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: 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 Introduction 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.