannot be entered into the test directly. Multiple main effects of independent variables or interactions cannot be tested. Additionally, the adoption of the p-value should be read only as evidence of dissociation and not as evidence of equivalence when the test is not significant.

Bootstrapped LMM was proposed as a statistical tool to compare single cases to a control group (Huber,

Klein, Moeller, & Willmes, 2015). However, this approach is limited to frequentist statistics and does not allow the researcher to see the performances of both the control group and single case, or the differences between them as clearly as the bmscstan tool.

The BMSC method was developed with a view to overcoming these limits. In the present study, the simulated experimental design was designed to allow a direct comparison between the Crawford method and Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 ours. Namely, we simulated basic cases of simple dissociation, and no-dissociation. However, it is crucial to remember that the BMSC allows much more complicated designs to be tested even at this stage of development with, hypothetically, no limitations to the number of main effects and interactions. Another plus is that it is done within a Bayesian framework, offering indexes (i.e. the Bayes Factor) that can provide evidence in support of the null hypothesis and not only against it.

The case involving real data showed that with bmscstan it is possible to compute a BMSC model in few lines of code. In this way, a data set that includes a single case and a control group can be analysed in depth. By means of posterior predictive p-value check, the Gelman and Rubin diagnostic and the Effective Sample

Size, the quality of a model can be kept under control, as well as the reliability of the estimates.

Thanks to the plot and the pairwise functions, it is also possible to analyse complex effects further, and identify those that are of interest for any specific hypotheses and/or experimental designs.

The specifications relating to the random effects in the bmscstan package currently have some limitations.

The correlations between the random effects are not shown, and it is not yet possible to specify uncorrelated random effects. However, the study of the correlations among random effects, and the use of uncorrelated random effects are not typically fundamental to single case studies. Therefore, the current version is potentially applicable to all single cases with linear dependent variables. Nonetheless, a solution to this limitation is currently being investigated and should be included in future versions of the package.

Furthermore, the random part of the model is only applied to the control group and not to the single case data. However, this is not a limitation but a methodological choice since it is not possible to estimate random effects from one individual. This can be done by using the same random effects as those estimated in relation to the control group and the single case (see Huber et al., 2015), but in our opinion, this strongly implies the assumption that the same random structure applies to a pathological condition as to healthy controls. This may potentially generate biases in estimates made relating to the effects on the single case, and therefore the random effects of the single case are not computed in the BMSC model.

Conclusions Peer-reviewed, published version:

Scandola, M., & Romano, D. (2021). Bayesian Multilevel Single Case Models using “Stan”. A new tool to study single cases in Neuropsychology. Neuropsychologia, 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834 The BMSC constitutes a new tool to be used in the study of single cases. We believe that it has the potential to inject new life into the field of neuropsychology as this method has been proved to be ten times more reliable than the Crawford t-test under simulation. Additionally, it is more flexible in terms of potential experimental designs and enables the researcher to test any design relating to main effects and interactions that he/she requires. It also makes it possible to estimate effects without assuming an equal number of trials for each participant (patient or control).

The BMSC methodology is derived from the work of Crawford which has been proved to be as reliable as stated and is easy to use and straightforward in its interpretation (Crawford et al., 2010, 1998). Our goal was to come up with a new, more flexible tool for those experimental designs that Crawford’s tests are not able to cover. Additionally, our method also provides an instrument that follows the recent trend of shifting the attention from p-values to other inferential indices and estimations, such as Bayes Factors or Bayesian

Credible Intervals. Peer-reviewed, published version:

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