Background: School plays an important role in transgender and non-binary (TNB) youths’ life and well-being. The aim of this study was to gain a better understanding of how the lived experiences, gender affirmation and challenges encountered by TNB youths in the school setting affect their well-being. Method: Our study was a qualitative secondary data analysis, based on the interviews of 12 Canadian TNB youths aged 15–17 years old. Results: We found that TNB students’ well-being was closely related to the acknowledgment of gender identity at school. Several factors, including school socio-cultural environment, teachers’ and peers’ attitudes and behaviours, school physical environments and the respect of confidentiality of gender identity were all found to impact TNB students’ well-being. To face adversity related to some of these factors, TNB youths used several contextually driven strategies such as compromising, educating, and sensitizing others about gender diversity and avoiding certain people or situations. Conclusion: Our results highlight the important influence of school climate and culture, as well as teachers’, school personnel’s and peers’ behaviours and attitudes on TNB youths’ well-being. Our findings can guide future interventions to help schools become more inclusive and supportive of gender diversity.  Gender minority youths experience high rates of adverse mental health outcomes during early adolescence . CompareThe mental health and well-being of individuals belonging to minority groups, including gender minorities are strongly related to social environmental factors ,6. Yet, Existing literature describing TNB youths’ experiences in schools often fails to distinguish TNB teenagers from their lesbian, gay and bisexual (LGB), cis-identifying peers. However, TNB young people experience challenges at school that are different from those encountered by their non-heterosexual, cisgender peers ,13. ThusFeeling safe at school is a major concern for many TNB youth according to cross-sectional studies ,14, and Literature on TNB students, has shown that TNB students are at higher risk of victimization  due to their gender identity ,18,19. TTNB students experiencing higher levels of victimization are less likely to feel supported by an adult at school . In addiMost studies focusing on TNB youth well-being at school rely on cross-sectional surveys or other quantitative approaches ,25. ThesDrawing from qualitative data, this article aims to provide a more thorough understanding of one of these factors that emerged as fundamental to youths’ wellbeing: the school environment. To achieve this goal, we will address the three following questions: (1) What are the lived experiences of TNB youths in the school setting, and how are these impacting their well-being and resilience? (2) How is the affirmation of gender identity supported (or undermined) among TNB students in the school setting? (3) What are the strategies employed by TNB students to face adversity and obstacles imposed on them by the school cultural, social and physical environments?As part of the larger project ‘Digging Beneath the Surface: An Intersectional Exploration of Trans Youth Experience’, led by the Canada Research Chair on Transgender Children and their Families, and funded by the Canadian Social Sciences and Humanity research Council, the team decided to specifically analyze minor (under the age of 18 years) TNB youth narratives regarding their experience in school settings. The project aimed to identify oppressive factors and structures that negatively influence TNB youths’ well-being, factors that contribute to TNB youths’ resilience and thriving, and to understand how oppression and resistance processes interact. This study used a community-based participatory research approach  and was The project included 54 interviews with 15–25-year-old TNB youths, living in different cities of the province of Québec. Participants were recruited with the help of community partner organizations via paper (posters and flyers) and online invitations. Recruitment efforts aimed to achieve participant diversity related to gender, age, citizenship status, cultural and socioeconomic backgrounds. According to the latest available census data , the majInterviews were conducted in person, in a location and language (French or English) chosen by the participant, by an interviewer identifying as TNB. Interviews were audio-recorded, with the participants’ authorization, and lasted approximately two hours. Interviews were transcribed by the interviewer to ensure accuracy of study materials. For this article, we selected the transcripts of participants under the age of 18 -years, for a total of twelve participants. This age range was selected to include youth who are currently in high school in Québec. We excluded older participants as their high school experience would not have been as recent and this could have had an impact on the relevance of our analysis. All the interviews included in this analysis were conducted between November 2016 and June 2018 which means that all participants had a high school experience with the same level of legal protection against gender identity related discrimination. Indeed, in June 2016, gender identity and expression became protected by the Quebec Charter of Rights and Freedoms and minors aged 14 years and older gained the civil right to change their gender markers on their birth certificate without parental consent. Hence, to focus on youth who only have a current or very recent experience within this framework of new protection allows us to have a better understanding of current TNB youth experience in high schools. Conversely, including older participants who would have discussed their high school experience retrospectively would have produced an analysis that is less accurate because they did not have the same legal protection than what is available today.A different pseudonym was chosen for each transcript to preserve participant anonymity. For this paper, the twelve interviews were reanalyzed, with a focus on school experiences and gender affirmation at school, on the way school impacted participants’ well-being and resilience and the strategies participants used when facing obstacles and adversity.Among participants, two identified as trans girls, four as trans boys and six as non-binary youth. All participants were White except for one participant who described themselves as being from a mixed heritage. This lack of cultural diversity in our sample is unfortunate as it does do not reflect the full population from which we selected the sample (15–25 years old) which includes youth from diverse cultural backgrounds. However, youth aged between 15 and 17 years were homogenous in terms of cultural background. One participant lived in a rural area while all others lived in large urban cities. All twelve participants mentioned that they had “come out” about their gender identity while in high school. At the time of the interview, nine participants were still in high school while three had recently graduated. Information about where participants went to school was not explicitly collected. However, the fact that youths came from different cities suggests that they also went to different schools. All three college students described their school experiences during the interview, which we drew from for our analysis of high school experience. For those who had transitioned to college at the time of the data collection, we also included some of their insights on strategies (theme 2) to navigate new environments as this seems particularly important and relevant to understand TNB youth experiences at school. Analyses were conducted in accordance with Meyer’s  minorityData were analyzed through Braun and Clark’s thematic analysis , which iOur first step was to familiarize ourselves with the data. To do so, JK conducted a first reading of the non-coded transcripts, originating from the initial study, and retained all sections of the interviews containing school-experiences-related content. Each of these sections were read multiple times. Then, for the second phase of our analysis, the interviews were manually and systematically coded by JK, generating an initial set of codes. In phase 3, codes expressing similar concepts were identified and combined to describe common themes. These groups of codes were discussed with all study members leading to the formation of thematic categories. After discussion with the research team, in phase 4 of our thematic analysis, all extracts and sections of the interviews discussing school issues were reread. Codes were revised to ensure they were exclusive to their category. Then, in phase 5, categories were named and assembled as the themes started to emerge. In phase 6, a final reorganization of codes and categories took place, in parallel with drafting of this manuscript. According to Braun and Clarke , writingYouth explained that their wellbeing was mostly conditioned through experiences of validation of their gender identity. Study results will be presented in two sections. First, through the discussion of impeding and protective factors for TNB youth wellbeing, and second, through the description of strategies used by TNB youth to face adversity. Most participants provided examples of how their identity was affirmed, ignored, or diminished at school according to three main dimensions, namely school climate and culture, direct behaviours and attitudes of school staff, and behaviours and attitudes of peers. School climate and culture was an important dimension influencing TNB students’ identity acknowledgment. A lack of knowledge about TNB youth identities was perceived by study participants as a major obstacle in getting their identity validated and their rights respected by students, teachers, and school personnel. Some participants felt they were schooled in an environment where the lack of understanding of gender diversity was generalized rather than stemming from a limited number of individuals. This was illustrated by Anto, a 17-year-old non-binary student, who mentioned how both students and staff had a limited understanding of gender diversity when they “came out”:“When I came out I said, ‘Hey, my name’s [chosen name] now,’ I don’t go by, like, my birth name. People were confused, they didn’t know what it was.”Aside from the overall lack of knowledge about gender diversity, participants felt that the over-representation of cisnormativity in school prevented adequate inclusion of TNB youth. Arthur, a 17-year-old non-binary teenager, explained how they were left aside in their gym class when sports teams were separated between boys and girls: “So I just sat on the bench. And then the teacher was like ‘Well, go play.’, and I was like ‘Yeah, but what side do I play on?”. * (Quotations followed by «*» have been translated from French).Overall, the lack of awareness and acceptance of gender diversity and the overabundance of gender stereotypes and cis-gendered assumptions at school were identified as very difficult to live with: “[…] people will judge that something is feminine then they’ll associate feminine with girl, that’s more difficult […] these things really hurt me, but others, you know, they can’t understand that, like, how we really feel.”*Inappropriate cis-normative comments and remarks made by teachers and classmates, even if made unintentionally, had a negative impact on TNB participants’ well-being. Restrictive cultural representations of gender are marginalizing for TNB youths. These manifest in many spaces at school, but particularly in gendered spaces, such as bathrooms and locker rooms. Gendered locations are an important dimension for validation of gender identity, but often standardize a binary vision of gender. These spaces were often perceived as uncomfortable by youth: “[…] Having to choose every time where I feel the most comfortable and like, I don’t want to make other people uncomfortable either […]” *In most cases, TNB students were not consulted about their preference regarding the use of gendered spaces or other trans-inclusive accommodations and had to endure measures that could be as discriminatory, or even more discriminatory, as no accommodation at all. Some youth could use the bathroom of their choice, for example a gender-neutral bathroom, but this preferred bathroom was sometimes far out of their way, which also can cause problems:‘’I had asked not to have to go to a boy or a girl bathroom, so they told me to go... to go to a toilet that is really really far!’’*However, having access to gender neutral bathrooms was not a solution that was desired by all. For example, Eric, a 15-year-old trans boy explained that he going to the boys’ bathroom until the school director realized that he was trans, and then forced him to go to a ‘mixed’ bathroom and changing room despite him not wanting to. This had negative consequences on his wellbeing.:“Interviewer: At the beginning, you were going to change in the guys’ locker room but after, he told you to go in the mixed locker room?Participant: In the mixed toilets.Interviewer: In the mixed toilets?Participant, Yeah.Interviewer: Ok. Ok... How did you feel about all that?Participant: Well badly. Because, you know …I want to-I want to go to the guys’ bathrooms, to the guys’ locker rooms.Interviewer: What effect did it have on you?Participant: Well that hurt me.”Therefore, the lack of inclusion of TNB students in school decision processes about gender diversity was perceived as a form of resistance against supporting their identity and was experienced negatively. Max, a 16-years-old, non-binary student, explained how difficult it was for them, to have their chosen name used at their school because of the teachers and principals’ attitude of cisnormativity which treats trans identities as needing justification to be valid and to benefit from accommodation: “Just for teachers to use it [chosen name] it was a big deal […] they asked me for a gender dysphoria diagnosis report and proofs that I was going to take hormones and everything when it shouldn’t even be necessary.” *Unjustified delays, discrepancies, or inequities in the attribution of accommodations were also cited as examples by participants as ways to resist supporting trans-inclusive rights.Some were also concerned about the way their personal information was treated by the school system, and about possible breaches in confidentiality. Administrative documents, containing students’ sex and birth name  were a major concern for confidentiality: “It’s really hard to show my papers to others, like people I just met. They have to check my deadname. For me, my deadname is something really private […].” *Students often need to show these documents in schools, forcing unwanted disclosure of one’s gender identity and, thus, undermining TNB students’ privacy. Yet, schools that implemented mechanisms to protect youth privacy and were able to maintain confidentiality of information were identified as a protective factor allowing participants to feel safe in their school. For example, Axel, a 15-year-old trans male, explains that despite his name not being changed legally, the school list shows his two names, so people call him correctly, which is in his experience, very helpful. “I am in a school which is quite organized too... I was the first [trans youth], but they are really open, so it helps […] [I had to persevere] but they saw that this was important. They said they would take steps [to adapt the school], and they did it with me... they have always been there to help me.’’*Participants clearly expressed that school staff’s attitudes and behaviours, whether they were teachers, school personnel or administrators had an impact on their experience of gender identity validation. Staff support was identified as a protective factor for TNB students’ well-being. Conversely, a lack of support by school staff was harmful for TNB students’ well-being:“[…] the first person I told [about my gender identity] was the social worker at my school, and since, she has always been super supportive […] she’s helped me a lot… it’s like a super important relationship for me.” *“Teachers have a closed mind, and they really influence the students a lot, so, I don’t know, if at least teachers were there to support me it would’ve been that already, like something, but not even.” *Indifference toward TNB students’ needs, as manifested by teachers and school personnel, was also seen as problematic by study participants. Students described feeling that school personnel did not seem concerned about their needs: “Let’s say I was talking about my needs and all, and school personnel, in moments like these are just like: ‘Ok, yeah, but it’s your problem’ like they don’t even care.”*“[My teacher said:] Yeah OK, but you’re just one student and there are 30 other students in the class.”*Feeling ignored was reported as an important barrier for the inclusion of TNB students at school. Participants also felt ignored by staff who misused their preferred name and pronouns. Conversely, when staff used the correct name and pronouns, participants felt validated.Teachers, principals, and school personnel’s incredulity about students’ gender identity was identified as a major obstacle by many students. Skyler, a 16-year-old, non-binary youth, explained that they were pressed with questions—sometimes irrelevant and intrusive—about their gender identity by their school vice-principal: “I had to explain almost everything to the vice-principal and then she asked me weird questions.” *Finally, some schools and adults adopted a more repressive approach. Zack, a 17-year-old non-binary student, provided examples of situations where they lost points or failed school assignments for expressing their gender identity: “I actually failed a French writing assignment in grade 9 because I wrote using male pronouns.”The lack of acknowledgment and denial of student gender diversity by an adult in a position of authority was a recurrent theme throughout the interviews, and participants detailed these situations extensively. Acceptance and acknowledgment of gender diversity by classmates was also an important component of TNB youths’ well-being. Students who reported a higher level of well-being at school felt that their gender identity was acknowledged and supported by their peers. Remi, a 17-year-old non-binary student, explained how they felt their affirmed identity was immediately accepted by their high school friends, because they spontaneously adopted their right name and pronouns: “[My friends], they accepted my transition really well and they switched my pronouns and my name pretty much on the spot, so that was really cool. And it had a really positive effect on me.” *On the contrary, the misuse or refusal to use preferred names and pronouns by peers was detrimental to several youth who explained they felt invalidated and that it had a negative impact on their well-being. Discussing how peers refused to affirm their gender identity through their name and pronoun, this youth show how their mental health deteriorated: “Because, you know, before people started using the right pronouns and the right name with me, I ended up on the psychiatry unit like three times.”*Student Associations and groups, such as Gay-Straight Alliances (GSAs) were a major source of peer support. Marlie, a 15-year-old and trans girl, explained how positive her experience at school was:“At my school, we have a GSA. […] So, all of the LGBTQ kids, at our school, we, like, go there.”The presence of LGBTQ+ organized activities at school provided youth with support, and a sense of belonging. Some youth also reported being involved with arts or other shared interest clubs, which was helpful. However, not all youth had positive experiences with their peers.Some participants experienced indifference which could be either positive, or negative. For some, indifference was a sign of tolerance, or even sometimes, acceptance: “And people were like, ‘Oh yeah. She’s trans, or whatever. And so, no one really cares that much, which is good”“It’s ok ‘cause people, they don’t care, but at the same time they accept it.”*Some participants described explicitly that they felt their identity was disrespected by other students. One manifestation of this lack of acceptance was the passive rejection experienced by some of them, which lead participants to feel isolated and lonely. This passive form of rejection was described as an insidious process: “And you know, even my friends at school, I am sitting at the table with them for lunch and I feel like I’m sitting with strangers, you know. I feel alone surrounded by people, so well”. *Others described situations where they were excluded from school social interactions, deliberately set aside by their peers and friends or even bullied after revealing their gender identity or coming-out. Bullying was mostly perpetrated through transphobic comments, invasion of privacy and deliberate misuse of preferred names and pronouns. Marlie, a 15-year-old trans girl, detailed how a friend rejected her after her coming-out, and then bullied her by deliberately disclosing her identity to others without her consent:“ […] And one of them was, like, one of my ex-friends, which is not that great. […] And he was making fun of me for it and stuff, and telling other people about that, so that’s not cool.”Being victimized and rejected at school, as reported by Marlie, has strong negative consequences on TNB youths’ well-being. Numerous strategies used by TNB students to face adversity at school were identified in the transcripts of the open interviews with study participants. Youth participants described several strategies to face their peers’ wrongful behaviours. Among these strategies, some TNB youths surrounded themselves with allies as a strategy to face bullies and oppressors. Olivier explained how having such close friends at school kept him from being bullied after he disclosed his gender identity: “For sure, if people had comments to make, I mean, I really have a lot of friends and I know that my friends are there to defend me […].” *When victimization and bullying were inevitable, some participants reported actively avoiding bullies by avoiding certain places at school or social interactions. Other participants described intentionally steering clear of unaccepting people and deliberately ignored oppressing situations or persons. Like Axel, a 15-year-old trans male, who described how he reacted when he faced someone’s contempt towards him: “I don’t react badly–these people they don’t accept me the way I am. It’s not worth it to stay with them […].” *Strategies were used to face issues with peers but also adults in positions of authority. Among these, to educate, to make people aware and to advocate for TNB rights and realities were the main strategies employed. As explained earlier by Skyler who had to answer intrusive questions about their gender identity, a lack of understanding of gender diversity appeared to contribute to school staff’s resistance to offer an inclusive environment. Accordingly, educating and raising awareness about gender diversity, and self-advocacy were key strategies used by students to face invalidation and other obstacles. However, as explained by Skyler, these strategies were not always protective of TNB youths’ well-being, as they were often imposed on them: “[…] we get the impression that we have to explain everything for them to feel comfortable, or for them to agree to [offer a trans-inclusive accommodation], but at the same time we don’t feel like doing it sometimes.” *Others used compromise as a strategy to face school personnel and principals’ resistance toward providing a trans-inclusive environment. Coralie, a 15-year-old trans girl, used this strategy when she asked to use the girls’ bathroom and was denied this right by the school’s administration. Coralie settled on using the gender-neutral teachers’ bathroom. She commented: “[…] At least we were able to meet halfway.”*All twelve TNB students publicly disclosed their gender identity during high school. However, some of them had to change schools- whether because they graduated or because they moved to a different area. Those subjects did not necessarily redisclose their non-cisgender identity at their new school. Depending on the context, TNB chose to either disclose, partially disclose or not disclose their gender identity to others. Their choice was often associated with a sense of safety in the new environment, again showing the importance of school climate and culture (discussed in Theme 1). As Skyler illustrates: “Sometimes I hide my identity […] let’s say I’m with an adult that might be in a position of authority or if I’m in an alley and it’s nighttime and I don’t feel like getting into trouble.”*Others, like Remi, sometimes chose to partially disclose their gender identity to avoid uncomfortable situations: “Even when I tell people I’m trans, I only say that I am a guy, I don’t say that I’m non-binary because it’s complicated and I don’t feel like explaining”*.Maintaining control over the privacy of their gender identity was important for participants. Thus, the choice of strategy often depends on the specific context and circumstances faced by TNB youths. Our data highlight some dimensions of school culture and climate around gender, and the impact they had on youth, especially regarding their experiences of validation or invalidation of their identity, such as the importance of supportive school associations or personnel . Ullman In our research, participants illustrated many situations where their identity was either validated and/or supported, or, conversely, ignored, rejected and, even, repressed. Youth were also victims of discrimination, which took many forms. As identified in this and other cross-sectional studies, discrimination from peers led TNB youth to experience rejection, victimization, and bullying ,9,13,20.Similar observations have been made in previous studies that have explored more widely how teachers, principals and school personnel can influence TNB students’ lives, both negatively and positively ,37,38,39Our study has also allowed to identify strategies used by TNB youth when facing adversity, such as to educate, to compromise, and to advocate. These are varied and contextually driven . While oPrevious research has shown that TNB youth will alternate strategies of resistance depending on the context and may sometimes avoid them altogether if they feel unsafe . While owith him. Conversely, Eric’s story regarding bathroom use illustrates how not listening to a young person’s needs may have detrimental consequences on them. No assumptions should be made regarding how to adequately address their needs as each TNB person and experience are different [Measures for Openness to and support for trans and non-binary students: a guide for educational institutions, for guidance in making adaptations to support TNB students in schools. See: https://familleslgbt.org/app/uploads/2022/03/Measures-for-openness-to-and-support-for-trans-and-non-binary-students.pdf, accessed on 8 September 2022). This could be done by schools inviting trans youth support organizations to provide education on gender identity and the needs of trans youth to all pupils, and plan specific training for school staff. As such, when a trans or non-binary youth enrolls in their school, they would be better prepared to meet their needs, and would already have access to tools and strategies to respect and affirm their students’ gender identity. By inviting trans youth support organizations to talk about trans youth experiences, they would sensitize all students to the possibility that one of their friends may one day affirm a gender-diverse identity and would facilitate the development of a more positive gender climate.Given individual differences and specific needs for each student, TNB youth should be involved in decision-making processes at school. In our research, Axel, for example, explained how it was helpful to have the school adapt to trans pupils, and doing it ifferent ,42 as reifferent ,44. Thusifferent . As manyifferent ,42,44. EThere are some limitations to the current study. We only selected 14 to 17-year-old teenagers to examine their specific experience in high school settings, resulting in a sample of only twelve participants. Given the limited size of our sample and the complexity of the issues at stake, it is unlikely that the present study achieves data saturation. Additionally, our sample was not culturally and geographically diversified, even though the sample of the overarching study was. Thus, important topics, such as intersectionality in school settings and differences between rural and metropolitan schools, may not have been appropriately captured in our data. Finally, interview questionnaires were not built specifically to assess the complete school experience of TNB youths, but rather to explore TNB youths’ well-being in general. Thus, this study is not fully representative of TNB students’ lived experiences at school and some topics might not have been explored. However, this specific limitation also represents a strength of our study given that TNB youths could spontaneously express which factors they felt had the strongest impact on their well-being. The fact that school-related aspects were discussed so thoroughly speaks strongly to the magnitude of the impact of school factors on TNB youths’ well-being.Our study reveals that openness, validation, and support of gender diversity at school can positively affect TNB youths’ well-being. Conversely, various forms of non-recognition of gender identity, victimization and bullying towards TNB youths impede their wellbeing and should not be tolerated at school. Schools should proactively ensure that they put in place measures that will facilitate the inclusion of gender diverse young people and adopt strategies that respect and affirm youth gender identities. Such measures should pay attention to confidentiality of information to protect students’ privacy regarding their gender identity. While these themes have already been explored in the literature, our qualitative data allowed us to illustrate how they are experienced by young people in the school setting. While our study methodology was not meant to identify all possible adaptations and improvements that could be made to support TNB youth in the school setting, our data nevertheless demonstrate the importance of improving schools’ gender climate. Peer, teacher, school personnel, and principal support is essential and contributes to TNB students’ well-being. While TNB youths should be included in decisions that concern them, schools have the responsibility and potential to make every one of their students, regardless of their gender identity, feel safe and included.


In the original publication, the first and last name of all the authors were inadvertently swapped. It is corrected in this correction.http://creativecommons.org/licenses/by/4.0/.The article “Ranking ecosystem services delivered by trees in urban and rural areas”, written by Patrycja Przewoźna, Krzysztof Mączka, Marcin Mielewczyk, Adam Inglot, Piotr Matczak, was originally published electronically on the publisher’s internet portal on 28 March 2022 without open access. With the author(s)’ decision to opt for Open Choice the copyright of the article changed on 02 May 2022 to @ The Author(s) 2022 and the article is forthwith distributed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit The original article has been corrected.


Recent years have seen a rapid increase in the number of studies using evidence-accumulation models  in the fields of psychology and neuroscience. These models go beyond observed behavior to extract descriptions of latent cognitive processes that have been linked to different brain substrates. Accordingly, it is important for psychology and neuroscience researchers to be able to understand published findings based on these models. However, many articles using (and explaining) these models assume that the reader already has a fairly deep understanding of (and interest in) the computational and mathematical underpinnings, which may limit many readers’ ability to understand the results and appreciate the implications. The goal of this article is therefore to provide a practical introduction to the DDM and its application to behavioral data – without requiring a deep background in mathematics or computational modeling. The article discusses the basic ideas underpinning the DDM, and explains the way that DDM results are normally presented and evaluated. It also provides a step-by-step example of how the DDM is implemented and used on an example dataset, and discusses methods for model validation and for presenting  model results. Supplementary material provides R code for all examples, along with the sample dataset described in the text, to allow interested readers to replicate the examples themselves. The article is primarily targeted at psychologists, neuroscientists, and health professionals with a background in experimental cognitive psychology and/or cognitive neuroscience, who are interested in understanding how DDMs are used in the literature, as well as some who may to go on to apply these approaches in their own work. An important domain in cognitive psychology and cognitive neuroscience is decision-making: the process of recognizing features of the situation in which we find ourselves, considering numerous possible alternative responses, selecting and executing one response, observing the outcomes, and adjusting our behavior accordingly. Disruption to any of these processes can affect decision-making, with real-world consequences; examples include addiction, where individuals make decisions to sacrifice long-term health for short term benefits of the addictive substance or behavior , but abnOne approach to understanding decision-making is through computational models, such as the drift diffusion model DDM; , which wAlthough first described over 50 years ago, the DDM has recently enjoyed widespread use, partly due to the development of powerful and freely-available software implementing computationally-intensive model-fitting algorithms, and partly due to an accumulating literature documenting that the DDM can indeed shed light on latent cognitive processes that are not necessarily evident from traditional hypothesis-driven methods of behavioral data analysis , and thaThe DDM is thus of broad interest in cognitive psychology and cognitive neuroscience , which hThe current article thus aims to provide a reader-friendly introduction to the DDM and its use. The article discusses the basic ideas underpinning the DDM, provides a step-by-step example of how the DDM is implemented and used on an example dataset, and discusses methods for model validation and conventions for presenting  model results. The goal is to provide sufficient background for a reader to critically read and evaluate articles using the DDM – without necessarily mastering the detailed mathematical underpinnings. However, for those readers who wish to go a little deeper, the Supplemental Material provides R script to allow readers to run the examples discussed in the text, and to generate the data figures and tables shown in the article see .Importantly, the current article is not meant to offer a comprehensive review of the DDM literature , Stroop tasks , and saccadic flanker tasks .On such tasks, even well-trained participants show a speed-accuracy tradeoff: they may increase accuracy at the expense of slower (more careful) responding, or make very quick decisions that are more likely to be erroneous . This spTo address these issues, a complementary approach to analyzing the observed behavioral data is to use computational models that attempt to extract latent cognitive parameters that, together, could produce the observed distribution of RT and accuracy data.evidence accumulation models. These models conceptualize decision-making as a process in which, on each trial, individuals accumulate evidence favoring one or another possible response, until enough evidence accumulates to reach a criterion or threshold, at which point a decision is made and the corresponding response is initiated. Evidence accumulation models are sometimes called sequential sampling models, reflecting the idea that the nervous system repeatedly  obtains bits of information (samples) from the environment, until a threshold of evidence is reached. The speed-accuracy tradeoff reflects a balance point determining when to stop sampling the environment and make a decision based on the data at hand.The drift diffusion model (DDM), first described by Ratcliff and colleagues , is one parameters, that can be assigned different values. An easy way to think of parameters is as dials (or control bars) on a stereo system that each control one aspect of the sound , and can be adjusted individually so that together they result in the desired effect. Similarly, parameters in an evidence accumulation model may control aspects such as how fast evidence is accumulated, a built-in bias for one response alternative over another, and a tendency to emphasize speed or accuracy. Each of these parameters can be adjusted in the model, affecting how the model behaves.Like all computational models, the DDM is defined by a series of mathematical equations, containing a number of In the sections below, we walk through major steps in the modeling process, following the flow-chart in encode the stimulus (often denoted Te), the time to reach a decision about how to respond to that stimulus (Td), and the time required for the nervous system to execute the chosen motor response (Tr). Thus, on a given trial, the observed reaction time is the sum of these three components: RT = Te + Td + Tr.The DDM starts with the assumption that the RT on each trial, defined as the time from stimulus onset until execution of the motor response, can be decomposed into three parts : the timTe and Tr separately, normally the encoding and response time are lumped together into a single parameter representing non-decision time (Ter): that portion of the RT that occurs independently of the decision-making process Td. Given this simplification, RT = Ter + Td. Typical values of Ter lie in the range 0.1–0.5 s, partly depending on the complexity of stimuli and the specific motor responses involved . It’s usually assumed that Ter may differ across individuals, but is relatively constant across trials for one individual performing one task.Although it might in principle be possible to measure Td), which is the time to make a decision . On a single trial, noisy information is accumulated across time while the process travels along a corridor bounded by the two possible responses . Larger values of a mean that the decision-making process must travel further (up or down) to reach a boundary. The effect of larger a is thus that decision-making will be slower and more cautious: slower because more evidence is required before a distant boundary is reached and a response is triggered, and higher accuracy because it will be rare for the decision process to “mistakenly” cross the wrong boundary (a) or accuracy (high a).By convention, the lower boundary is assigned a value of 0 on the boundary . Boundarz) that ranges from 0 (lower axis) to 1 (upper axis). If z = 0.5, the starting point is equidistant from the two boundaries. However, if z approaches 1 (or 0), the decision process starts off close to the upper (or lower) boundary on every trial, meaning that less information is required in order to reach that boundary and initiate the corresponding response. The starting point z therefore reflects a response bias in favor of one or the other response.On each trial, the decision process starts from a location on the y-axis defined by a parameter denoting a relative starting point (Td are schematized as multiple red lines in d), schematized as the slope of the heavy black line in The decision-making process in the DDM is assumed to be noisy (schematized by the jagged red line in a), response bias (starting point z), information processing speed (drift rate d), and non-decision time (Ter). These parameters are sometimes called “free parameters,” in the sense that they can take on different values (“freely”) – and just like the knobs on a stereo, changing each parameter affects DDM behavior.As summarized in For example, let us consider a task, schematized in z) will move the starting point closer to the upper boundary, meaning that the evidence accumulation process has farther to travel to reach the r1 boundary than to the r2 boundary, making it easier (and faster) to decide in favor of r2 on any trial. Such a prepotent response bias for r2 might be created if, say, r2 responses are much more frequent or highly-rewarded in the task.As shown in a) would make both responses faster, without necessarily favoring one over the other. It also increases the error rate, because it’s easy to for noise to push the decision-making process across either boundary. A reduced boundary separation might happen if, say, the subject had been instructed to respond quickly, even at the expense of reduced accuracy. Increasing a would have the opposite effect of increasing response caution and producing slower RTs.As shown in d.s2) would result in faster evidence accumulation on s2 trials, while decreasing the other  would result in slower evidence accumulation on s1 trials. Drift rates are typically slower (less steep) under more difficult task conditions, with decreased stimulus discriminability, or in the presence of distracting stimuli.As shown in Ter would affect overall RT, without otherwise affecting the decision-making process. For example, patients with motor dysfunction might have increased Ter , independent of decision-making considerations.Finally, increasing or decreasing non-decision time Ter, a, z, and d interact to affect overall task performance, including both accuracy and RT. To use the stereo example again, the auditory effect of changing one parameter (bass) may be very different depending on whether the value of another (volume) is low or high. To understand these complex interactions, rather than using schematics such as Ter, a, z, and d. The model is then applied to the behavioral task: On each trial, the model is presented with a stimulus , and the diffusion process starts from z and is allowed to run , resulting in reaction time RT = Ter + Td. Over a series of such trials, usually the same stimuli in the same order as experienced by a human participant, the model’s responses and RTs are recorded, producing “simulated data.” The accuracy and RT distributions in the simulated data can then be compared against the accuracy and RT distributions in the empirical data, to see how well the model replicates or “fits” the empirical data. Typically, the goal is to fine-tune the parameter values until the model’s accuracy and RT distributions are as close as possible to the empirical data – a process called model-fitting or parameter estimation.Together, the values of the DDM parameters Ter, a, z, and d. More elaborate versions can be considered. For example, in many cases, it makes sense to consider different drift rates for different trial types or conditions. For example, a lexical decision task might have two types of stimuli (s1 = nonwords and s2 = words) but also have easy and hard trials, depending on whether the trigrams are visually degraded or not, or whether the non-words are pronounceable or not. In such cases, it may make sense to allow different drift rates for each combination of stimulus conditions, with the expectation that (for most participants), there will be a steeper drift rate for trials under the easy condition than the harder condition. In the schematic of d.s1 and d.s2), we might have one for each configuration of stimulus and condition .Before going on, it’s worth noting that the above description considers a “standard” DDM with four free parameters Versions of the DDM have also been considered that include additional free parameters specifying the amount of trial-by-trial variation in the DDM parameters ; howeverThe purpose of performing model-fitting to estimate parameter values is to provide some insight into the underlying cognitive processes. These processes are latent in the sense that we cannot observe them directly, only impute them based on the participant’s pattern of behavior. These latent processes may be very important for understanding the cognitive neuroscience of decision-making. For example, they may map onto different brain systems, and may vary in principled ways in patient groups with different neuropsychological disorders.In a computational model, we have the advantage that those cognitive processes are made explicit, in the form of the parameter values governing the model’s behavior. This can provide a way to identify specific cognitive mechanisms that underlie group differences.a, corresponding to greater response caution , affects Ter but not the other parameters  but also with increased response caution  do not discriminate groups . The DDMIn sum, there is a considerable and growing body of literature using the DDM to elucidate cognitive processes that affect decision-making.Suppose that we are going to collect data from human participants on a simple task with two stimuli (or classes of stimuli) s1 and s2 that map to two responses r1 and r2, respectively. The task has 500 trials, including 150 s1 and 350 s2 trials, so that s2 (and r2) occur more frequently than s1 (and r1). We will assume that participants could realistically achieve 100% accuracy, but the requirement to respond as quickly as possible introduces errors due to the speed-accuracy tradeoff. We might be interested in comparing two groups of participants , and we might plan to analyze the behavioral data using one ANOVA  to compare accuracy across groups, and another to compare RTs across groups. However, being aware of the speed-accuracy tradeoff, and also because we are interested in the underlying cognitive processes that produce any observed group differences, we also plan to apply a computational model, the DDM.Ter, boundary separation a, response bias z, and (at least one) drift rate d. Given that our task involves two different stimuli (or classes of stimuli) s1 and s2, we likely want to allow separate drift rates (d.s1 and d.s2) for trials with our two types of stimuli.Having decided to use the DDM, the first question is: which free parameters will we consider? As noted in d.s1 = −d.s2: i.e., the two drift rates are of equal steepness, but slope in opposite directions, one down to r1 and the other up to r2). Our resulting model would therefore have only three free parameters that could vary across subjects: Ter, a, and a single drift rate d. Similarly, if we think there is no reason to assume a response bias, we might consider consider fixing z at 0.5. In this case, z would no longer be “free” to vary across participants, but would have the same value for everyone.Both simpler and more complicated models are possible. For example, we already noted above an example where we might want multiple drift rates corresponding to multiple task conditions. On the other hand, if we think s1 and s2 should be equally discriminable, we might assume only a single drift rate  would add (at least) tens of msec to this lower bound. However, empirical data files often contain a few “very fast” RTs of <100 msec, which could be the result of anticipatory responding initiated before stimulus onset, or even a very delayed response from a prior trial. There may also be “extremely slow” RTs, which could occur because the participant is briefly inattentive . Unfortunately, such outlier RTs can strongly influence the outcome of hypothesis tests on RT data . The choice of appropriate cutoff criteria for a specific study will of course vary, but common cutoffs are often RT < 100 or 200 msec and RT > 1 or 1.5 s; studies have suggested that findings may be relatively robust to minor differences in the exact cutoff values e.g., .ad hoc, but may greatly reduce power and can introduce biases of their own ; given that RTs in the data seem to range from about 0.25–1.0 s, we might estimate non-decision time at Ter = 200 msec ; for drift rate, we might set d.s1 < 0, reflecting that the evidence accumulation on s1 trials should proceed downward (toward r1), and d.s2 > 0, so that evidence accumulation on s2 trials should proceed upward (toward r2). Here, for illustration, we’ll set d.s1 = −1 and d.s2 = +1.25.We start by proposing some “reasonable” values for the DDM parameters. For example, we might set boundary separation at an arbitrary value of z with a boundary separation of a and drift rate of d.s1, allow the diffusion process to operate until a boundary is crossed at time Td, and then record both accuracy (was the correct response “r1” generated?) and overall RT (Ter + Td) for this trial. In this example, we find that the response r1 is indeed chosen, with an RT of 0.382 ms.To simulate a single trial with stimulus s1, we plug these parameter values into the DDM equations, start the evidence process at Since the evidence accumulation process in the DDM is noisy, we would repeat a second time, and get a slightly different RT . In all, we repeat 500 times , resulting in distributions of predicted RTs for correct and incorrect responses, and a predicted accuracy rate (how often the correct response was selected). These predictions (from the DDM) are then compared to the accuracy and RT distributions in the empirical data (from the participant) to determine how well the model fits the empirical data.For example, In this example, on s1 trials, the DDM chose the correct response (r1) 77% of the time – very close to the 76% accuracy in the empirical data. On s2 trials, the DDM chose the correct response (r2) 75% of the time – a little lower than the 83% accuracy in the empirical data. The histograms for the predicted data also look similar to those from the empirical data, although not perfect: for example, the mean for correct responses to s1 is about 0.44 s, which is a little faster than the mean 0.54 s in the empirical data.We could also plot PDFs and CDFs of the predicted data, to see how closely these overlap the PDFs and CDFs from the empirical data. For example, the empirical data could be divided into quantile “bins,” for example, based on the RT at which the fastest 10%, 30%, 50%, 70%, 90%, and 100% of RTs have occurred. For our test dataset, this would result in an approximation to a degraded CDF shown by the black lines in a from 1 to 1.01 – and run the DDM again to see if the new combination of parameter values provided a better approximation to the participant’s data. After iterating over a large number of possible parameter values, we would obtain the best possible fit: a set of values for a, z, Ter and drift rates d.s1 and d.s2 that together allow the DDM to most closely replicate the empirical data.All of the above has allowed us to assess how well the DDM reproduces a single participant’s data, given a single set of parameter values in the model. We could then slightly perturb one or more of our parameter values – say, changing Needless to say, estimating the optimal values for multiple free parameters is a formidable computational challenge, one that is complicated by the fact that changes to one parameter may affect the optimal value of another parameter. Fortunately, several methods have been devised, many of which are currently available as open-access software packages.This section reviews several methods of parameter estimation for DDM that have been widely used in the literature, and that have been implemented as freely-available software.χ2 method , then the model is said to provide a good fit to the empirical data.An early, and mathematically tractable, approach to estimating DDM parameters is the od e.g., , which cod e.g., . For thoχ2 – minimizing the difference between empirical data and DDM predictions. Without delving too deeply into the mathematical methods, suffice to say that this can be done via an optimization routine. Many optimization routines involve iterative search: start with an initial rough estimate of the parameter values, calculate how well the model using those values fits the data, and then iteratively perturb one or more of the values, to see if this provides a better fit: if yes, then the new values are adopted for the next pass; if no, then the old values are retained and a different perturbation is tried. Early in the search, when the fit is likely to be poor, minor perturbations of parameter values may produce large improvements in fit; but the search is said to converge when values for all parameters have stabilized and no further perturbations can be identified that improve fit by more than a very small value . At this point, the value of χ2 is returned as a metric of goodness-of-fit, and the corresponding parameter values are taken as the optimal or “best-fitting” parameter estimates for the empirical data.The goal is to identify a set of DDM parameter values that, together, minimize χ2 method is one of the estimation methods instantiated in the freely available fast-dm package  would not dramatically affect the results. However, the χ2 approach requires a large number of trials to produce a reliable estimate  and may be especially problematic if there are relatively few error responses  to generate estimates for each parameter. MLE may be most familiar to readers as a means to identify parameter values (beta weights) in regression models, to minimize difference (error) between the predicted and empirical outcomes. The principle is the same here.L; more commonly, researchers report the log of that value, which is called the log-likelihood estimate LLE.Formally, MLE tries to find a set of parameter values that, together, maximize the probability that the outcome of the model matches the empirical data on all trials. This probability is referred to as the likelihood estimate, LLE for the model applied to a given dataset. Again, this is normally done by optimization routines that work by iterative search: constructing a (possibly random) set of “starting values” for the parameters and evaluating LLE, then perturbing one or more parameters by a small amount and re-evaluating LLE, until no further improvements in LLE can be obtained by perturbing any of the parameters. The goal then becomes to find the set of parameter values that, together, maximize For example, Freely-available software implementing MLE for DDM includes the fast-dm package  as well LLE, even though this may not be the optimal solution. For this reason, it’s often a good idea to run the MLE procedure multiple times, with different starting values, to make sure the same solution is found each time.MLE approaches have been successfully used in a large number of DDM studies, and can be used even when there are relatively few  trials available from each participant ; howeverRecently, the advent of open-source platforms for Bayesian statistics have given rise to a number of Bayesian approaches for estimating DDM parameter values  and a measure of confidence in that estimate . For example, y = 0. For example, whereas the prior estimate for z treated all values in the range from 0 to 1 as equally plausible, the posterior estimate for z has mean 0.60, indicating a mild but definite response bias (toward the upper boundary and r2); the narrow width (SD = 0.2) indicates high confidence that the true value lies in the body of the posterior. In this case, we can say that information from the data has “overwhelmed” the priors, resulting in much more closely specified posteriors.Whereas the Calculating posterior distributions is extremely computationally intensive, and direct solution is generally intractable: i.e., there is no known mathematical way to directly calculate the posteriors from the priors and the data. Instead, approaches such as Markov Chain Monte Carlo (MCMC) methods leverage computer power to generate approximate solutions . At the next iteration, the distribution of one of the parameters is perturbed slightly, perhaps by slightly altering its mean or SD, and new parameter values are drawn at random from the distributions, and the model is run again. If the result is an improvement , then the updated parameter values are used for the next step in the chain; otherwise, the old parameter values are retained. The process is repeated hundreds or thousands of times, until no further improvements are discovered, at which point the process is said to have converged on a solution: a set of distributions (posteriors) for each parameter.In brief, MCMC methods estimate a distribution by repeatedly drawing “sample values” to form a “chain” of values for each parameter. A simple version of MCMC might run as follows: At the first step or iteration of the chain, a value is selected at random for each parameter from the prior distributions, and the resulting model is evaluated , then typically all the parameter values are updated at each iteration . Therefore, each sample in the chain contains updated values for all the parameters.If we are examining a DDM with five free parameters  across iterations is known as convergence. Convergence can be visually assessed by inspecting the traceplots of the chains, which should look like the “fat, flat, hairy caterpillars” of Convergence can also be assessed quantitatively. One widely-used measure of convergence is the Gelman-Rubin Rˆ (“R-hat”) statistic  which asGiven successful convergence, the posterior distributions can be reported e.g., , and/or Freely-available software implementing Bayesian approaches to DDM includes the python-based HDDM  and the χ2 methods, when a limited numbers of trials are available  in the estimation routines and also due to scaling factors adopted by different software packages.via the RWiener package in R) and Bayesian MCMC (via the DMC package in R) were used to estimate the five DDM parameters for our test dataset. As shown in a, z and Ter; estimated values of drift rate differ slightly, but the relative relationships are preserved: d.s1 is in the opposite direction to, and steeper than, d.s2 in both methods.For example, both MLE  is used to generate parameter estimates. For many purposes, the choice will simply reflect the approach and/or software package with which the investigator is most comfortable.a = 1.2, d.s1 = −1.25 (sloping down toward the lower boundary), d.s2 = +1.1 (sloping toward the upper boundary), z = 0.6, and Ter = 0.2. That means that we have the luxury of knowing the correct answer: if the DDM is functioning as advertised – if it is truly able to “infer” parameter values by inspecting the data generated – then the parameter estimates should be very close to the true (or “generating”) parameter values. In fact, Now it’s time for a big reveal: the original “empirical” test dataset shown in  was itseOf course, if this dataset had been generated by a human being, we would not have the luxury of knowing the generating parameters – the entire point of using the DDM would be to attempt to infer these latent parameter values from the empirical data. But the ability of the DDM to accurately recover parameters from a test dataset greatly increases our confidence that that the methods can be applied to this type of data.n = 10 empirical data files obtained from our simple task  using MLE (via the RWiener package in R) to find best-fitting parameters for each data file. The results are shown in LLE obtained for each data file, given those best-fitting parameters.All the work we have done so far has rested on attempting to fit the DDM to a single data file, and a “simulated” data file at that. Now let us assume we were going to fit the model to a group of data files obtained from multiple participants an experiment. For the purposes of example, let us consider a, z, d.s1, d.s2 and Ter) that, together, allow the DDM to most closely approximate each individual participant’s behavior on the task. At this point, there are a few ways in which we should validate our model, bolstering confidence that the DDM is actually discovering parameter values that describe the underlying processes that generated the data.The estimated parameters in Before going any further, the importance of a simple sanity check cannot be overstated. Do the model results even make sense?One type of sanity check has already been mentioned: in the context of Bayesian MCMC methods, researchers often report Rˆ and/or show “hairy caterpillars” to document convergence e.g., , and mayIf the parameter estimation process did not converge, then obviously the parameter estimates cannot be trusted. Sometimes, failure to converge just indicates an unlucky choice of starting point, and if the optimization routine is re-run with a different randomly selected starting point, convergence may be achieved. In this case, it’s customary for authors simply to note how many attempts (re-starts) were required before convergence was achieved.Ter cannot be lower than the empirically-observed behavioral RT (since RT = Ter + Td and Td cannot be less than zero), and boundary separation a is in arbitrary units but cannot be <0 (since it represents a distance or separation). In a, z and Ter all meet these minimal criteria.Assuming convergence, the next step should always be a sanity check of the parameter estimates obtained. For example, non-decision time d.s1 < 0 and d.s2 > 0, meaning that in each case the drift rate slopes toward the correct boundary. The magnitude  of the drift rates suggests that the evidence accumulation process on s1 trials is somewhat steeper than on s2 trials. This might reflect something about the underlying nature of the task . In any case, the drift rates in If there is more than one drift rate, sign (direction) and steepness (slope) should be consistent with observed accuracy and relative response speeds. In In sum, then, our parameter estimates all appear plausible. In general, any parameter values that violate common sense likely indicate that the model has failed, regardless of what fit metrics may be reported.A next important step to establish model validity is a predictive check, in which the parameter estimates obtained from the empirical data are used to generate simulated datasets. Specifically, a DDM with the estimated parameter values is run on the same task  and the model’s predicted response and RT recorded for each trial. This could be done one or more times for each data file, or for a few representative data files selected at random, or even using the group means for the parameter estimates.The simulated data should mimic key features of the behavioral data, such as accuracy and mean/SD of RTs for correct and incorrect responses to each stimulus. PDFs (or RT histograms) and CDFs for simulated data can also be visually compared against the empirical data ; for RT on incorrect responses, the correlation is lower, particularly for s1, reflecting the relatively low number of error responses on which these calculations are based.If the model predicts the empirical data well, the plots for simulated and empirical data for each participant will be highly overlapping. If we have a large dataset, rather than inspect each simulated data file individually, it may be enough to “spot-check” a few representative cases, and then look at some summary statistics, such as mean percent accuracy and median RT in the simulated vs. empirical data. Yet another possibility is shown in prove that they were generated in the same way – but failure would almost certainly argue that the model is not valid. So, predictive checks are a conventional first step in model validation.The fact that the simulated data share many of the features with the empirical data cannot, of course, After generating the simulated data, a parameter recovery study can be conducted, in which the DDM is applied to the simulated data, to see whether the parameter values which generated those simulated data can be correctly recovered by the DDM e.g., .r) between generating and recovered parameters is considered “good” if r > 0.75 or “excellent” if r > 0.90 (r > 0.9).r > 0.90 . For thecapable of correctly recovering parameters given data such as these. If the model cannot even replicate the generating parameters for simulated data, where we know what the true values for each parameter are, then we certainly cannot trust that it is accurately estimating the parameters for human participants, where the generating parameters are not known!A successful parameter recovery study confirms that the model-fitting procedure is able to reliably estimate (“recover”) the parameter values that generated the simulated data. This cannot, of course, guarantee that the model has accurately estimated parameter values from the empirical data, but it does increase our confidence that the model is (at least) The previous section focused on how we find a set of estimated parameter values that provide the “best possible” fit for each file in our empirical data. It’s also important to ask just how good that “best possible” fit actually is. Model selection typically refers to a process of systematically comparing different instantiations (or versions) of that model, with different free parameters, to determine which provides the best way of understanding the data.a, z, Ter and separate drift rates d.s1 and d.s2 for s1 and s2 trials, and we obtained pretty good results, validated both by predictive check and parameter recovery study; but could we have obtained  as good results with a simpler model, say, assuming only a single drift rate d, regardless of stimulus?For example, in the preceding sections, we used a DDM with five free parameters, including Other things being equal, we would typically favor the simpler model with fewer free parameters, as a simpler way to describe the data, and also because models with more free parameters have a higher risk of overfitting the data. Overfitting refers to situations where a model can describe an existing dataset with high accuracy, but does not generalize well to other datasets. This is a concern not only in the DDM, but in all kinds of model fitting, such as linear regression: adding a large number of predictor variables to make a more complex model may result in overfitting the sample data, such that the regression equation obtained makes very accurate predictions on the sample data, but not for new datasets. A simpler regression model, with fewer predictor variables may sacrifice some accuracy but provide better generalization.So, do we really need two separate drift rates in our model? Are the benefits (better model fit) worth the costs ? To answer this question, we need to evaluate two things: first, exactly how “good” a fit does each version of the model provide to the data? Second, is one fit “meaningfully” better than the other?LLE. The maximal LLE is simply the value of LLE that was obtained using the best-fitting parameter values  were presented in LLE obtained for each data file, using those “best-fitting” parameters.For purposes of discussion, let us use the nickname DDM−5 to refer to our DDM with five free parameters: a, z, and Ter but only a single drift rate d to be used on all trials regardless of the stimulus.  For purposes of discussion, let us call this version DDM-4 (DDM with four free parameters).We might then consider a DDM with only four free parameters: LLE obtained under DDM-4 with that obtained DDM-5  be less than or equal to that obtained by DDM-5. This is because any solution explored by DDM-4 (which constrains d.s1 = −d.s2) should also be explored by DDM-5 .A first, important point is that the maximal sufficiently better fit than DDM-4, enough to justify its added complexity?So, the question here is not whether DDM-4 can provide a better fit: we know that it cannot. The question is: can DDM-5 provide a d.s1 and d.s2 that were nearly equal in magnitude, though oppositely signed, and so DDM-4 (where the two drift rates are forced to have the same magnitude) provides just as large LLE as does DDM-5. And so, at least for these three participants, there does not appear to be much “advantage” to using the more complex model.For example, LLE) for participants #1, #2 and #3. Averaged across all 10 participants, DDM-5 does provide numerically better mean LLE than DDM-4: 181.8 vs. 163.5. What we need is a way to quantify whether this 20-unit improvement in LLE is “significant” or “meaningful” – enough to justify our use of the more complex model.On the other hand, the larger DDM-5 provides a much better fit . The smaller AIC, the better; therefore, the addition of 2k to the LLE results in a larger “penalty” (increasing AIC) for models with more free parameters. Using this formula, the mean AIC for DDM-5 is −354, and that for DDM-4 is −319, so we would conclude that the larger model, despite its added complexity, is a better description of the dataset.There are a number of metrics that can be used to address this question. One of the most commonly used is Akaike’s Information Criterion (AIC), which is an attempt to compare x models : specifik as well as the number of trials n in the dataset (BIC = k*ln(n) − 2*LLE; again, lower (more negative) is better. BIC is only valid if n >> k . A nice feature of BIC is that there are conventions for interpreting BIC values , considers number of parameters  dataset : BIC = kC values : as a ruIn our example, DDM-5 has mean BIC of −333 and DDM-4 has mean BIC of −302. The difference is >30, so we conclude that there is very strong evidence favoring the more complex model with separate drift rates.via MCMC methods , which is a generalization of AIC for use when posterior distributions have been obtained  methods , or the  methods , which iIf Bayesian methods have been used, it is also possible to report a Bayes Factor (BF), which is a ratio of the marginal likelihood of the two models, interpretable as the relative strength of the evidence that each model is correct. Values of BF = 1 mean the two models are equally likely, while larger values of BF make us increasingly confident in supporting the first hypothesis (or model). As a rule of thumb, BF > 3 is considered weak evidence and BF > 10 is considered strong evidence e.g., .All these metrics – AIC, BIC, DIC, WAIC, BF – are used to compare how well two models describe the same data file(s). As discussed earlier, there may be some participants for whom one model has a much lower metric than the other model, but some participants where the reverse is true. Often, a decision favoring one model or the other is based on a simple majority vote: which model results in best metrics for a majority of participants. Always, the burden is on the more complex model to justify its use, so if the complex model does not clearly provide a better fit, then the simpler model would likely be preferred.And remember: model selection methods only tell us which of the models under consideration fits the data best. It does not guarantee that any of them are correct .So far, we have estimated best-fitting parameters for our data; we have conducted predictive checks and parameter recovery studies to assure ourselves that DDM-5 can accurately recover parameters for this type of data; and we have compared DDM-5 vs. DDM-4 , and concluded that DDM-5 is the best description of our data, providing excellent fits to the data with no unnecessary complexity .At this point, the parameter estimates from our DDM-5 can  be reported for individual data files and/or summarized for each group . These could be the value estimates for each parameter returned by MLE; or, if Bayesian methods were used, the results could be presented as medians or means of the posterior distributions for each parameter, or even plots of the posterior distributions for each parameter. For example, perhaps subjects #1–5 constitute the control group and subjects #6–10 constitute the experimental group in our study; a, z, and Ter are generally consistent across groups, but that one group has much higher drift rate for s1 than s2, whereas the other group has equivalent drift rates for the two stimulus types. This apparent interaction could be confirmed by ANOVA or other inferential statistics.The point estimates for each parameter can be also subjected to statistical analysis, using analogous methods to those that were used to analyze the behavioral data (accuracy and RT). To illustrate, in our example study, it appears that In the real world, we might be interested in comparing parameter estimates between patients vs. controls . Some researchers have suggested that model parameters can be used as classifiers to distinguish patients from controls, and that addition of parameter estimates can improve classification beyond standard demographic and clinical variables and/or behavioral variables alone e.g.,  July 28.The final step in our study would be publishing the results and, as with any scientific method, it is important that the report be complete enough to allow the reader to critically evaluate the methods, the results, and the authors’ interpretation, including several key questions.χ2, MLE, Bayesian MCMC). Ideally, the authors should provide code for validation/replication. The reader should be able to evaluate whether the model design was appropriate for the cognitive task (and research hypothesis) under study; for example, is there an existing literature using these methods with the study population? If the authors present a new or modified approach, did they explain and justify this process?Reporting requirements for the DDM should, at a minimum, state what free parameters were considered, and what method of model-fitting was used  have different requirements, such as number of responses, number of total trials, and minimum number of correct/incorrect responses per trial type see ; were thBefore presenting parameter estimates from the model, the authors should validate their model. First, did the authors present any theoretical justification for the free parameters being considered? Did they conduct predictive tests to show that the model can, in fact, generate simulated data that at least superficially captures important aspects of the behavioral data? Did they conduct a parameter recovery study to demonstrate that their model, as described, can accurately recover generating parameters? Did they conduct any model selection studies, to examine the effect of adding/deleting free parameters and did these results convincingly support the version of the model that the authors eventually reported?Sometimes, this information is relegated to an appendix or an online supplement, but it is an important part of the modeling process, and the reader should be assured that this was done.Turning now to the model results, which are often parameter estimates such as , usuallypost-hoc, rarely to test an a priori hypothesis. As such, overfitting is always a concern. Ideally, modeling results obtained in one sample should be validated in a new sample, but failing this (or until follow-up studies are conducted), authors can try techniques such as cross-validation (or “out-of-sample” testing), to fit the model to different subsets of the data and test how well it generalizes. At a bare minimum, the possibility of overfitting should be addressed when interpreting model results.A good modeling paper  will be honest about its limitations. At present, the DDM is almost always used sufficient to explain the empirical data; but can never prove that this is the case. Latent cognitive processes remain latent. Ideally, results from the model identify candidate cognitive processes that could then form the basis of future hypothesis-driven studies.A second general limitation of DDM is that it can show that estimated parameters are The DDM also makes many simplifying assumptions about the process of evidence accumulation and decision-making. Simplicity is a virtue, but of necessity leaves out complicating factors that can include variations in attention, emotion, and other processes that may influence decision-making. Standard versions of the models assume that the empirical RT distribution reflects a large number (dozens if not hundreds) of repeated measurements of a well-learned response under constant conditions. If the subject learns new response strategies, or loses attention, as the session proceeds, this assumption may not be valid. This issue is often partially remediated by having a long practice phase before “real” data collection starts, so that the RT and accuracy measurements reflect performance of a well-learned response.The main point of using computational models is (we hope) to uncover information about latent cognitive processes, and possibly to link these latent cognitive processes to brain substrates. So, did the current results actually provide any insights that would not be obvious from the behavior alone? For example, given that one group performed more slowly than another, can we understand this in terms of specific mechanisms such as increased boundary separation, reduced drift rate, and/or increased non-decision time – and if so, does this tell us anything interesting about the group in question? Even more interesting, can these parameters be mapped onto brain substrates or physiological processes?Ter due to motor dysfunction, “compensated” by reducing boundary separation).Importantly, model results can sometimes be informative even in the presence of non-significant behavioral results. For example, even if two groups performed similarly in terms of accuracy and RT, perhaps the model can suggest qualitatively different ways in which the groups solved the speed-accuracy tradeoff  from the U.S. Department of Veterans Affairs Clinical Sciences Research and Development Service. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


This manuscript deals with the qualitative study of certain properties of an immunogenic tumors model. Mainly, we obtain a dynamically consistent discrete-time immunogenic tumors model using a nonstandard difference scheme. The existence of fixed points and their stability are discussed. It is shown that a continuous system experiences Hopf bifurcation at one and only one positive fixed point, whereas its discrete-time counterpart experiences Neimark–Sacker bifurcation at one and only one positive fixed point. It is shown that there is no chance of period-doubling bifurcation in our discrete-time system. Additionally, numerical simulations are carried out in support of our theoretical discussion. CTL), and natural killer (NK) cells show a leading character. Various scientific models of the interactions between the increasing tumor and immune system have been established h passes through h crosses the horizontal line at First, assume that m system  we have h, in h certainly passes through In this case, we have 7952 see c. Secondb as the control parameter, it can be observed from b lies between b crosses the horizontal line at Then, from system , we haved system  is proviFinally, to discuss the dynamics of system , we takeh is taken as a bifurcation parameter, the discrete-time version, which is obtained using a nonstandard finite difference scheme, experiences Neimark–Sacker bifurcation about its positive fixed point h passes throughHere, we have considered an immunogenic tumor model for discretization and qualitative study. The original model  was pres