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ERROR: type should be string, got "https://doi.org/10.15362/ijbs.v27i0.395  W. Fielding & V. Ballance, 2021.\nJournal compilation International Journal of Bahamian Studies, 2021. University of The Bahamas University of The Bahamas Abstract This study investigates the mental health of victims of sexual abuse, particularly rape, in college \nstudent populations. The study identifies the connection between sexual abuse/rape and elevated \nscores of mental health. It indicates that even with other stressful events occurring in the lives of \nrespondents, sexual abuse/rape has a detrimental effect on the mental health of both males and \nfemales. A death in the family was the most commonly reported stressful event for males and \nfemales, and females were more likely than males to have suffered from a sexual attack. Early \nnegative experiences of sexual intercourse can apparently have long lasting negative effects on the \nvictim’s mental health. These findings require society to look beyond the physical consequences \nof sexual abuse and rape to ensure that the long-term mental health of victims, both male and \nfemale children and adults, is not overlooked. ORIGINAL ARTICLES \nMental Health and Negative Sexual Experiences of \nBahamian University Students Elizabeth J. Williams \nWilliam J. Fielding \n 0000-0001-5433-9673 \nVirginia C. Ballance \n 0000-0003-1067-8205 Introduction Bahamas highlighted that many victims of \nrape are unaware that they have been raped \nand that females can engage in unwanted \nsexual intercourse due to being afraid of their \nintimate partner. Bethel and Fielding (2020) \nalso found that men can be forceful in getting \ntheir intimate partners to participate in sexual \nintercourse, even when it is unwanted. They \nalso found that college students still \nsubscribe to myths with regard to sexual \nintercourse, such as strangers are most likely \nto rape a woman. Although there have been many studies do\nin The Bahamas on risky sexual behavio\n(Deveaux & Rolle, 2016), other aspects\nsexual abuse and rape appear to have be\nless studied. Gender-based violence in T\nBahamas is a national concern. The relativ\nhigh incidence of rape has been recogniz\nthrough the Bahamas National Task Force \nGender-based \nViolence \nreport \n(201\nAlthough women in particular are worr\nabout being victims of rape, their fear be\nno relationship to the reported crime figu\non rape. This suggests that even a singu\nevent causes great worry, and/or th\npresume that rape is more common than \nofficial figures suggest, a position that\nconfirmed by both Aranha (2016) and Bet\nand Fielding (2020). The study by Bethel a\nFielding (2020) on college students in T\nTable 1 \nNumber of Rapes, by Sex, Reported t\nYear \nMale \nn = \nFe\n2017 \n0 \n2018 \n4 \n2019 \n3 \nNote. Figures subject to change, Royal Baham\nBethel and Fielding (2020) found in th\nstudy that none of the men who were rap\nreported their victimization, so they would\nunlikely to benefit from any formal supp\nto cope with their trauma. Even though m\nrape is less common than female rape\nshould not be overlooked because it may\neven more traumatic for men than wom\nMusevenzi and Musevenzi (2018) illustr\nthis in their study of men in Zimbabwe w\nwere raped and fear being seen as femin\nand no longer being masculine. Afric\nchildren’s experiences of physical violen Although there have been many studies done \nin The Bahamas on risky sexual behaviours \n(Deveaux & Rolle, 2016), other aspects of \nsexual abuse and rape appear to have been \nless studied. Gender-based violence in The \nBahamas is a national concern. The relatively \nhigh incidence of rape has been recognized \nthrough the Bahamas National Task Force for \nGender-based \nViolence \nreport \n(2015). Dedication Dr. Elizabeth Williams joined The College of The Bahamas on September 1, 2004 as a lecturer in \nthe Critical Care Nursing programme. Her doctoral dissertation was entitled The Influence of \nKnowledge and Self-efficacy on Bahamian Women’s Adherence to a Hypertensive Medication \nRegime (2011) which recognized the importance of hypertension in our community. Her gentle \nmanner made her a favourite with students, who voted her as the best lecturer in the School of \nNursing and Allied Health Professions. Her love of research was evident in the way she taught the \nhigher-level research classes and her enthusiasm and dedication made her a popular teacher of \nresearch. She encouraged her students to study important topics, and this last research project of \nhers is a testament to her willingness to address “hard” issues which impact women’s health. This paper, dedicated to the life and memory of Dr. Elizabeth Williams, is the project that she was \nworking on before her untimely death. The completion of this project would not have been possible \nwithout the exceptional support of the co-authors. I hope that the findings of this study will fill an \nimportant gap in our knowledge of mental health in The Bahamas and add to the current body of \nknowledge. Ingrid Mobley, Ph.D., ARNP https://doi.org/10.15362/ijbs.v27i0.395  W. Fielding & V. Ballance, 2021. Journal compilation International Journal of Bahamian Studies, 2021. 2 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. Bahamas highlighted that many victims of \nrape are unaware that they have been raped \nand that females can engage in unwanted \nsexual intercourse due to being afraid of their \nintimate partner. Bethel and Fielding (2020) \nalso found that men can be forceful in getting \ntheir intimate partners to participate in sexual \nintercourse, even when it is unwanted. They \nalso found that college students still \nsubscribe to myths with regard to sexual \nintercourse, such as strangers are most likely \nto rape a woman. Method poorer mental health among women (Potter \net al., 2020) and may result in depression and \nanxiety (Nickerson et al., 2013). The long-\nterm effects of child sexual abuse on the \nmental health of adult females include anger, \ndepression, and sexual problems (Hailes et \nal., 2019). An Internet-based survey was devised that \nincorporated some of the questions asked by \nBethel and Fielding (2020). These included \nstandard \nitems \non \nrespondents’ \ndemographics as well as questions about their \nlifetime experiences of sexual intercourse \nand their attitudes towards rape. The \nquestionnaire also included 19 aspects \nrelating to the mental health of the \nrespondent. These \naspects \nincluded \nstatements such as, “I wish I were somebody \nelse,” “I feel like I am inferior to others,” and \n“I blame myself for being sexually abused.” \nThe writing authors believe that these \nquestions may have been drawn from a \nnumber of stress/trauma related scales (such \nas \nAmirkhan, \n1990; \nSharma, \n2018; \nSwahnberg & Wijma, 2003) and possibly \nreworded to be culturally appropriate for a \nBahamian student population. The scale was \na Likert response scale with a frequency \nresponse scale from 1 (never) to 5 (always), \nso the minimum score in this scale was 19 \nand the maximum score 95. In this study, the \nCronbach’s α for the scale was .94. The \nsurvey included additional questions about \nevents in the life of participants that may have \nbeen stressful, such as a death in the family \nor being a victim of crime. An Internet-based survey was devised that \nincorporated some of the questions asked by \nBethel and Fielding (2020). These included \nstandard \nitems \non \nrespondents’ \ndemographics as well as questions about their \nlifetime experiences of sexual intercourse \nand their attitudes towards rape. The \nquestionnaire also included 19 aspects \nrelating to the mental health of the \nrespondent. These \naspects \nincluded \nstatements such as, “I wish I were somebody \nelse,” “I feel like I am inferior to others,” and \n“I blame myself for being sexually abused.” In the Caribbean, Pilgrim and Blum (2012) \nidentified a number of environmental factors, \nincluding peers who engaged in violence, as \nhaving negative influences on mental health; \nthe importance of mental health issues has \nresulted in the Caribbean medical literature \ndevoting special issues to the topic (Barton, \n2012). Introduction Although women in particular are worried \nabout being victims of rape, their fear bears \nno relationship to the reported crime figures \non rape. This suggests that even a singular \nevent causes great worry, and/or they \npresume that rape is more common than the \nofficial figures suggest, a position that is \nconfirmed by both Aranha (2016) and Bethel \nand Fielding (2020). The study by Bethel and \nFielding (2020) on college students in The Another rape related myth that persists in The \nBahamas is that only females can be victims \nof rape (Rolle, 2020). Unpublished statistics \nprovided by the Royal Bahamas Police Force \n(see Table 1) confirm that male rape occurs, \nalthough it is accepted that male and female \nrape is under-reported (Bethel & Fielding, \n2020). Table 1 \nNumber of Rapes, by Sex, Reported to the Royal Bahamas Police Force \nYear \nMale \nn = \nFemale \n n = \nMales raped \n% \n2017 \n0 \n56 \n0% \n2018 \n4 \n56 \n6.7% \n2019 \n3 \n44 \n6.4% \nNote. Figures subject to change, Royal Bahamas Police Force (unpublished). Number of Rapes, by Sex, Reported to the Royal Bahamas Police Force and sexual violence have been related to risky \nsexual behaviour and mental health concerns \n(Smith et al., 2020). This demonstrates that \nadverse sexual experiences in both boys and \ngirls can be detrimental to their mental \nhealth. It has also been demonstrated in the \nUnited States that forced sexual intercourse \nhas detrimental effects beyond physical and \nmental trauma, such as negative academic \noutcomes—effects that can be life-long and \nhave implications for the development of a \ncountry (Rees & Sabia, 2013). In the United \nStates, the experience of rape was linked to Bethel and Fielding (2020) found in their \nstudy that none of the men who were raped \nreported their victimization, so they would be \nunlikely to benefit from any formal support \nto cope with their trauma. Even though male \nrape is less common than female rape, it \nshould not be overlooked because it may be \neven more traumatic for men than women; \nMusevenzi and Musevenzi (2018) illustrate \nthis in their study of men in Zimbabwe who \nwere raped and fear being seen as feminine \nand no longer being masculine. African \nchildren’s experiences of physical violence International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences 3 Findings There were 1,240 starts in the survey. One \nperson who entered the survey declined to \nparticipate. After cleaning the data so that \nonly respondents who were members of the \ntarget \npopulation \nwere \nincluded, \n865 \nrespondents were retained. Not all surveys \nwere completed, so this is the maximum \nnumber of respondents. Two respondents \ngave their sex as “other” and this group was \nomitted due the small number of responses. Of the remainder, most respondents were \nfemale (82.1% of N = 860). Although most of \nthe results are presented disaggregated by \nsex, in the case of males, in some cases the \nnumber of responses was small, so we then \njust present the results from the female \nparticipants. The modal age group was 21-24 \nyears (42.3%). Overall, 84.4% of participants \nhad participated in sexual intercourse; this Method After cleaning the data so that \nonly respondents who were members of the \ntarget \npopulation \nwere \nincluded, \n865 \nrespondents were retained. Not all surveys \nwere completed, so this is the maximum \nnumber of respondents. Two respondents \ngave their sex as “other” and this group was \nomitted due the small number of responses. Of the remainder, most respondents were \nfemale (82.1% of N = 860). Although most of \nthe results are presented disaggregated by \nsex, in the case of males, in some cases the \nnumber of responses was small, so we then \njust present the results from the female \nparticipants. The modal age group was 21-24 \nyears (42.3%). Overall, 84.4% of participants \nhad participated in sexual intercourse; this \nTable 2 \nMean Ranked Fear of Being a Victim of Se\nCrime, mean rank \nBeing raped \nBeing shot \nBeing sexually abused \nBeing held up and robbed \nBeing physically attacked by another \nHaving your home broken into \nHaving your car stolen \nHaving something stolen from your car \nNote. 1= most fearful, 8 least fearful; p values from \nAttitudes Toward Rape \nFemales were more aware than men of the \nimportance of consent with regard to sexual Context of the Fear of Rape Fear of rape is one of many anxieties, so \nrespondents were asked to rank their \nconcerns; they were more concerned about \ncrimes against the person than property (see \nTable 2). The fear of being a victim of crime \nwas dissimilar between the sexes. Table 2 \nshows that rape is the crime most feared by \nwomen, whereas men are most fearful of \nbeing shot. As demonstrated by Bethel and \nFielding (2020), the fear of crime does not \nreflect the statistical occurrence of crimes \nreported by the Royal Bahamas Police Force \n(2020). Table 2 \nMean Ranked Fear of Being a Victim of Selected Crimes Table 2 \nMean Ranked Fear of Being a Victim of Selected Crimes \nCrime, mean rank \nMale \nN ≈ 147 \nFemale \nN ≈ 665 \np = \nM \nM \nBeing raped \n4.2 \n2.6 \n< .001 \nBeing shot \n3.0 \n3.6 \n< .001 \nBeing sexually abused \n5.3 \n3.8 \n< .001 \nBeing held up and robbed \n3.5 \n4.2 \n< .001 \nBeing physically attacked by another \n4.2 \n4.3 \n.47 \nHaving your home broken into \n4.0 \n4.7 \n< .001 \nHaving your car stolen \n5.2 \n5.9 \n< .001 \nHaving something stolen from your car \n6.6 \n6.8 \n.081 \nNote. 1= most fearful, 8 least fearful; p values from t-test. International Journal of Bahamian Studies Vol. 27 (2021) \nTable 2 \nMean Ranked Fear of Being a Victim of Selected Crimes \nCrime, mean rank \nMale \nN ≈ 147 \nFemale \nN ≈ 665 \np = \nM \nM \nBeing raped \n4.2 \n2.6 \n< .001 \nBeing shot \n3.0 \n3.6 \n< .001 \nBeing sexually abused \n5.3 \n3.8 \n< .001 \nBeing held up and robbed \n3.5 \n4.2 \n< .001 \nBeing physically attacked by another \n4.2 \n4.3 \n.47 \nHaving your home broken into \n4.0 \n4.7 \n< .001 \nHaving your car stolen \n5.2 \n5.9 \n< .001 \nHaving something stolen from your car \n6.6 \n6.8 \n.081 \nNote. 1= most fearful, 8 least fearful; p values from t-test. Attitudes Toward Rape \nFemales were more aware than men of the \nimportance of consent with regard to sexual \nintercourse, and men were more likely to \nview rape as requiring physical resistance by Table 2 \nMean Ranked Fear of Being a Victim of Selected Crimes Method The 2017 World Health Organization \nreport on mental health indicates that, in The \nBahamas, many of its indicators have not \nbeen reported on, and there appears to be a \nlack of mental health programmes (World \nHealth Organization, 2017). The literature on violence and rape often \nfocusses on university students living on \nresidential campuses. However, the study \npopulation in this research project, although \nuniversity-aged students, typically live in \ndomestic \nsettings, \nnot \nstudent \naccommodation, so their experiences can be \nexpected to reflect those of non-resident \ncampus populations, such as that of students \nattending community colleges (Voth Schrag \n& Edmond, 2018; Potter et al., 2020). Therefore, the responses of participants \nshould not be assumed to reflect sexual \nexperiences in a specific place. The target population was enrolled college-\nlevel students in The Bahamas. Students from \na nursing research class used their social \nmedia contacts to recruit students, with credit \nbeing given to students for getting students to \nparticipate in the survey via a link to the \nSurveyMonkey™ questionnaire. Although \nsuch a student population does not reflect the \nwider population, its members are drawn \nfrom across the country, particularly New \nProvidence, so it predominately represents an \nurban population. The students reflect a range \nof economic backgrounds, and many receive \nfinancial aid from the government to attend \nuniversity. The study was approved by the The purpose of this cross-sectional study was \nto focus on the mental health of victims of \nrape or sexual abuse in college students. It \naimed to identify links, if any, between being \na victim of rape and the victim’s mental \nhealth. It also aimed to assess the impact of \nother traumatic events to put the event of rape \nin a wider context. International Journal of Bahamian Studies Vol. 27 (2021) 4 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences Institutional Review Board of University of \nThe Bahamas. percentage was similar for both sexes (χ2 = \n.54, df = 1, N = 858, p = .816). However, \nfemales were more likely than males to be \nparticipating in stable relationships (sexually \ninvolved with the same partner for 12 months \nor more): 59.1% compared with 47.3% of \nmales (χ2 = 7.03, df = 1, N = 853, p = .008). International Journal of Bah\nInstitutional Review Board of University of \nThe Bahamas. Findings \nThere were 1,240 starts in the survey. One \nperson who entered the survey declined to \nparticipate. Table 3 Table 3 \nPercentage of Respondents, Within Sex, Agreeing to Various Aspects of Rape \nAspects of rape \nMale Female \nχ2 \n \n% \n% \np = \nRape occurs when one of those engaged in the sexual intercourse \ndid not consent \n92.1 \n96.3 \n.019 \nRape occurs when one of those engaged in sexual intercourse \nconsents due to threats or fear of bodily harm from the other person \n86.8 \n85.3 \n.483 \nA married couple cannot rape each other \n39.6 \n43.8 \n.221 \nRape only occurs when the victim tries to fight off their attacker \n15.9 \n9.7 \n.011 \nOnly females can be raped \n4.0 \n1.3 \n.085 p = .82). Males tended to engage in sexual \nintercourse at an earlier age than females (see \nFigure 1). We also note that 30.7% of males \nand 15.7% of females who had participated \nin sexual intercourse did so before age 16. p = .82). Males tended to engage in sexual \nintercourse at an earlier age than females (see \nFigure 1). We also note that 30.7% of males \nand 15.7% of females who had participated \nin sexual intercourse did so before age 16. Attitudes Toward Rape intercourse, and men were more likely to \nview rape as requiring physical resistance by Females were more aware than men of the \nimportance of consent with regard to sexual International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 5 the victim to make the unwanted sex rape (see \nTable 3). Close to 40% of both males and \nfemales agreed that rape cannot occur within \nmarriage; this indicates the considerable \ndisagreement on this topic in a society where \nthe law recognizes rape only outside of \nmarriage, as demonstrated in debates on \nmartial rape (Benjamin & LeGrand, 2012). Some respondents in this study indicated that they may have been sexually abused by their \nhusbands. Although this study did not ask \nabout marital status directly, when we \ngrouped respondents who were aged over 30 \nand in long-term relationships (one year or \nmore) together with those who were married, \n7.2% of these 69 females reported having sex \nagainst their will with their partner in their \nmost recent sexual encounter. Participation in Sexual Intercourse Similar percentages of males (16.2%) and \nfemales (15.5%) had never participated in \nsexual intercourse (χ2 = .054, df = 1, N = 858, International Journal of Bahamian Studies Vol. 27 (2021) \nFigure 1 \nPercentage of Males and Females by Age at First Experience of Sexual Intercourse \n \n0%\n5%\n10%\n15%\n20%\n25%\nUnder 14\n14\n15\n16\n17\n18\n19-24\nOver 24\nMale\nFemale International Journal of Bahamian Studies Vol. 27 (2021) 6 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. Males and females reported having different \nexperiences of their first participation in \nsexual intercourse. Females were more likely \nthan males to agree that their first experience \nof sexual intercourse was not agreeable and, \non reflection, was abuse (see Table 4). However, it should be noted that both males \nand females felt that they had been abused \nwhen they first had sexual intercourse and \nboth males and females had various degrees \nof negative experiences. Table 4 \nPercentage of Respondents Agreeing to Various Aspects of their First Experience of \nSexual Intercourse \nFirst experience of sexual intercourse \nMale \nFemale \nΧ2 \n \n% \n% \np = \nI had sexual intercourse against my will \n4.8 \n12.6 \n.048 \nI found the experience enjoyable \n73.6 \n41 \n< .001 \nI did not enjoy it but my partner did \n13.6 \n38.1 \n< .001 \nI clearly agreed to have sexual intercourse \n93.5 \n78.3 \n.001 \nAfterwards, I wished I had not agreed to the sexual \nintercourse \n9.6 \n37.0 \n< .001 \nI knew the person well with whom I had sexual intercourse \n76.6 \n88.0 \n.017 \nAfterwards, I wanted to repeat the experience \n80.6 \n45.8 \n< .001 \nOn reflection, I feel that I was sexually abused when I first \nhad sexual intercourse \n5.6 \n14.1 \n.009 Respondents Agreeing to Various Aspects of their First Experience of Although clear consent prior to having sex \nwas given by most males and females, \nrespondents indicated that they sometimes \nwere unable to give consent or gave consent \nout of fear of their partner (see Table 5). Several differences in the experiences \nsurrounding sexual intercourse were reported \nby males and females as seen in Table 5. Females reported being hurt by their partners, \n23.6% of females had been hit by their \nintimate partners. Participating in sexual \nintercourse under circumstances of fear or \nincapacitation invalidates the consent and \nmakes it rape. Participation in Sexual Intercourse Table 6 \nPercentage of Respondents Agreeing to Various Aspects of their Most Recent \nExperience of Sexual Intercourse \nAgreeing to \nMale Female \nΧ2 \n \n% \n% \np = \nI had sexual intercourse against my will \n4.1 \n5.7 \n.273 \nI found the experience enjoyable \n87.9 \n82.4 \n.163 \nI did not enjoy it but my partner did \n11.3 \n12.5 \n.197 \nI clearly agreed to have sexual intercourse \n96.8 \n91.0 \n.072 \nAfterwards, I wished I had not agreed to the sexual intercourse \n7.4 \n11.4 \n.055 \nI knew the person well with whom I had sexual intercourse \n91.0 \n92.0 \n.63 \nAfterwards, I wanted to repeat the experience \n81.8 \n80.3 \n.202 \nOn reflection, I feel that I was sexually abused when I had sexual \nintercourse \n2.5 \n3.7 \n.299 \nTable 7 \nChoice Sex of Intimate Partner by Sex of Respondent. Sex of those with whom respondents had sexual intercourse \nMale % Female % \nOnly male \n4.2 \n87.0 \nOnly female \n89.2 \n0.7 Table 5 \nPercentage of Respondents Agreeing to Various Aspects of their Experiences of Sexual \nIntercourse \nAspect \nMale \nFemale \nΧ2 \n \n% \n% \np = \nDo you always give clear consent (verbal or non-verbal permission) \nbefore having sexual intercourse? 79.8 \n81 \n.61 \nHave you ever had sexual intercourse with an individual because \nyou were afraid of them? 1.7 \n15 \n< .001 \nHave you ever had sexual intercourse when you were physically or \nmentally unable to give consent? (e.g. drunk or high) \n24.4 \n25.6 \n.032 \nHave you ever had sexual intercourse when your partner was \nphysically or mentally unable to give consent? (e.g. drunk or high) \n18.5 \n15.5 \n.44 \nHave you ever had sexual intercourse with a person under the age \nof 16? 6.7 \n1.4 \n< .001 \nHave you been hit or physically hurt by your intimate partner? 10.1 \n23.6 \n< .001 \nNote. Options include Not sure except when marked §. Respondents Agreeing to Various Aspects of their Experiences of Sexual International Journal of Bahamian Studies Vol. Participation in Sexual Intercourse Table 5 also indicates that \n6.7% of males raped females due to having \nsex with underage partners. In their most recent experience of sexual \nintercourse, there were no statistically \nsignificant differences between the sexes in \ntheir attitudes towards their experience (see \nTable 6). This is a marked contrast to their \nrecollections of their first experience of \nsexual intercourse as seen in Table 4. In their most recent experience of sexual \nintercourse, there were no statistically \nsignificant differences between the sexes in \ntheir attitudes towards their experience (see \nTable 6). This is a marked contrast to their \nrecollections of their first experience of \nsexual intercourse as seen in Table 4. Significantly \nmore \nfemale \nthan \nmale \nrespondents reported having sexual partners \nof both sexes (χ2 = 564.9, df = 2, N = 674, p \n< .001; see Table 7). This finding is \nconsistent with Bethel and Fielding (2020) \nand broadly in line with female choice of \npartner in the United States (Tansill et al., \n2012). International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 7 Table 5 \nPercentage of Respondents Agreeing to Various Aspects of their Experiences of Sexual \nIntercourse \nAspect \nMale \nFemale \nΧ2 \n \n% \n% \np = \nDo you always give clear consent (verbal or non-verbal permission) \nbefore having sexual intercourse? 79.8 \n81 \n.61 \nHave you ever had sexual intercourse with an individual because \nyou were afraid of them? 1.7 \n15 \n< .001 \nHave you ever had sexual intercourse when you were physically or \nmentally unable to give consent? (e.g. drunk or high) \n24.4 \n25.6 \n.032 \nHave you ever had sexual intercourse when your partner was \nphysically or mentally unable to give consent? (e.g. drunk or high) \n18.5 \n15.5 \n.44 \nHave you ever had sexual intercourse with a person under the age \nof 16? 6.7 \n1.4 \n< .001 \nHave you been hit or physically hurt by your intimate partner? 10.1 \n23.6 \n< .001 \nNote. Options include Not sure except when marked §. Participation in Sexual Intercourse 27 (2021) \nTable 6 \nPercentage of Respondents Agreeing to Various Aspects of their Most Recent \nExperience of Sexual Intercourse \nAgreeing to \nMale Female \nΧ2 \n \n% \n% \np = \nI had sexual intercourse against my will \n4.1 \n5.7 \n.273 \nI found the experience enjoyable \n87.9 \n82.4 \n.163 \nI did not enjoy it but my partner did \n11.3 \n12.5 \n.197 \nI clearly agreed to have sexual intercourse \n96.8 \n91.0 \n.072 \nAfterwards, I wished I had not agreed to the sexual intercourse \n7.4 \n11.4 \n.055 \nI knew the person well with whom I had sexual intercourse \n91.0 \n92.0 \n.63 \nAfterwards, I wanted to repeat the experience \n81.8 \n80.3 \n.202 \nOn reflection, I feel that I was sexually abused when I had sexual \nintercourse \n2.5 \n3.7 \n.299 \nTable 7 \nChoice Sex of Intimate Partner by Sex of Respondent. Sex of those with whom respondents had sexual intercourse \nMale % Female % \nOnly male \n4.2 \n87.0 \nOnly female \n89.2 \n0.7 \nBoth male and female \n6.7 \n12.3 Respondents Agreeing to Various Aspects of their Most Recent \nS\nl I t International Journal of Bahamian Studies Vol. 27 (2021) \nTable 7 \nChoice Sex of Intimate Partner by Sex of Respondent. Sex of those with whom respondents had sexual intercourse \nMale % Female % \nOnly male \n4.2 \n87.0 \nOnly female \n89.2 \n0.7 \nBoth male and female \n6.7 \n12.3 Table 7 International Journal of Bahamian Studies Vol. 27 (2021) \nTable 7 \nChoice Sex of Intimate Partner by Sex of Respondent. Sex of those with whom respondents had sexual intercourse \nMale % Female % \nOnly male \n4.2 \n87.0 \nOnly female \n89.2 \n0.7 \nBoth male and female \n6.7 \n12.3 8 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 8 International Journal of Bahamian Studies Vol. 27 (2021) Events Impacting Mental Health Females were more likely than males to \nreport being a victim of physical violence \nfrom their intimate partner: 23.6% of females \nand 10.1% of males (χ2 = 10.8, df = 1, N = \n673, p = .001). A larger percentage of female \nthan male participants (47.9% of 368 \nresponses, compared to 33.1% of 103 \nresponses; χ2 = 7.15, df = 1, N = 471, p = \n.007) had participated in sexual intercourse \nagainst their will; in the case of females who \nhad engaged in unwanted sexual intercourse, \n27.8% had done so in the last month. Similarly, 23.7% of 536 females and 7% of \n113 males admitted to having been raped at \nleast once (χ2 = 16.2, df = 4, N = 649, p = \n.003). Only one of 11 male respondents \nobtained medical help for physical reasons \nfollowing the sexual abuse. In the case of \nfemales, 5.6% of 88 respondents sought help \nfor mental health reasons after being sexually \nabused, and 4.3% sought help for physical \nreasons following the abuse. Of the 24 males \nwho had suffered sexual abuse/rape, 58.3% \nagreed that it had affected their mental health; \nin the case of females, 81.7% of 241 \nrespondents agreed that it has affected their \nmental health. Various events in our lives can have negative \nimpacts upon us and some of these events \nwere included in the study. Here we consider \nevents that respondents thought had impacted \ntheir mental health. Females were more likely \nthan males to have suffered at least one event \nin Table 8 that they thought negatively \naffected their mental health in the previous 10 \nyears: 54.8% of females and 40.3% of males \n(χ2 = 10.77, df = 1, N = 856, p = .001). The \nmost commonly reported stressful event \nrelated to a death of a friend or family \nmember. However, the negative impact \nassociated with being a victim of a sexual \nattack was clearly different for males and \nfemales. Overall, it was apparent that females \nwere more likely than males to report events \nin their lives that affected their mental health \n(χ2 = 46.6, df = 1, N = 860, p < .001, see Table \n8). The other category included natural \ndisasters, such as Hurricane Dorian and the \nCovid-19 pandemic, and life events, such as \nending a relationship (break-up) and taking \nuniversity examinations. Table 8 Table 9 \nPercentage of Female Respondents Reporting Being a Victim of Rape and Also \nExperiencing Events that Had Negatively Affected Their Mental Health, Percentages \nWithin Victim of Rape \nVictim of rape \nA death of a \nfriend/ family \nmember \nVictim of \nproperty \ncrime \nVictim of a \nphysical \nattack \nVictim of a \nsexual attack \nVictim of \nbullying \n \n \n% \n% \n% \n% \n% \nN \nNo, never \n27.9 \n4.9 \n4.9 \n4.2 \n8.1 \n409 \nOnly once \n43.9 \n10.6 \n10.6 \n33.3 \n12.1 \n66 \nSometimes \n26.7 \n4.4 \n15.6 \n40.0 \n15.6 \n45 \nOften \n33.3 \n8.3 \n41.7 \n75.0 \n33.3 \n12 \nFrequently \n50.0 \n0% \n25.0 \n25.0 \n0 \n4 Table 9 \nPercentage of Female Respondents Reporting Being a Victim of R\nExperiencing Events that Had Negatively Affected Their Mental Healt\nWithin Victim of Rape Female Respondents Reporting Being a Victim of Rape and Also \nvents that Had Negatively Affected Their Mental Health, Percentages Table 8 Table 8 \nPercentage of Respondents Reporting Events that Had Negatively Affected Their Mental \nHealth \nNegative event \nMale \nFemale \n \n \n% \n% \np = \nA death of a friend/family member \n27.9 \n29.6 \n.68 \nVictim of bullying \n7.1 \n10.9 \n.16 \nVictim of a sexual attack \n2.6 \n10.8 \n.002 \nVictim of a physical attack \n3.2 \n6.9 \n.087 \nVictim of property crime \n5.2 \n5.1 \n.96 \nOther \n9.7 \n18.1 \n.006 \nN \n154 \n706 \n \nNote. Multiple answers allowed. Table 8 \nPercentage of Respondents Reporting Events that Had Negatively Affected Their Mental \nHealth \nN\nti\nt\nM l\nF\nl Respondents Reporting Events that Had Negatively Affected Their Mental E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 9 9 reverse may be the case. In Table 9, we only focused on females being \nvictims of rape due to the relatively small \nnumber of males who were victims of rape. Being a victim of rape is associated with \nelevated chances of experiencing other \nevents \nthat \nrespondents \nconsidered \ndetrimental to their mental health (χ2 = 55.5, \ndf = 16, N = 350, p < .001, see Table 9). Using \nthe responses of participants who had never \nbeen victims of rape as a benchmark, it can \nbe appreciated that as the occurrence of rape \nincreases, so do other stressful events, or the Being a victim of a sexual attack resulted in \nthe largest differences between the mental \nhealth scores of victims and non-victims of \nany of the negative events in Table 9 \n(victims, M = 48.5, non-victims, M = 35.9, \nt(630) = 7.48, p < .001). Analysis of \ncovariance, \ntaking \ninto \naccount \nthe \noccurrence of the events in Table 9, resulted \nin adjusted means (victims, M = 45.9, non-\nvictims, M = 36.3, p < .001). Mental Health Scores It would appear that there may be long lasting \neffects on the mental health of females who \nfelt abused when they first participated in \nsexual intercourse, as those who felt sexually \nabused when first initiated into sex had a \nhigher mean mental health score than those \nwho did not feel abused (see Table 10). This \nwould suggest that even though other events \nhad a negative impact on the mental health of \nfemale participants, it was still possible to \ndetect the residual negative effect of their \nfirst sexual encounter. The 19 questions on mental health were used \nto calculate a mental health score, with lower \nscores indicative of better mental health than \nhigher scores. The minimum score in this \nscale was 19 and the maximum score 95. The \noverall mean mental health score was 37.3 \n(SE = .54). However, females had a higher \nmental health score than males (females, M = \n37.9, SE = .6, males, M = 34.3, SE = 1.26, t = \n-2.57, df = 627, p = .01). This result is \nconsistent with other literature from the \nCaribbean (Pilgrim & Blum, 2012). International Journal of Bahamian Studies Vol. 27 (2021) 10 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 10 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 10 Table 10 \nFemale Experience of First Sexual \nIntercourse and Mental Health Score \nSexually abused at \nfirst sexual \nintercourse \nM \nSE \np = \n1 Strongly agree \n44.8 2.05 \n<.001 \n2 \n46.2 6.3 \n3 Cannot say \n39.9 1.86 \n4 \n40.2 2.54 \n5 Strongly disagree \n36.2 .65 \nNote. p value from analysis of variance. Table 10 \nFemale Experience of First Sexual \nIntercourse and Mental Health Score partner was associated with a higher mean \nmental health score (t = 3.81, df = 629, p < \n.001). partner was associated with a higher mean \nmental health score (t = 3.81, df = 629, p < \n.001). In the case of rape, both males and females \nhad higher mean mental health scores, even \nwhen the scores were adjusted for other \nnegative events in their lives, other than \nsexual abuse (see Table 12). Mental Health Scores Although the \nmean mental health score for males who had \nbeen raped sometimes has a high standard \nerror due to the small sample size, the fact \nthat it is the largest mean in the table may still \nindicate the detrimental effect rape has on \nmales. This is more evident when we \nappreciate that, overall, males had a lower \nmental health score than females (males, \n34.3, SE = 1.26, females, 37.9, SE = .60), a \nmean difference of 3.6. Females who had experienced unwanted \nsexual intercourse reported having higher \nmental health concerns the more frequently \nthey had had unwanted sex (see Table 11); \ntherefore, within the group of females \nparticipating in unwanted sex, the repetition \nof the event is linked with a higher mean \nmental health score. In the case of females, \nthe more frequently they had had sex against \ntheir will, the higher their mental health \nscore: r = .16 (p < .001). Although the mental health scores between \nrespondents who had had sexual relations \nwith others of the same sex were not \nstatistically different (ANOVA, F(2, 622) = \n1.21, p = .3), the interaction between sex of \nrespondent and sexual partner (F(2, 622) = \n2.2, p = .112) was suggestive of greater \ntrauma associated with respondents who \nparticipated exclusively in non-heterosexual \nsex (see Table 13). The relatively small \nsample size of the number of respondents \nwho reported sexual intercourse with both \nsexes or the same sex as themselves may \naccount for the lack of statistical significance. Table 11 \nMean Mental Health Score and Lifetime \nExperience of Unwanted Sex of Female \nRespondents \nHad sex against \nyour will: \nM \nSE \np = \nOnly once \n37.4 \n.89 \n \n.001 \nA few times \n41.6 \n1.44 \nOften \n46.4 \n5.14 \nFrequently \n49.8 \n6.09 \nNote. p value from analysis of variance. N = 324 Table 11 \nMean Mental Health Score and Lifetime \nExperience of Unwanted Sex of Female \nRespondents Female study participants who had been \nsexually abused indicated that the most likely \nvictimizers were people known to them, in \nparticular current or ex-boyfriends or friends. However, it should be noted that relatives and \nthose with trusted access to the victim were \nreported as victimizers by about a quarter of \nthe abused respondents. Relatively few \nfemales had been victimized by strangers \n(see Table 14). This finding is consistent with \nBethel and Fielding (2020) and in line with a \nstudy from the United States (Brooks, 2001). Mental Health Scores In the case of their most recent sexual \nencounter, those females who felt that they \nhad been abused had higher mental health \nscores than those who did not feel abused \n(Kruskal-Wallis H = 12.6, df = 4, p = .013). Also, having been hit by their intimate International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 11 Table 12 \nMean Mental Health Score and Respondent Lifetime Experience of Rape, by Sex of \nRespondent \nSex \nRaped \nM \nSE \nN \np = \nMale \nNever \n33.6 \n1.18 \n101 \n.003 \nOnly once \n35.5 \n5.98 \n4 \nSometimes \n58.4 \n6.96 \n3 \nFemale \nNever \n36.2 \n.66 \n389 \n.007 \nOnly once \n40.3 \n1.67 \n62 \nSometimes \n43.1 \n1.99 \n44 \nOften \n46.6 \n4.06 \n11 \nFrequently \n40.6 \n6.35 \n4 \nNote: Means adjusted for occurrence of negative events in the lives of the respondents in Table 7, excluding \nsexual abuse. p values from analysis of variance. Health Score and Respondent Lifetime Experience of Rape, by Sex of Note: Means adjusted for occurrence of negative events in the lives of the respondents in Table 7, excluding \nsexual abuse. p values from analysis of variance. International Journal of Bahamian Studies Vol. 27 (2021) \nTable 13 \nMean Mental Scores of Respondents, by Their Sex and the Sex of Their Intimate Partners \nSex of those with whom respondents had sexual intercourse \nMale \nFemale \n \nM \nM \nOnly male \n49.6 \n37.3 \nOnly female \n33.0 \n38.7 \nBoth male and female \n40.6 \n42.3 \nTable 14 \nAssociation Between the Victimizer and Female Sex Abuse Victim \nFemale participants having had sex against their will with: \n% reports \nGroup % \nBoyfriend \n35.9 \n \nBest friend \n0.5 \n \nBoyfriend (first time) \n0.5 \n \nBoyfriend and uncle \n0.5 \n \nBoyfriend, brother \n0.5 \n \nBoyfriend/Guy I had frequent sex with \n0.5 \n \nBoyfriends, cousins, uncles \n0.5 \n \nIntimate partner/ friends with benefits \n0.5 \n \nEx-boyfriend \n4.6 \n \nFirst ex-boyfriend \n0.5 \n \nNow ex-boyfriend \n0.5 \n \nDate \n1.5 \n46.7 \nGroup n = \n91.0 \n \nFriend \n1.8 \n \nFamily friend \n3.1 \n \nFriend, brother \n1.0 Table 13 \nMean Mental Scores of Respondents, by Their Sex and the Sex of Their Intimate Partners \nSex of those with whom respondents had sexual intercourse \nMale \nFemale \n \nM \nM \nOnly male \n49.6 \n37.3 \nOnly female \n33.0 \n38.7 \nBoth male and female \n40.6 \n42.3 International Journal of Bahamian Studies Vol. Mental Health Scores 27 (2021) \nTable 14 \nAssociation Between the Victimizer and Female Sex Abuse Victim \nFemale participants having had sex against their will with: \n% reports \nGroup % \nBoyfriend \n35.9 \n \nBest friend \n0.5 \n \nBoyfriend (first time) \n0.5 \n \nBoyfriend and uncle \n0.5 \n \nBoyfriend, brother \n0.5 \n \nBoyfriend/Guy I had frequent sex with \n0.5 \n \nBoyfriends, cousins, uncles \n0.5 \n \nIntimate partner/ friends with benefits \n0.5 \n \nEx-boyfriend \n4.6 \n \nFirst ex-boyfriend \n0.5 \n \nNow ex-boyfriend \n0.5 \n \nDate \n1.5 \n46.7 \nGroup n = \n91.0 \n \nFriend \n1.8 \n \nFamily friend \n3.1 \n \nFriend, brother \n1.0 12 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 12 Female participants having had sex against their will with: \n% reports \nGroup % \n“Friend” and boyfriend \n1.0 \n \n“Friend” \n0.5 \n \n“Trusted” friend \n0.5 \n \nFriend, we were never actually together, we never make it official \n0.5 \n \nFriend/Neighbor \n0.5 \n \nA guy who was my friend but isn’t my friend anymore \n0.5 \n \nEx Best friend \n0.5 \n \nFiancé and Old Friend \n0.5 \n \nNot intercourse, but assaulted or touched against my will by someone I \nwas friends with \n0.5 \n20 \nGroup n = \n39 \n \nHusband \n4.6 \n \nUncle \n1.5 \n \nUncle, close friend at former high school \n0.5 \n \nMy children [sic] daddy \n0.5 \n \nFather \n0.5 \n \nStepfather \n2.6 \n \nGodfather \n0.5 \n \nBrother \n2.6 \n \nCousin \n6.7 \n \nCousin (male) \n1.0 \n \nCousin and girl in primary school \n0.5 \n \nRape and Sexual Abuse- Male cousin \n0.5 \n \nRelatives \n0.5 \n \nGod brothers \n0.5 \n23.1 \nGroup n = \n45 \n \nNeighbour \n1.5 \n \nA situationship I was in [verbatim, not clear to authors] \n0.5 \n \nBabysitters [sic] son, uncle, brother \n0.5 \n \nCan't say \n0.5 \n \nCo worker \n0.5 \n \nDa dog outside \n0.5 \n \nEx friend’s uncle \n0.5 \n \nJust sexually assaulted/abused \n0.5 \n \nSexually abused never raped. First time it was my cousin. Second a \npolice officer. Third a priest. 0.5 \n \nOlder neighbour \n0.5 \n \nThe entire world, police and family \n0.5 \n6.7 \nGroup n = \n13 \n \nStranger \n3.1 \n \nStranger I met once prior \n0.5 \n3.6 \nGroup n= \n7 \n \nN = \n195 International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 13 Bethel and Fielding (2020), who reported on \nthe life-time sexual experiences of college \nstudents in The Bahamas. The fear of rape \nwas again found to be the crime of greatest \nconcern to females, even though, according \nto police statistics, it is not a commonly \nreported crime (Bethel & Fielding, 2020). Mental Health Scores The current study confirms that sexual \nintercourse is an activity that is not without \nrisk of violence, particularly for females, and \nsome females, even if not admitting to being \nraped, clearly agreed to participate in sexual \nintercourse out of fear. This may help to \nexplain the fear that females have of rape, \neven if they do not feel that they have been \nraped. The violence to which they are \nsubjected by their intimate partner may \nultimately be manifested by rape. Therefore, \nviolence appears to be a normative aspect of \nfemale life. The fact that about 25% of the \nfemale participants had unwanted sex in the \nprevious month indicates that unwanted \nencounters are not uncommon. As Bethel and \nFielding (2020) found, circumstances such as \nthis increase the percentage of individuals \nwho are legally raped beyond the percentage \nof people who admit to being raped. Discussion Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 14 As might have been expected, respondents \nhad experienced a number of events that they \nfelt had negatively affected their mental \nhealth; the most common of these was a death \nof someone whom they knew. Respondents \nwho had reported being raped were also more \nlikely to have suffered other events in their \nlives that had negative impacts on their \nmental health. As noted in Norway, \nchildhood experiences of violence increase \nthe risk of these children being victimized as \nadults and increase their risk of suffering \nfrom mental health concerns (Thoresen et al., \n2015). This study cannot explain why this \nmight be, but it may be associated with \npersons who have less coping capability than \nothers with respect to negative events. This \nresult warrants further study. Although the first sexual experience of \nfemales was less satisfactory than for males, \nit is apparent that the most recent sexual \nencounters of males and females were \nexperienced with equal approval, even \nthough some respondents may have felt \nabused. The study confirmed that persons \nknown to the victim were the most likely \ngroup to inflict sexual abuse/rape. Family \nmembers accounted for about 25% of \nvictimizers identified by respondents (Table \n14), indicating that family members used \ntheir privileged position in the family to \nvictimize others. This helps to explain why \nsexual abuse/rape is under reported, as it is \nconcealed behind the walls of the home, \nwhich hinders reporting and investigation of \ncases. Dating violence in adolescent women \nhas been demonstrated to be associated with \ngreater mental health concerns (Hébert et al., \n2008). What is of interest, and merits further study \nin the Bahamian context, is that it appears \nthat the negative aspects of the respondents’ \nsexual initiation appear to be detectable today \nthrough their mental health score, a finding \nconsistent with Dovran et al. (2016) and \nBurgić Radmanović (2020). This suggests \nthat a person’s early sexual experience can \nhave life-long consequences. This would \nsuggest that every effort should be made to \nensure that a person is appropriately prepared \nfor engaging in sexual intercourse, even if the \nact is consensual, or how a participant might \nprotect themselves from abuse, if it is not \nconsensual. Discussion In considering the results, we should be \naware that the target population was college-\nlevel students, not the wider population, so, \nalthough the study provides useful insights \nwith regard to sexual experiences and mental \nhealth, it may not reflect what occurs in the \nwider and more diverse population of The \nBahamas, particularly with respect to older \npeople. Notwithstanding these limitations, \nthis study goes some way to contributing to \nthe need identified in the Strategic Plan to \nAddress Gender-Based Violence (Bahamas \nNational Task Force for Gender-based \nViolence, 2015) to undertake further research \non gender-based violence in The Bahamas. Although interpersonal violence has many \nnegative \nphysical \nimpacts \non \nvictims \n(Campbell, 2002) that cannot be separated \nfrom mental trauma, this study only aimed to \nidentify possible links between the mental \nhealth of Bahamian college students and their \nnegative experiences of sexual intercourse; \nalso, it allows additional linkages to be made, \nas it considers other traumas that may \ninfluence mental health. Although this is a \nlimitation of the study, it does help to \ncomplement the study on physical trauma in \nThe Bahamas by Burnett-Garraway (2001). Although it is easy to focus on female rape \nand abuse, this study confirms the results of \nBethel and Fielding (2020) and unpublished \ndata provided by the Royal Bahamas Police \nForce (see Table 1) that males are also \nsubjected to rape. As indicated by others, for \nexample, Musevenzi and Musevenzi (2018), \nrape can be even more traumatic for male \nvictims than for female victims due to its \nassociated stigma. Additionally, as has been \nreported elsewhere, for example, Pakistan \n(Ali et al., 2013), it is not unusual for victims \nto be unwilling to seek help. Consequently, \nthis suggests that there should be increased \ngreater concern for appreciating that only \nfemales are victims of rape and need help. The methodology of this study is not \nnecessarily the most appropriate to identify \nlinkages \nbetween \nnegative \nsexual \nexperiences and mental health, but in the \nabsence of a longitudinal study, which could \nhave identified the changes in mental health \nover time, this study may still be useful in \nproviding preliminary information on how \nnegative sexual experiences affect mental \nhealth. The study findings are similar to those \nreported by researchers in the United States \n(Voth Schrag & Edmond, 2018) and by International Journal of Bahamian Studies Vol. 27 (2021) 14 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 14 E. Discussion This may mean that additional \nefforts need to be made with respect to sex \neducation for school children due to the fact \nthat some 25% of female respondents’ first \nexperience was during their school years, \nand, for females in particular, this experience \nwould most certainly be classified legally as \nrape (Table 4). Dating violence may be what is being \ndescribed in Table 14, where boyfriends, etc. were \nreported \nas \nbeing \ncommonly \nresponsible for the violence inflicted upon \nour study respondents, a figure consistent \nwith the range reported by Bergen and \nBarnhill (2006). Beyond dating violence, we \nshould note that in the United States, 14% of \nmarried women have reported being raped by \ntheir husbands (Brooks, 2001). Therefore, \ngiven the expected small number of married \nfemales in a college population, the fact that \n7.2% of older female respondents reported \nbeing victimized by their husband/long-term \npartner is noteworthy, and the occurrence of \nmartial rape requires further study in The \nBahamas. It is apparent that intimate partners \ncan be violent towards those with whom they \nhave relationships, and this violence is also \nassociated with increased mental health \nscores. Consequently, \nvarious \nnegative \naspects of imitate relationships, violence, \nsexual abuse, and rape can contribute to \nhigher mental health scores. As others have indicated, victims who fail to \nseek help from such a traumatic event may \nfind that their mental health problems \nbecoming worse (Tansill et al., 2012). These \nnegative impacts can also affect the children International Journal of Bahamian Studies Vol. 27 (2021) E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 15 of sexually abused mothers, as they are less \nlikely to praise their children, i.e., use \npositive parenting techniques (Fujiwara et al., \n2012). elevated mental health concerns (Pereira & \nCosta, 2016). These observations reflect in \npart the stigma of being what is loosely \ntermed “gay” or not engaging solely in \nheterosexual sex (“The Truth About Being \nGay,” 2016), a stigma that has resulted in \npersons being attacked in the streets \n(Turnquest, \n2016). Consequently, \nthis \nmarginalized group may be in even greater \nneed of mental health care than the \nheterosexual group. Although most participants were more \npositive about their most recent sexual \nexperience, those who felt abused or raped \nhad higher mental health scores than those \nwho had not experienced this trauma. Discussion Although the study methodology does not \nallow for national estimates of the occurrence \nof sexual abuse or rape to be made, the fact \nthat study participants reported having sexual \nintercourse against their will in the previous \nfour weeks starts to give an idea of the \npossible frequency of sexual abuse or rape. There was also an indication that persons \nwho were not “straight” in their sexual \nbehaviour had higher mental scores, a result \nconsistent with Szalacha et al. (2017), who \nalso found differences in the mental health of \nfemales who reported having sex with other \nfemales. In Portugal, the stigma of being non-\nheterosexual has been associated with Overall, \nthe \nresults \nfrom \nthis \nstudy \ndemonstrate the link between negative sexual \nencounters and mental health and its long-\nterm nature in the Bahamian context. Consequently, the recommendations of \nNaylor et al. (2012), which focus on \nintegrating mental health care with primary \nhealth care, may be beneficial. The authors \nhope that these results will assist with \nincreasing the awareness of the hidden harm \nof sexual trauma and encourage society to \nsensitively engage victims of negative sexual \nexperiences. Dr. Williams died on November 26, 2020, aged 55. International Journal of Bahamian Studies Vol. 27 (2021) Acknowledgments We would like to acknowledge the students in Nursing Research 409 Spring 2020 class, which \nwas taught by Dr. Elizabeth Williams, who participated in the data collection and shared their final \npapers with us. Dr. Williams died before she could write up this investigation, so we have had the \nhonour of writing this paper on her behalf and have attempted to present the data in a way \nconsistent with her IRB submission. We are grateful for the feedback of Dr. Michelle Bettin on an \nearlier draft and for the comments of the referees. International Journal of Bahamian Studies Vol. 27 (2021) 16 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 16 References National Resource Center on Domestic \nViolence, Pennsylvania Coalition Against \nDomestic Violence. http://www.ncdsv.org/images/VAWnet_\nMaritalRapeNewResearchDirections_2-\n2006.pdf National Resource Center on Domestic \nViolence, Pennsylvania Coalition Against \nDomestic Violence. http://www.ncdsv.org/images/VAWnet_\nMaritalRapeNewResearchDirections_2-\n2006.pdf Ali, T. S., Mogren, I., & Krantz, G. (2013). 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Forced \nintercourse, mental health, and human \ncapital. Southern Economic Journal, \n80(2), 324–344. https://doi.org/10.4284/0038-4038-\n2013.015 Naylor, C., Parsonage, M., McDaid, D., \nKnapp, M., Fossey, M., & Galea, A. (2012). Long-term conditions and mental \nhealth: The cost of co-morbidities. The \nKing's Fund. https://www.kingsfund.org.uk/sites/defaul Rolle, R. (2020, November 3). UB students \nsex assaults shock: Campus survey reveals International Journal of Bahamian Studies Vol. 27 (2021) 18 E. Williams, W. Fielding & V. Ballance. Mental Health and Negative Sexual Experiences. 18 one-in-three are victim of some form of \nrape: Comments. The Tribune. Journal of Bahamian Studies, 22, 91–98. \nhttps://doi.org/10.15362/ijbs.v22i0.276 http://www.tribune242.com/news/2020/no\nv/03/ub-students-sex-assaults-shock-\ncampus-survey-revea/ Tansill, E. C., Edwards, K. M., Kearns, M. C., \nGidycz, C. A., & Calhoun, K. S. (2012). 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ERROR: type should be string, got "https://doi.org/10.1007/s42113-020-00082-y\nComputational Brain & Behavior (2020) 3:322–340 https://doi.org/10.1007/s42113-020-00082-y\nComputational Brain & Behavior (2020) 3:322–340 ORIGINAL PAPER Abstract Multidimensional scaling (MDS) models represent stimuli as points in a space consisting of a number of psychological\ndimensions, such that the distance between pairs of points corresponds to the dissimilarity between the stimuli. Two\nfundamental challenges in inferring MDS representations from data involve inferring the appropriate number of dimensions\nand the metric structure of the space used to measure distance. We approach both challenges as Bayesian model-\nselection problems. Treating MDS as a generative model, we define priors needed for model identifiability under metrics\ncorresponding to psychologically separable and psychologically integral stimulus domains. We then apply a differential\nevolution Markov-chain Monte Carlo (DE-MCMC) method for parameter inference, and a Warp-III method for model\nselection. We apply these methods to five previous data sets, which collectively test the ability of the methods to infer an\nappropriate dimensionality and to infer whether stimuli are psychologically separable or integral. We demonstrate that our\nmethods produce sensible results, but note a number of remaining technical challenges that need to be solved before the\nmethod can easily and generally be applied. We also note the theoretical promise of the generative modeling perspective,\ndiscussing new and extended models of MDS representation that could be developed. Keywords Multidimensional scaling · Bayesian model selection · MDS dimensionality · Separable and integral stimuli ·\nW\nIII b id\nli Keywords Multidimensional scaling · Bayesian model selection · MDS dimensionality · Separable and integral stimuli ·\nWarp-III bridge sampling Bayesian Inference for Multidimensional Scaling Representations\nwith Psychologically Interpretable Metrics Quentin F. Gronau1 · Michael D. Lee2 © The Author(s) 2020\nPublished online: 8 July 2020 © The Author(s) 2020\nPublished online: 8 July 2020 \u0002 Quentin F. Gronau\nQuentin.F.Gronau@gmail.com 2\nDepartment of Cognitive Sciences, University of California\nIrvine, Irvine, CA 92697, USA 1\nDepartment of Psychological Methods, University of\nAmsterdam, 1012 WX Amsterdam, Netherlands Introduction knowledge, and the capability of the mind to make adaptive\npredictions about properties and consequences (Shepard\n1987). For these reasons, mental representations found\nvia MDS methods have been and remain widely used in\ncognitive process models of identification, categorization,\nand decision making (e.g., Nosofsky 1992). Multidimensional scaling (MDS) was developed in the\n1950s in cognitive psychology as a statistical method for\nmaking inferences about human mental representations\n(Shepard 1957, 1962; Kruskal 1964) MDS models the\nsimilarities or psychological proximities between pairs\nof stimuli, representing each stimulus as a point in a\nmultidimensional space, such that more similar stimuli are\nnearer each other. The core psychological motivation is\nthat the similarities reflect the basic cognitive process of\ngeneralization. Generalization can be thought of as the\nability to treat two stimuli as being the same, and has been\nargued to serve as a basis for the mental organization of Soon after its development in cognitive psychology,\nhowever, MDS algorithms found application as a statistical\nmethod that produces a low-dimensional representation of\na set of objects, based on a measure of the similarities\nbetween them. As a data-reduction or visualization method,\nMDS has been applied in the natural, biological, and human\nsciences, with application areas as diverse as representing\nthe similarities of skulls in archaeology, the tastes of\ncolas in marketing, and the voting patterns of senators in\npolitics (e.g., Borg and Groenen 1997; Cox and Cox 1994;\nSchiffman et al. 1981). \u0002 Quentin F. Gronau\nQuentin.F.Gronau@gmail.com Whether viewed as a model of psychological represen-\ntation or a data-reduction method, a foundational challenge\nin MDS modeling is determining the dimensionality M of\nthe representational space. In his 1974 Presidential Address\nto the Psychometric Society, Roger Shepard identified six 323 Comput Brain Behav (2020) 3:322–340 Integral Fig. 1 MDS representations\nwith integral and separable\nmetric structures\nIntegral\nSeparable Integral\nSeparable Fig. 1 MDS representations\nwith integral and separable\nmetric structures Separable Integral Separable 1991) emphasizes the role that the metric structure of\nthe space plays in capturing key psychological properties\nof the stimuli. In particular, the idea is that different\nmetrics capture the theoretical and empirical distinction\nbetween separable and integral stimuli (Attneave 1950;\nGarner 1974). Separable stimuli are those for which the\ncomponent dimensions can be attended to separately. An\nexample is different shapes of different sizes, since it is\npossible for people to attend selectively to either shape or\nthe size. Introduction Integral stimuli, by contrast, are those for which the\ncomponent dimensions cannot be attended to independently. The standard example is color, since it is typically not\npossible for people to attend selectively to the underlying\nhue, saturation, and brightness components. basic challenges for MDS, the third of which was “The\nproblem of determining the proper number of dimensions\nfor the coordinate embedding space” (Shepard 1974, p. 377). A number of methods for solving the problem of\nMDS dimensionality have been developed in both statistics\nand psychology. The most common approach is a scree test\nthat aims to identify an “elbow” in the goodness-of-fit as\ndimensionality increases (Cox and Cox 1994; Kruskal 1964;\nSchiffman et al. 1981). Steyvers (2006) suggests the use of\ncross-validation methods, although this approach does not\nseem to be widely used. Since choosing the correct dimensionality of an MDS\nis naturally regarded as a model-selection problem—\nthat is, choosing between a one-dimensional versus two-\ndimensional versus three-dimensional representation, and\nso on—the statistically principled approach offered by\nBayes factors should provide a solution (Kass and Raftery\n1995). Along these lines, Lee (2001) implements an\napproach based on the Bayesian Information Criterion\n(BIC). The difference between BIC values for representa-\ntions with different dimensionality provides a crude approx-\nimation to the Bayes factor. Oh and Raftery (2001) provide\na different approach to approximation by computing the\nmarginal likelihoods of different representations using plug-\nin point estimates for the stimulus locations. This is an\napproximation because the exact Bayes factor requires an\nintegration across the stimulus location parameters. Oh\n(2012) developed a method based on spike-and-slab priors,\nin which the dimensionality is determined by the marginal\nposterior probabilities for each dimension that the coordi-\nnate locations are not zero for all stimuli. Figure 1 shows how different metric structures are used\nto represent integral and separable stimuli. In the left\npanel, there are four stimuli, represented by the points\np1, . . . , p4. The pairwise distances between these points,\nsuch as d12 between the first point and the second point,\nare modeled using the Euclidean metric, and so correspond\nto standard straight lines. In the right panel, there are\nthree stimuli, and the pairwise distances between them are\nmodeled according to the city-block metric. Intuitively, this\ncorresponds to comparing the stimuli on each underlying\ndimension independently, then adding those dissimarilities\nto get an overall measure of dissimilarity. Introduction Admittedly, this account of integrality and separability\nis a theoretical and empirical caricature, and much more\nnuanced and detailed accounts are possible (Shepard 1991;\nTversky and Gati 1982). The point is that psychological\nrepresentations based on MDS need to make assumptions\nabout the metric structure of the space, and use metrics\nother than the Euclidean metric. As J¨akel et al. (2008, p. 2)\npoint out, from the origins of MDS as a psychological\nmodel, “There was no a priori reason to believe that mental From the perspective of MDS as psychological models\nhowever, none of these approaches qualifies as being\nprincipled and complete. The key issue is that the theory of\nmental representation developed by Shepard (1957, 1987, 324 Comput Brain Behav (2020) 3:322–340 the development of joint prior distributions on the stimulus\nlocation parameters for the Euclidean metric, and non-\nEuclidean metrics other than the city-block metric. With\nthese priors established, we apply an approach to Bayesian\ninference using differential evolution Markov-chain Monte\nCarlo (DE-MCMC) computational sampling methods. The\nDE-MCMC method helps address the difficulties inher-\nent in inferring MDS representations, which are especially\nevident in non-Euclidean cases. We then use the Warp-\nIII bridge sampling method to approximate the marginal\ndensities needed to determine Bayes factors. We apply the\nmethod to five previously studied data sets, differing in\nthe type of stimuli and expected dimensionality of their\nMDS representation. For all five applications, the method\nmakes sensible inferences about dimensionality, and pro-\nduces interpretable stimulus representations. We conclude\nwith a discussion of remaining statistical and computa-\ntional challenges, and potential directions for refining and\nextending the approach. representations should be Euclidean.” Previous methods\nfor determining the dimensionality of MDS representations\nusing Bayesian model selection, however, have either been\ninsensitive to the metric structure of the representation (Lee\n2001) or have focused on the Euclidean metric (Oh 2012;\nOh and Raftery 2001). The use of non-Euclidean metrics raises another chal-\nlenge, related to inferring MDS representations themselves. There is evidence that it can be computationally difficult\nto find multidimensional city-block MDS representations\n(Groenen et al. 1998; Hubert et al. 1992), as well as find-\ning unidimensional MDS representations (Mair and Leeuw\n2014). Given that these difficulties stem from basic geomet-\nric properties of the MDS representations, it seems likely\nthey will continue to present an issue for Bayesian methods\nof inference. Introduction Finally, there is the challenge of inferring the appropriate\nmetric structure for an MDS representation. Shepard (1991)\nreviews the original statistical approach to this problem,\nwhich involved applying non-metric MDS algorithms for\na large number of different metrics, and choosing the one\nwith the best goodness-of-fit. As Lee (2008) pointed out,\nthis approach neglects to account for the component of\nmodel complexity that arises from the functional form\nof parameter interaction (Pitt et al. 2006), which is\noften the only difference between MDS models using\ndifferent psychologically interpretable metrics. Lee (2008)\ndeveloped a Bayesian approach in which the possible\nmetrics correspond to a parameter that is inferred jointly\nwith the coordinate location parameters that represent the\nstimuli. Okada and Shigemasu (2010) developed and tested\nthis approach further, and showed it is capable of recovering\nthe correct metric in simulation studies. Both the Lee (2008)\nand Okada and Shigemasu (2010) methods, however, failed\nto resolve basic challenges in model identifiability that arise\nfrom treating the choice of metric structure as a parameter\ninference problem. It is possible these identifiability issues\ncould be addressed by considering the choice as a model-\nselection problem, and restricting the set of possibilities to\na few interpretable metrics. The Identifiability Problem σ ∼TruncatedGaussian\n\u0006\n0.15,\n1\n0.22\n\u0007\nT (0, ) ,\n(3) (3) where the T (0, ) indicates the sampled value is truncated to\nbe a positive real number. This is an informative prior (Lee\nand Vanpaemel 2018), consistent with previous data and\nmodeling. Intuitively, σ corresponds to the average standard\ndeviation of different individual ratings of the same pair\nof stimuli. Empirical estimates of this standard deviation\nin previous data tend to range from about 0.1 to about 0.2\n(Lee 2001; Lee and Pope 2003).2 Accordingly, the prior is\ncentered on 0.15, but allows a wide range of possibilities. Most other methods, in contrast, assume the MDS\nspace is Euclidean. The post-processing of the coordinate\nlocation parameters used by both Oh and Raftery (2001)\nand Oh (2012) assumes a Euclidean space and controls for\ntranslation, reflection, and rotation. Okada and Mayekawa\n(2018) extend the approach developed by Okada (2012),\nwhich relies on Procrustes analysis. Their post-processing\nuses a loss function to align posterior samples of the\ncoordinate location, but again assumes a Euclidean space. We note that this MDS model does not incorporate\nindividual differences. It is assumed that the same point\npi represents the ith stimulus for all participants. We also\nemphasize, however, that individual-level proximity data\ndijk are modeled, rather than averaged or aggregated data\nacross participants. The problems inherent in averaging\ndata have long been understood (Estes 1956), and have\nbeen studied in the specific cognitive modeling context\nprovided by MDS representations (Lee and Pope 2003). Our approach is to require the same underlying MDS\nrepresentation to provide an account of each individual\nproximity matrix. Besides the lack of flexibility in the nature of the\ndistance metric, post-processing methods have the effect\nof implementing modeling assumptions without explicitly\nspecifying those assumptions as part of the model. While\nthis is often practical, it is theoretically inelegant, and\ncontrary to the goals of generative modeling. Ideally, the\nconstraints required for model identifiability should be part\nof the model itself. In the case of MDS models, these\nconstraints are naturally imposed through the specification\nof a joint prior over the coordinate location parameters\nthat addresses the transformational invariances, removes\nthe need for post-processing, and makes bridge sampling\nfeasible. To complete the generative model, a straightforward\napproach would be to give all of the coordinate locations\nfor the representational points uniform priors pim\n∼\nUniform\n\b\n−1, 1\n\t\n. The Identifiability Problem Formally, suppose there are N stimuli to be represented,\nbased on observed proximity data from P participants,\nwith dijk measuring the proximity between the ith and jth\nstimulus provided by the kth participant. We assume these\nobserved proximities are normalized to lie between 0 and 1. The point representing the ith stimulus in a M-dimensional\nspace is pi = (pi1, . . . , piM) and the distance between\npoints pi and pj is measured by the Minkowski metric with\nmetric parameter r, so that ˆdijk =\n\u0002 M\n\u0003\nm=1\n\u0004\u0004pim −pjm\n\u0004\u0004r\n\u00051/r\n. (1) (1) The Minkowski metric has special cases of the city-block\nmetric when r = 1 and the Euclidean metric when r = 2. Values of r between 1 and 2 can potentially be interpreted\nas intermediate assumptions about the independence of\nstimulus dimensions between the end point of complete\nseparability and complete integrality. ˆ Accordingly, the goals of this article are to examine the\nimplementation of MDS models that use psychologically\ninterpretable metrics, including both the Euclidean and a\nnon-Euclidean metric, and explore the possibility of infer-\nring the appropriate dimensionality and metric structure\nof these representations using Bayesian model-selection\nmethods. The structure of the remainder of the article is\nas follows. In the next section, we define MDS models,\nand address the issue of model identifiability under differ-\nent metrics. Consistent with previous literature, we argue\nthat the city-block metric presents fundamental problems\nin making MDS representations identifiable. This leads to The goal of MDS is for the modeled distances ˆdijk to\ncorrespond to the observed proximities dijk. We use the\nprobabilistic model dijk ∼Gaussian\n\u0006\nˆdijk, 1\nσ 2\n\u0007\n,\n(2) (2) 325 Comput Brain Behav (2020) 3:322–340 where σ is the standard deviation with which the observed\nproximities are measured.1 It is assumed to be the same for\nall of the proximities, and is given a prior location parameters to control for translation, reflection,\nand permutation. For example, to control for translation,\nthe method zero centers every posterior sample of the\nsets of coordinate location. The Lee (2008) method does\nnot control for rotation, which is problematic, because the\nmethod also attempts to infer the r metric parameter, and\nso the inferred representational space can have a Euclidean\nmetric, which requires rotational invariance. The Identifiability Problem These priors, however, made the model\nnon-identifiable, because the distances between points\nare invariant under transformations (Borg and Groenen\n1997, Ch. 2). The distances between points are preserved\nunder translation, reflection, axes permutation (for non-\nEuclidean metrics), and rotation (for the Euclidean metric). A principled Bayesian approach for controlling these\ninvariances to ensure model identifiability constrains the\ncoordinate location parameters through a joint prior\ndistribution that depends on the assumed metric. This generative approach is used by the “parameter\nfixing” method considered by Okada and Mayekawa\n(2018), who evaluate it as a contrast with the Procrustes\nmethods that are their focus. Parameter fixing corresponds\nto setting a structured joint prior over the coordinate\nlocation parameters. Okada and Mayekawa (2018) define\nthe appropriate prior for a Euclidean space using results\nprovided by Bakker and Poole (2013), which were derived\nusing an analytic method based on matrix properties. Our goal is to extend this approach to include non-\nEuclidean representations. We start by considering one-\ndimensional MDS representations, before considering mul-\ntidimensional representations in both Euclidean and non-\nEuclidean metric spaces. We take a geometric approach\nto identifying the required joint priors for invariance con-\nstraints, complementing the non-geometric approach of\nBakker and Poole (2013) for the Euclidean metric. Previous Approaches Existing MDS modeling methods that use Bayesian\ninference almost always rely on post-processing to address\nthe issue of identifiability. The method developed by Lee\n(2008) post-processes posterior samples of the coordinate 1We parameterize the Gaussian distribution in terms of mean and\nprecision parameters, for consistency with the JAGS graphical\nmodeling language. 2See also the data repository at https://osf.io/ey9vp/ One-dimensional Representation 1We parameterize the Gaussian distribution in terms of mean and\nprecision parameters, for consistency with the JAGS graphical\nmodeling language. For a one-dimensional representation, all of the psycho-\nlogically interpretable metrics we consider give the same 326 Comput Brain Behav (2020) 3:322–340 Fig. 2 Identification constraints\nfor a one-dimensional\nrepresentation coordinate location to zero, “+” denotes constraining it to be\npositive, and “R” denotes imposing no constraint. coordinate location to zero, “+” denotes constraining it to be\npositive, and “R” denotes imposing no constraint. distances. The required constraints on the points are shown\nin Fig. 2, with one point fixed at the origin to control\ntranslation, and second point restricted to be positive to\ncontrol reflection. These constraints can be formalized by a joint prior with p1 = 0\np2 ∼Uniform\n\b\n0, 1\n\t\np3, . . . , pN ∼Uniform\n\b\n−1, 1\n\t\n. (4) (4) p3, . . . , pN ∼Uniform\n\b\n−1, 1\n\t\n. (4) Non-Euclidean Multidimensional Representations emphasized in the seminal text by Borg and Groenen (1997,\npp. 369–372). Finding constraints for invariance in non-Euclidean metrics\nis more complicated, and is especially difficult for the\ncity-block metric. The basic geometric problem was noted\nas early as Arnold (1971), and discussed in Shepard’s\n(1974) presidential address. A simple demonstration of the\nfundamental problem is provided by Fig. 4. The three panels\ncorrespond to Euclidean (r = 2), city-block (r = 1), and\na general non-Euclidean (r = 1.5) metric, and show unit\niso-distance contours around the same two points in each\nmetric, shown as black dots. These iso-distance contours are\nthe “unit circles” of each metric, showing all the points in\nthe space that are the same distance from the two points. For the Euclidean metric, these contours are familiar circles,\nand coincide at only one point, shown by the white dot. This\nmeans that there is a unique point in the space that is equally\ndistant from the two points shown by black dots. In the\ncontext of an MDS representation, a stimulus that is equally\ndifferent to both of the points can be uniquely identified. Figure 5 provides a concrete example, based on the\nmore general configuration examined by Borg and Groe-\nnen (1997, Figure 17.6). Each panel shows a representa-\ntion of six fictitious people in terms of two underlying\ndimensions. The city-block distance between each pair of\npeople is identical in both configurations. This means,\nof course, that this proximity matrix is equally consis-\ntent with both representations, and either could be inferred\nfrom the data. But, the two representations are substan-\ntively different, in non-trivial ways. The representations\ndo not differ simply by changing the axes, and have\nbasic structural differences. For example: Cedric, Ding-\nbats, and Ethelred are co-linear in the first representation,\nbut not in the second, where Dingbats, Ethelred and Fiona\nbecome co-linear; the ordering of Albert and Beowulf\nchanges on both dimensions between the configurations;\nand so on. In fact, once the lack of invariance revealed\nby the Borg and Groenen (1997, Figure 17.6) analysis is\nunderstood, it is clear that many additional representations\nfor the proximity between the six people could be con-\nstructed, supporting a wide range of different meaningful\ninterpretations. For the city-block case, however, the iso-distance\ncontours are diamonds, and there are infinitely many points\nthat are equally different. Euclidean Multidimensional Representations Figure 3 shows the constraints needed to identify Euclidean\nMDS representations in two and three dimensions. In\nthe two-dimensional case, the first point p1 is fixed at\nthe origin, to control translation, the second point p2 is\nconstrained to the positive x-axis, to control reflection in\nthe y-axis and rotation, and the third point p3 is constrained\nto have a positive y-value to control reflection in the x-\naxis. The same logic is applied in the three-dimensional\ncase, with p1 controlling translation, p2 and p3 controlling\nreflection and rotation in successive axes, and p4 controlling\nthe final reflection. Formally, these constraints in D dimensions correspond\nto the joint prior p11, . . . , p1D = 0\np21 ∼Uniform\n\b\n0, 1\n\t\np22, . . . , p2D = 0\np31 ∼Uniform\n\b\n−1, 1\n\t\np32 ∼Uniform\n\b\n0, 1\n\t\np33, . . . , p3D = 0\np41, p42 ∼Uniform\n\b\n−1, 1\n\t\np43 ∼Uniform\n\b\n0, 1\n\t\np44, . . . , p4D = 0\n. . . p11, . . . , p1D = 0\np21 ∼Uniform\n\b\n0, 1\n\t\np22, . . . , p2D = 0\np31 ∼Uniform\n\b\n−1, 1\n\t\np32 ∼Uniform\n\b\n0, 1\n\t\np33, . . . , p3D = 0\np41, p42 ∼Uniform\n\b\n−1, 1\n\t\np43 ∼Uniform\n\b\n0, 1\n\t\np44, . . . , p4D = 0\n. . . (5) These are the first two cases of a general pattern, clear by\ninduction, that applies to a M-dimensional representation,\nand corresponds to the matrix result provided by Bakker\nand Poole (2013). An intuitive presentation of the inductive\npattern is shown below, where “0” denotes fixing a (5) Fig. 3 Identification constraints for Euclidean representations in two dimensions (left) and three dimensions (right) Fig. 3 Identification constraints for Euclidean representations in two dimensions (left) and three dimensions (right) 327 Comput Brain Behav (2020) 3:322–340 Fig. 4 The nature of iso-distance curves and the identifiability of mid-points for the three Minkowski metrics corresponding to r = 2 (Euclidean),\nr = 1 (city-block), and r = 1.5 tance curves and the identifiability of mid-points for the three Minkowski metrics corresponding to r = 2 (Euclidean)\n1 5 Fig. 4 The nature of iso-distance curves and the identifiability of mid-points for the three Minkowski metrics corresponding to r = 2 (Euclidean),\nr = 1 (city-block), and r = 1.5 Non-Euclidean Multidimensional Representations Three specific possibilities are\nshown by white dots, but clearly any point along the line\nwhere the iso-distance contours coincide is possible. In the\ncontext of an MDS representation, this means that there is\na fundamental difficulty in identifying a stimulus that is\nequally different to both of the points. This basic problem\nis not, in general, solved by the introduction of additional\nstimuli that provide additional constraints. Indeed, the\nproblem compounds for potential city-block representations\nwith many stimuli. Bortz (1974, see, especially, Figures 2\nand 3) provides compelling examples, and the same point is A practical approach for identifying city-block repre-\nsentations, used by Nosofsky (1985), relies on determining\nthe values of some stimuli on some dimensions, by means\nexternal to the MDS modeling. Ultimately, this strategy\ncan solve the problem, if it is possible to find the val-\nues of every stimulus on every dimension. But, Fig. 5\nsuggests the strategy may not be effective in situations\nwhere the identification of just a few stimuli is possible. In both representations, Dingbats is at the same location, 328 Comput Brain Behav (2020) 3:322–340 Dimension 1\nDimension 2\n Albert\n Beowulf\n Cedric\n Dingbats Ethelred\n Fiona\nDimension 1\nDimension 2\n Albert\n Beowulf\n Cedric\n Dingbats\n Ethelred\n Fiona\nFig. 5 Two city-block representations of six fictitious people in terms of two dimensions. Both representations have identical proximity matrices Albert Fig. 5 Two city-block representations of six fictitious people in terms of two dimensions. Both representations have identical proximity matrices Fig. 5 Two city-block representations of six fictitious people in terms of two dimensions. Both representa coincide at only one point. The asymmetry of these contours\nmakes clear they do not have the rotational invariance of the\nEuclidean r = 2 metric. In this way, general non-Euclidean\nmetrics, such as r\n=\n1.5, capture the psychological\nidea that the dimensions in an MDS representation have\nmeaning and allow selective attention, while avoiding the\ndegenerate lack of identifiability inherent in the city-block\nmetric. consistent with values on dimensions having been exter-\nnally determined, yet the locations of the remaining stimuli\nare under-determined. In addition, if, for example, Albert\nwas additionally identified as being located in the position\nshown in the first representation, that would constrain the\ninference about Beowulf and Cedric, but would not con-\nstrain Ethelred and Fiona, who could still be inferred to be at\neither of the possibilities shown in the two representations. 3These order constraints can be imposed either in decreasing manner,\nas shown in Fig. 6 for easier visualization, or in an increasing manner,\nas they are in our code. Non-Euclidean Multidimensional Representations Thus, while the addition of stimuli, or the identification of\ndimension values for some stimuli, may work in some spe-\ncific circumstances, we do not believe either represents a\ngeneral approach to making city-block MDS representations\nidentifiable. Figure 6 shows the constraints needed to identify these\nsort of non-Euclidean MDS representations in two and three\ndimensions. In the two-dimensional case, the first point p1\nis once again fixed at the origin, to control translation, the\nsecond point p2 is constrained to the positive quadrant to\ncontrol reflection. In addition, the constraint that p22 ≤p21\nis imposed, requiring the value of the second stimulus on\nthe y-axis not to be larger than its value on the x-axis. This constraint controls for axis permutation, preventing the\ntwo dimensions from being swapped, and so allocates a We do not know how to solve the problem of MDS\nmodel invariance for the city-block metric. As the right-\nmost panel of Fig. 4 makes clear, however, the problem\ndoes not occur for Minkowski-metric parameters r > 1. For the r = 1.5 metric, the iso-distance contours again Fig. 6 Identification constraints for non-Euclidean representations in two dimensions (left) and three dimensions (right) Fig. 6 Identification constraints for non-Euclidean representations in two dimensions (left) and three dimensions (right) 329 Comput Brain Behav (2020) 3:322–340 Marginal Likelihood Comparing MDS models with different dimensions and\nmetrics via Bayes factors and posterior model probabilities\nrequires the computation of the marginal likelihood for all\nof the models, Mm,r, being considered where m denotes\nthe dimensionality and r the metric. Let D denote the\nobserved data (i.e., the pairwise dissimilarity ratings dijk)\nand P denote the N × m matrix with the latent stimulus\ncoordinates for each stimulus. The marginal likelihood for\nmodel Mm,r corresponds to the normalizing constant of the\njoint posterior distribution for θ = (P , σ): These first two cases once again make clear a general\npattern, in which the coordinate values of the second\npoint are positive and order constrained.3\nFormally,\nthe constraints for non-city-block but non-Euclidean D\ndimensions are p11, . . . , p1D = 0\np21, . . . , p2D ∼Uniform\n\b\n0, 1\n\t\n:\np21 ≥. . . ≥p2D\np31, . . . , p3D ∼Uniform\n\b\n−1, 1\n\t\n. . . (6 (6) p(D | Mm,r) =\n\nq(θ | D, Mm,r) dθ\n=\n\n \np(D|P, σ, Mm,r)\n\u000b\n\f\r\n\u000e p(P |Mm,r)\n\u000b\n\f\r\n\u000e p(σ |Mm,r)\n\u000b\n\f\r\n\u000e dPdσ, p(D | Mm,r) =\n\nq(θ | D, Mm,r) dθ\n=\n\n \np(D|P, σ, Mm,r)\n\u000b\n\f\r\n\u000e\nLikelihood\np(P |Mm,r)\n\u000b\n\f\r\n\u000e\nJoint Prior on\nStimulus Locations\np(σ |Mm,r)\n\u000b\n\f\r\n\u000e\nPrior on\nImprecision\ndPdσ Bayesian MDS Inference via DE-MCMC (7) When posterior samples for MDS models are obtained\nusing conventional Markov-chain Monte Carlo algorithms\n(MCMC; e.g., Gamerman & Lopes, 2006), it can occur\nthat chains get stuck in local maxima. In our experience,\nthe reason is typically that the stimuli that are constrained\nare similar to each other. To prevent local maxima, we\nimplemented a heuristic to order the stimuli in a way that\nthose defining the constraints are dissimilar. We motivate\nand describe this heuristic in detail in Appendix 1. In\naddition, to improve sampling, we used the differential\nevolution Markov-chain Monte Carlo algorithm (DE-\nMCMC; e.g., Heathcote et al. in press; Turner et al. 2013)\nthat helps to guide the chains to regions of high posterior\ndensity. where q(θ\n| D, Mm,r) denotes the unnormalized joint\nposterior density. where q(θ\n| D, Mm,r) denotes the unnormalized joint\nposterior density. Bayesian Model Comparison via Bridge\nSampling specific underlying stimulus dimension to each axis. The\nthree-dimensional case extends this logic by requiring that\nthe z-axis value of the second point be positive, to prevent\nreflection, and be less than the value of the second point on\nthe y-axis, to prevent permutation. Bridge Sampling Since the marginal likelihood in Eq. 7 is not available\nanalytically, we use Warp-III bridge sampling (Meng\nand Schilling 2002) to estimate this potentially high-\ndimensional integral. Bridge sampling (Meng and Wong\n1996; for a recent tutorial, see Gronau et al. 2017) is based\non the following identity: p(D | Mm,r) = Eg(θ)\n\u000f\nh(θ) q(θ | D, Mm,r)\n\u0010\nEp(θ|D,Mm,r) [h(θ) g(θ)]\n,\n(8) (8) DE-MCMC is a population-based MCMC algorithm that\ngenerates efficient proposals via a population of interacting\nchains (Turner et al. 2013). One strength of the algorithm\nis that it works well for highly correlated target distri-\nbutions. However, we used DE-MCMC primarily for the\nreason that the interacting chains can guide each other to\nregions of high posterior density which helps to avoid the\nissue of chains getting stuck in local maxima. Specifically,\nduring burn-in, we used a migration step that remedies\nthe problem of outlier chains in an effective manner (for\ndetails, see Turner et al. 2013, Appendix 2). We found that\nthe combination of the ordering heuristic and DE-MCMC\nprovides effective sampling consistently for the Euclidean\nmetric,\nand\nis\npartially\neffective\nfor\nnon-Euclidean\nmetrics. where the numerator is an expected value with respect\nto a proposal distribution g(θ), the denominator is an\nexpected value with respect to the parameter posterior\ndistribution p(θ | D, Mm,r), and h(θ) is a function such\nthat 0 <\n\u0004\u0004\nh(θ) p(θ | D, Mm,r) g(θ) dθ\n\u0004\u0004 < ∞. The\nbridge sampling estimate is obtained by sampling from\nthe proposal distribution g(θ) and the posterior distribution\np(θ | D, Mm,r) to approximate the two expected values. Meng and Wong (1996) showed that the optimal choice for\nh(θ) is given by ho(θ) ∝\n\u000f\ns1 q(θ | D, Mm,r) + s2 p(D | Mm,r) g(θ)\n\u0010−1 ,\n(9) (9) where si\n= ni/(n1 + n2), i ∈{1, 2}, n1 denotes the\nnumber of samples from the posterior p(θ | D, Mm,r),\nand n2 denotes the number of samples from the proposal\ng(θ). The optimal choice for h(θ) depends on the marginal 330 Comput Brain Behav (2020) 3:322–340 likelihood of interest. Therefore, in practice, the bridge\nsampling estimate is obtained via an iterative scheme,\npresented below, that updates an initial guess of the marginal\nlikelihood until convergence. first moment of the proposal and the posterior distribution,\nas shown in the upper-right panel. Bridge Sampling To compute the\nWarp-III estimate, one obtains 2n1 posterior samples: the\nfirst half of these samples is used to approximate μ and\nC with their sample versions ˆμ and ˆC, the second half\nof the posterior samples is used in the iterative scheme\n(i.e., Eq. 11). We use the bridgesampling R package\n(Gronau et al. in press) to compute the bridge sampling\nestimate in Eq. 11. η = b C−1 (ζ −μ) ,\n(10) (10) where b\n∼\nBernoulli (0.5) on {−1, 1}, μ denotes the\nexpected value vector of the posterior samples, and \u0005 =\nCC⊤denotes the posterior covariance matrix (i.e., C is\nobtained via a Cholesky decomposition). Figure 7 illustrates the warping approach for the\nunivariate case. In the upper-left panel, the solid line\ncorresponds to the standard Gaussian proposal distribution\nand the gray histogram depicts synthetic posterior samples. Subtracting the posterior mean from all samples matches the 5The function f needs to be one-to-one and its inverse f −1 needs to\nhave a well-defined Jacobian. 4Note that other proposal distributions are conceivable. The only\nconstraints are that the proposal has a zero mean vector, an identity\ncovariance matrix, and exhibits no skewness. 4Note that other proposal distributions are conceivable. The only\nconstraints are that the proposal has a zero mean vector, an identity\ncovariance matrix, and exhibits no skewness.\n5The function f needs to be one-to-one and its inverse f −1 needs to\nhave a well-defined Jacobian.\n6We use a function f that applies a log transformation to σ and\n(scaled) probit transformations to the non-zero elements of P . The\ntransformation for the ordered coordinates of the second stimulus for\nthe non-Euclidean case is described in Appendix 2. Note that it is\nirrelevant whether the coordinates are ordered as decreasing, as shown\nin Fig. 6 for easier visualization, or increasing, as implemented in\nour code. The transformation described in the appendix assumes the\nlatter. These transformations can be applied after having obtained\nposterior samples for θ. Furthermore, where necessary, the expressions\nare adjusted by the relevant Jacobian term |det Jf −1(ζ)|. Bridge Sampling Dividing all samples by\nthe posterior standard deviation matches the second moment\nof the two distributions, as shown in the lower-right panel. Finally, attaching a minus sign with probability 0.5 to the\nposterior samples achieves symmetry and thus matches the\nthird moment of the proposal and the posterior distribution,\nas shown in the lower-left panel. The variability of the bridge sampling estimate is\ngoverned not only by the number of samples but also,\nby the overlap between the proposal and the posterior\ndistribution. To obtain estimates with low variability, it\nis therefore prudent to maximize the overlap between\nthese two distributions. The Warp-III approach attempts to\ncreate a large overlap by fixing the proposal to a standard\nmultivariate Gaussian distribution and then manipulating\n(i.e., “warping”) the posterior in a way that matches the\nfirst three moments of the two distributions.4 Crucially, the\nwarping procedure retains the normalizing constant of the\nposterior (i.e., the marginal likelihood of interest). The Warp-III bridge sampling estimate based on ho(θ)\nis computed via an iterative scheme where the value of the\nestimate at iteration t is given by (for more details see,\nGronau et al. 2019): ˆp(D | Mm,r)(t+1) =\n1\nn2\nn2\n\u0011\ni=1\nl2,i\ns1 l2,i+s2 ˆp(D|Mm,r)(t)\n1\nn1\nn1\n\u0011\nj=1\n1\ns1 l1,j +s2 ˆp(D|Mm,r)(t)\n,\n(11) (11) A prerequisite for the warping procedure is that all\nelements of the parameter vector are allowed to range\nacross the entire real line. This can be achieved via a\nchange-of-variables of the form ζ\n=\nf (θ), where f\nis a suitable5 vector-valued function that transforms the\nconstrained elements of θ so that all elements of ζ are\nunconstrained.6 The Warp-III procedure is based on the\nfollowing stochastic transformation of the unconstrained\nparameter vector ζ: l1,j =\n| ˆC|\n2\n\u0012\nq(2 ˆμ−ζ ∗\nj |D,Mm,r)+q(ζ ∗\nj |D,Mm,r)\n\u0013\ng\n\u0014\nˆC\n−1\u0014\nζ ∗\nj −ˆμ\n\u0015\u0015\n,\n(12)\nand\nl2,i =\n| ˆC|\n2\n\u0012\nq( ˆμ−ˆC ˜ηi|D,Mm,r)+q( ˆμ+ ˆC ˜ηi|D,Mm,r)\n\u0013\ng(˜ηi)\n. (13) (12) (13) In Eqs. 12–13, q(· | D, Mm,r) denotes the unnormalized\nposterior density with respect to the unconstrained parame-\nter vector ζ, {ζ ∗\n1, ζ ∗\n2, . . . , ζ ∗\nn1} denote n1 posterior samples,\nand {˜η1, ˜η2, . . . , ˜ηn2} denote n2 samples from the standard\nmultivariate Gaussian proposal distribution. Applications 4Note that other proposal distributions are conceivable. The only\nconstraints are that the proposal has a zero mean vector, an identity\ncovariance matrix, and exhibits no skewness. In this section, we present applications of our method to\nfive existing data sets. For each application, we describe the\nstimuli and the nature of the data, as well as make clear\nour expectations about the MDS representation that will be\ninferred. In particular, we state our expectations about both\nthe dimensionality and metric structure of the representation\nwhenever possible. The results we present are based on\nconsidering MDS models up to and beyond this expected\ndimensionality, so that the inference our method makes\nis clear. Where possible, we apply our method under the\nassumption that the metric space is both Euclidean (r = 2)\nand non-Euclidean (r = 1.5) so that an inference can also 5The function f needs to be one-to-one and its inverse f −1 needs to\nhave a well-defined Jacobian. 6We use a function f that applies a log transformation to σ and\n(scaled) probit transformations to the non-zero elements of P . The\ntransformation for the ordered coordinates of the second stimulus for\nthe non-Euclidean case is described in Appendix 2. Note that it is\nirrelevant whether the coordinates are ordered as decreasing, as shown\nin Fig. 6 for easier visualization, or increasing, as implemented in\nour code. The transformation described in the appendix assumes the\nlatter. These transformations can be applied after having obtained\nposterior samples for θ. Furthermore, where necessary, the expressions\nare adjusted by the relevant Jacobian term |det Jf −1(ζ)|. 331 Comput Brain Behav (2020) 3:322–340 Fig. 7 Illustration of the Warp-III procedure. The black solid line shows the standard Gaussian proposal distribution and the gray histogram shows\nsynthetic posterior samples. Available at https://tinyurl.com/y7owvsz3 under CC license https://creativecommons.org/licenses/by/2.0/ Fig. 7 Illustration of the Warp-III procedure. The black solid line shows the standard Gaussian proposal distribution and the gray histogram shows Fig. 7 Illustration of the Warp-III procedure. The black solid line shows the standard Gaussian proposal distribution and the gray histogram shows\nsynthetic posterior samples. Available at https://tinyurl.com/y7owvsz3 under CC license https://creativecommons.org/licenses/by/2.0/ assuming a Euclidean metric.7 As for all of our applications,\nwe used 15 chains and 500 burn-in samples. During burn-\nin, the probability of a migration step was set to 0.05. 7We note, however, for completeness that we had difficulty with\nconvergence using the r = 1.5 metric for these data. Applications After\nburn-in, migration was switched off, and the algorithm was\nrun for 9000 iterations. We only retained every third sample\nso that we ended up with 3000 samples per chain for further\nuse (i.e., a total of 45,000 samples collapsed across chains). be made about the integrality or separability of the stimulus\ndomain. For some applications, we were unable to generate\nsamples with acceptable convergence for the r = 1.5 metric. In those cases, we only report results assuming the r = 2\nmetric. Line Length Because\nof the assumptions of equal prior probabilities, the ratio of\nany pair of posterior probabilities is naturally interpreted\nas a Bayes factor. The key result is that the expected\none-dimensional representation is inferred, with a posterior\nprobability near one. We expect the MDS representation to use the Euclidean\nmetric, consistent with the integral nature of the color\nstimulus domain. We also expect a two-dimensional\nrepresentation, following the color circle found by previous\nMDS analyses of these and other color similarity data,\nsuch as the Shepard (1962) original MDS analysis of data\nreported by Ekman (1954). The right panel of Fig. 8 shows the inferred one-\ndimensional MDS representation. The black lines show\nthe stimuli in terms of their physical line lengths,\nlocated at the posterior mean of their location in the\npsychological\nspace. The\nblue\nhistograms\nshow\nthe\nmarginal posterior distributions for each line stimulus. The\nMDS representation arranges the line stimuli in order of\ntheir length, but they are not evenly spaced, despite the\nlines increasing in constant physical increments. Instead,\nthe psychological representation shows compression for the\nlonger lines, consistent with basic psychophysics (Fechner\n1966). This compression is large enough that the posterior\ndistributions begin to overlap for the longest line stimuli. Figure 9 shows the results of applying our method,\nassuming a Euclidean metric. This was a case in which\nwe were unable to generate samples with acceptable\nconvergence for the r = 1.5 metric. For the Euclidean\nmetric, there is uncertainty regarding the dimensionality,\nwith a three-dimensional representation having probability\na little over 0.6 and a two-dimensional representation having\nalmost all of the remaining probability. The inferred three-\ndimensional representation is shown by pairing the first two\ndimensions as a two-dimensional plot in the center of Fig. 9,\nand showing the remaining third dimension separately to\nthe right along an axis. Because of our ordering heuristic,\nthe yellow and purple-blue stimuli were fixed at the origin\nand on the first axis. These assignments mean that the\nfirst two dimensions effectively represent the expected color\ncircle that “bends” the visible physical spectrum from red\nto purple colors into a circle that reflects the psychological\nsimilarity between the end points. Line Length The left panel of Fig. 8 shows posterior model\nprobabilities, assuming equal prior probabilities, for one-,\ntwo-, and three-dimensional MDS representations. To\nassess the stability of the posterior model probability\nestimates, we ran the Warp-III procedure five times based on\nnew samples from the proposal distribution (we always used Our first application involves the similarity judgments\nbetween nine lines of equally increasing length provided\nby 27 participants, as reported in Cohen et al. (2001). We expect these stimuli to have a one-dimensional MDS\nrepresentation, corresponding to line length. Because the\nMinkowski metrics are all equivalent in a one-dimensional\nspace, we do not have any expectations about the metric\nstructure. Thus, we applied our method to these data by 332 Comput Brain Behav (2020) 3:322–340 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 8 Results for line-length similarity data from Cohen et al. (2001). The left panel shows the posterior model probabilities for one- through\nthree-dimensional MDS representations. The right panel shows the 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 8 Results for line-length similarity data from Cohen et al. (2001). The left panel shows the posterior model probabilities for one- through\nthree-dimensional MDS representations. The right panel shows the\ninferred one-dimensional representation with black lines showing the\nline stimuli at their inferred locations and blue histograms showing the\nmarginal posterior distributions for these locations 0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability inferred one-dimensional representation with black lines showing the\nline stimuli at their inferred locations and blue histograms showing the\nmarginal posterior distributions for these locations inferred one-dimensional representation with black lines showing the\nline stimuli at their inferred locations and blue histograms showing the\nmarginal posterior distributions for these locations Fig. 8 Results for line-length similarity data from Cohen et al. (2001). The left panel shows the posterior model probabilities for one- through\nthree-dimensional MDS representations. The right panel shows the been considered in the MDS literature (e.g., Borg and\nGroenen 1997; Carrol and Wish 1974). We consider only\nthe data from the ten participants with normal color vision. the same set of posterior samples). These five repetitions\nare drawn as separate lines but, in this case, the results are\nso similar that they are visually indistinguishable. Line Length The third dimension,\nwhich we did not expect, could correspond to something\nlike luminance, since low luminance purple-like colors are\ngenerally located at one end of the dimension and high\nluminance yellow-like colors are generally located at the\nother end. Colors Our second application considers classic data reported\nby Helm (1964), involving the similarities between ten\ncolors. The experimental procedure involved trials in which\nparticipants were presented with physical tiles of three\ndifferent colors, and moved one of the tiles to reflect their\nperceived overall similarity of the color of this tile to the\ncolors of the other two tiles. Based on these responses, Helm\n(1964) calculated measures of pairwise similarities between\nthe colors, and the resulting proximity data have previously 333 Comput Brain Behav (2020) 3:322–340 yell\n pur\n green-yellow-2\n blue\n green\n green-yellow\n1\n2\n3\n4\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 9 Results for color similarity data from color-normal subjects\nreported by Helm (1964). The left panel shows the posterior proba-\nbilities for one- through four-dimensional MDS representations. The\nright panel shows the inferred three-dimensional representation, with\ntwo dimensions shown as a two-dimensional plot in the center, and the\nthird dim\nlabels s\n95% cre\ndimensi yellow\n purple-blue\n red-purple\n green-yellow-2\n red-orange\n blue\n purple-2\n green\n green-yellow-1\n purple-1\n purple-1\n red-orange\n purple-2\n yellow\n purple-blue\n red-purple\n blue\n green\n green-yellow-1\n green-yellow-2\n1\n2\n3\n4\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability yellow\n purple-blue\n red-purple\n green-yellow-2\n red-orange\n blue\n purple-2\n green\n green-yellow-1\n purple-1\n purple-1\n red-orange\n purple-2\n yellow\n purple-blue\n red-purple\n blue\n green\n green-yellow-1\n green-yellow-2\n1\n2\n3\n4\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 9 Results for color similarity data from color-normal subjects\nreported by Helm (1964). The left panel shows the posterior proba-\nbilities for one- through four-dimensional MDS representations. The\nright panel shows the inferred three-dimensional representation, with\nthird dimension shown along an axis to the right. Circular markers and\nlabels show the inferred locations of each stimulus and error bars show\n95% credible intervals for the marginal posterior distribution for each\ndimension Dimensions third dimension shown along an axis to the right. Circular markers and\nlabels show the inferred locations of each stimulus and error bars show\n95% credible intervals for the marginal posterior distribution for each\ndimension Fig. 9 Results for color similarity data from color-normal subjects\nreported by Helm (1964). The left panel shows the posterior proba-\nbilities for one- through four-dimensional MDS representations. The\nright panel shows the inferred three-dimensional representation, with\ntwo dimensions shown as a two-dimensional plot in the center, and the 8For these stimuli, we did not have access to information about the\nprecise physical values of the radius and angles, and so the depictions\nin Fig. 11 are approximate. Rectangles with Line Segments analyses of these data, we expect a two-dimensional MDS\nrepresentation. We also expect the two stimulus dimensions\nto be psychologically separable. Our third application involves data reported by Kruschke\n(1993) involving the similarity between eight geometric\nstimuli. These stimuli consisted of rectangles with interior\nline segments, and varied in terms of the height of the\nrectangle and the horizontal location of the line segment. A\ntotal of 50 participants provided similarity ratings on a nine-\npoint scale for all 28 stimulus pairs. Based on the original\n(Kruschke 1993) and subsequent (e.g., Lee 2001, 2008) Figure 10 shows the results of applying our method\nassuming both the r = 1.5 and r = 2 metrics. It is clear\nthat a two-dimensional representation with the separable\nr = 1.5 metric is inferred. It has essentially all of the\nposterior probability, with one- and three-dimensional r =\n1.5 representations, and all of the r = 2 representations\nhaving essentially no posterior probability. The inferred 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 10 Results for rectangles with interior line segments data reported\nby Kruschke (1993). The left panel shows the posterior probabilities\nfor one- through three-dimensional MDS representations, for both the\nMinkowski metrics with r = 1.5 and r = 2. The right panel shows\nthe inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension the inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability Fig. 10 Results for rectangles with interior line segments data reported\nby Kruschke (1993). The left panel shows the posterior probabilities\nfor one- through three-dimensional MDS representations, for both the\nMinkowski metrics with r = 1.5 and r = 2. The right panel shows the inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension 334 Comput Brain Behav (2020) 3:322–340 the inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. Rectangles with Line Segments 11 Results for the Shepard circles data collected by Treat\net al. (2001). The left panel shows the posterior probabilities for\none- through three-dimensional MDS representations, for both the\nMinkowski metrics with r = 1.5 and r = 2. The right panel shows\nthe inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension 1\n2\n3\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability Dimensions the inferred two-dimensional representation. The stimuli are shown at\ntheir inferred locations and error bars show 95% credible intervals for\nthe marginal posterior distribution for each dimension Fig. 11 Results for the Shepard circles data collected by Treat\net al. (2001). The left panel shows the posterior probabilities for\none- through three-dimensional MDS representations, for both the\nMinkowski metrics with r = 1.5 and r = 2. The right panel shows Colored Shapes representation closely matches the ways in which the stimuli\nphysically vary, with each psychological axis corresponding\nto an interpretable stimulus dimension. The horizontal axis\ncorresponds to the position of the line segment and the\nvertical axis corresponds to the height of the rectangle. Our final application considers similarity data for nine\ncolored shape stimuli collected by Lee and Navarro (2002). The stimuli were circles, squares, and triangles that were\ncolored red, green, and blue. The data were collected from\n20 participants, each of whom rated the similarity of each\npair of stimuli on a five-point scale. Shepard Circles The middle and right panels show the inferred four-dimensional\nrepresentation, with two dimensions shown in each panel. The col-\nored shapes show the inferred locations of each stimulus and error bars\nshow 95% credible intervals for the marginal posterior distribution for\neach dimension 1\n2\n3\n4\n5\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability Dimensions representation, with two dimensions shown in each panel. The col-\nored shapes show the inferred locations of each stimulus and error bars\nshow 95% credible intervals for the marginal posterior distribution for\neach dimension Fig. 12 Results for colored shapes data reported by Lee and Navarro\n(2002). The left panel shows the posterior probabilities for one-\nthrough five-dimensional MDS representations for the Euclidean met-\nric. The middle and right panels show the inferred four-dimensional stimulus set. The corresponding approximately equilateral\ntriangles could be equally well accommodated by any of\nthe Minkowski metrics we are considering. Thus, from\na statistical perspective—without regard to the theory of\nseparable and integral stimuli—we expect the simplest\nmetric to be inferred. Since all metrics should be able\nto fit the data, the one with the smallest functional form\ncomplexity should be preferred. expectations, with the exception of the color application. In addition, where inferences about whether a Euclidean or\nnon-Euclidean metric structure were made, they matched\ntheoretical expectations. It is interesting to note that all\nof the applications for which non-Euclidean metrics made\ninference difficult involved stimulus domains for which the\nexpectation was that the Euclidean metric was appropriate. We also think that the five applications serve to\ndemonstrate the usefulness of our approach to determining\ndimensionality and metric structure. Our approach is to\ntreat these determinations as Bayesian model-selection\nproblems and use Bayesian posterior probabilities to make\ninferences. Complete Bayes posterior probabilities have\nnot been used in this way previously to determine either\ndimensionality or metric structure, and our introduction of\nthe Warp-III method to solve the difficult computational\napproximation problems involved represents progress on\nthese long-standing challenges in MDS modeling. We found that this was a third case in which we were\nunable to generate samples with acceptable convergence for\nthe r = 1.5 metric. Accordingly, Fig. 12 shows the results\nof applying our method assuming the Euclidean metric. A four-dimensional representation is clearly favored. This\nrepresentation is shown in terms of two two-dimensional\nsubspaces, and has the expected structure. The middle panel\nof Fig. Shepard Circles Our fourth application involves data collected by Treat et al. (2001), involving the similarity between nine geometric\nstimuli known as “Shepard circles.” These stimuli consist\nof a closed semi-circle with an interior ray from the\ncenter to the perimeter. The nine stimuli are constructed\nby exhaustively varying three different radius lengths and\nthree different angles for the internal ray. As for the\nrectangles with line segments, we expect a separable two-\ndimensional MDS representation. For these stimuli, we\nexpect the dimensions to correspond to the radius and angle\ndimensions. Following the previous analysis in Lee and Navarro\n(2002), we expect a four-dimensional representation. This\nrepresentation is best understood as being the product\nof a pair of two-dimensional representations, with one\nrepresenting the similarities between the shapes, and the\nother representing the similarities between the colors. There\nare only three shapes and three colors, and neither set of\nthree has a natural ordering. Instead, the circle, square, and\ntriangle are all approximately equally different from one\nanother, and the same is true of the red, green, and blue\ncolors. These equal similarities are naturally represented by\ntwo-dimensional approximately equilateral triangles. The\nfour-dimensional representation we expect is simply the\nindependent combination of these two two-dimensional\nsubspaces. Figure 11 shows the results of applying our method\nassuming both the r = 1.5 and r = 2 metrics.8 It is\nclear, once again, that a two-dimensional representation\nwith the separable r\n=\n1.5 metric is inferred. The\ninferred representation also again closely matches the ways\nin which the stimuli physically vary, with the horizontal\naxis corresponding to the radius of the semi-circle and the\nvertical axis corresponding to the angle of the ray. Our expectations for the metric structure of the MDS\nrepresentations are less straightforward. Theoretically, the\ninteraction between the shape and color dimensions is a\nclassic example of a separable relationship. The metric\nstructure within the color subspace, however, is theoretically\nintegral, as for the previous application. Countering these\ntheoretical expectations is the fact that there are only three\nvalues for the color and shape dimensions present in the Comput Brain Behav (2020) 3:322–340 335 1\n2\n3\n4\n5\nDimensions\n0\n0.2\n0.4\n0.6\n0.8\n1\nModel Probability\nFig. 12 Results for colored shapes data reported by Lee and Navarro\n(2002). The left panel shows the posterior probabilities for one-\nthrough five-dimensional MDS representations for the Euclidean met-\nric. Shepard Circles 12 shows a subspace that captures the similarity\nrelationships between the red, green, and blue colors. The\nright panel shows a subspace that captures the similarity\nrelationships between the circle, square, and triangle shapes. These subspaces were found using an orthogonal Procrustes\nmethod (Borg and Groenen 1997, p. 162). In particular,\nwe solved for the orthogonal transformation matrix that\nmost closely mapped the inferred coordinate locations\nto the expected representational structure, defined as the\nproduct of two subspaces each with an equilateral triangle\nconfiguration. Despite this progress, we think the greatest contribution\nof the current work is to highlight fundamental challenges\nin MDS models of mental representation, and suggest new\navenues for theoretical development. The challenges largely\nstem from our insistence on fully Bayesian inference,\nwhich has enormous advantages in terms of reaching\ncomplete, coherent, and principled conclusions, but also\nraises technical hurdles. The opportunities largely stem\nfrom our adoption of a generative modeling approach (Lee\n2018). In particular, we think there are many remaining\npossibilities relating to the use of different metrics in MDS\nrepresentations, and that there is an opportunity to extend\nthe generative approach to develop more complete cognitive\nprocess models for inferring MDS representations. We\nconclude by discussing some of these challenges and\nopportunities. Other Representations We did not consider Minkowski metrics with r < 1. This\npossibility has been proposed as a way of representing\nstimulus domains in which the component dimensions\ncompete for attention (Shepard 1987, 1991; Tversky and\nGati 1982). The identifiability constraints for this metric\npresent an open research challenge, and it is not clear\nhow well DE-MCMC sampling methods will perform in\ninferring representations. Our current approach to determining the appropriate\nmetric treats this inference as a model-selection problem,\nand only considers the possibilities r\n= 1.5 and r\n=\n2. Allowing for other metrics is theoretically interesting,\nbut computationally difficult. One obvious cost is the\nneed to generate posterior probabilities across a larger set\nof candidate models. But it also seems likely that some\nmodels will be difficult to make inferences about. We\ntried our DE-MCMC approach for r = 1.1 on a number\nof data sets, and were not able to achieve satisfactory\nconvergence. Furthermore, as explained above, for a few\nof the applications, we were also not able to achieve\nsatisfactory convergence for r = 1.5. These challenging\ncases involved stimulus domains for which the expectation\nwas that the Euclidean metric was appropriate, which\nleads to a speculative suggestion that failure is related\nto model mis-specification. This is a potential example\nof a general aspect of Bayesian model comparison that\ncan be computationally challenging: in order to rule out\nmodels that are likely mis-specified, one needs to be able\nto infer them well enough that they can be part of the\nmodel comparison. Although we believe that DE-MCMC\nis a powerful sampling algorithm which substantially helps\nalleviate the issue of non-converging chains, future research\nshould explore different sampling algorithms that may\nperform better, particularly for non-Euclidean metrics. There is also the possibility of moving beyond the\nMinkowski family of metrics. In his presidential address,\nShepard (1974, Figure 11) presented a taxonomy of\nmetric spaces, each of which makes different fundamental\nrepresentational assumptions that could be appropriate for at\nleast some stimulus domains. There has been relatively little\nwork in exploring these possibilities. Lindman and Caelli\n(1978) investigated MDS representations using Riemannian\nspaces with constant curvature, and Cox and Cox (1991)\npresented compelling applications for a special case of this\napproach involving MDS representations on a sphere. Discussion Collectively, the five applications demonstrate that our\nmethod is able to make reasonable inferences about MDS\nrepresentations. The inferred number of dimensions, and the\ninferred stimulus locations, generally matched theoretical 336 Comput Brain Behav (2020) 3:322–340 Technical Challenges insight into how people represent the real-world stimuli. The successful applications we presented—in which there\nwere clear expectations about dimensionality, metric, and\nrepresentational structure—provide a basis for believing the\nBayesian framework can provide this insight to situations\nwhere, because there are no clear theoretical expectations,\nanswers must be inferred from data, if and when the\ncomputational technical hurdles are overcome. Developing a generative MDS model in a Bayesian setting\nrequired the key issue of identifiability and invariance to\nbe solved in terms of prior information, rather than more\nheuristically through post-processing. We used an exist-\ning solution to this challenge for the Euclidean metric,\nand proposed a solution for psychologically interpretable\nnon-Euclidean metrics with 1 < r < 2. We also high-\nlighted, however, the fundamental intractability of MDS\nrepresentations using the city-block metric. This intractabil-\nity has been documented before (Bortz 1974; Frank 2006,\nFigure 5.4; Shepard 1974, Figure 11), but has not pre-\nvented the use of MDS representations inferred based on the\ncity-block metric in the cognitive modeling literature (e.g.,\nKruschke 1993; Lee and Wetzels 2010). Conclusion One example, involving the line-length application, was\npresented in a preliminary form by Lee (2014). A simple\nplot of the raw behavioral data suggests that one of the\n27 participants appears to have reversed the scale that was\nused to judge similarity. This means that their judgments\ncontaminate the inference of the MDS representation. Lee\n(2014) used a simple latent-mixture model extension of the\nbasic MDS generative model, in which either the scale was\nused correctly or reversed. One participant was inferred\nto have reversed the scale, as expected. Perhaps more\nimportantly, however, the resulting inference about the one-\ndimensional MDS representation was shown to have less\nuncertainty than the one shown in Fig. 8. In this way,\nthe introduction of individual differences in the cognitive\nprocess of similarity judgment helped decontaminate the\ninference about the representation of stimuli. We adopted a Bayesian model-selection approach to the\nproblem of determining the dimensionality and metric struc-\nture of MDS representations, while considering psycholog-\nically interpretable Euclidean and non-Euclidean metrics. Our methods for inferring the representations and choosing\ntheir dimensionality and metric structure show the promise\nof the approach, but computational challenges remain a\nbarrier in terms of an easy-to-use general capability. Our\nmethods and applications also show the promise of plac-\ning MDS representations in a generative cognitive modeling\nframework, offering the possibility of new models of how\npeople represent stimuli, and how those representations help\nguide behavior. Acknowledgments All code is available on the Open Science\nFramework: https://osf.io/82g3r/. We thank Rob Nosofsky and Mike\nD’Zmura for helpful discussions. QFG acknowledges the support\nby a Netherlands Organization for Scientific Research (NWO)\ngrant (406.16.528). Correspondence should be sent to Quentin F. Gronau, University of Amsterdam, Nieuwe Achtergracht 129 B,\n1018 WT Amsterdam, The Netherlands. E-mail may be sent to\nQuentin.F.Gronau@gmail.com. Acknowledgments All code is available on the Open Science\nFramework: https://osf.io/82g3r/. We thank Rob Nosofsky and Mike\nD’Zmura for helpful discussions. QFG acknowledges the support\nby a Netherlands Organization for Scientific Research (NWO)\ngrant (406.16.528). Correspondence should be sent to Quentin F. Gronau, University of Amsterdam, Nieuwe Achtergracht 129 B,\n1018 WT Amsterdam, The Netherlands. E-mail may be sent to\nQuentin.F.Gronau@gmail.com. The same basic generative approach could support much\nmore general cognitive process modeling using MDS rep-\nresentations. Other Representations A new idea raised by our application to the colored shape\nstimuli involves the possibility of different metric structures\nwithin the same representation. These stimuli involved\ntwo sorts of stimulus dimensions: those representing color,\nwhich are usually considered to be integral, and those\nrepresenting qualitatively different shapes, which seems\nmore separable. Certainly the interaction between the color\ndimensions and the shape dimensions would be expected\nto be separable, since it seems likely people can selectively\nattend to either the color or the shape of a stimulus,\ndepending upon the cognitive context. This suggests a\ngeneralization of the MDS models in which each pair of\ndimensions is associated with a metric. Collectively, these technical challenges mean that our\napproach cannot currently be applied to large naturalistic\nstimulus domains. For example, Nosofsky et al. (2018)\nconsider MDS representations based on sparse matrices\nof pairwise similarity judgments for a set of 360 images\nof rocks, and Hebart et al. (2020) report extensive\ncrowd-sourced triadic comparison similarity data for 1854\nimages of real-world objects. Being able to determine\nthe dimensionality, metric structure, and psychological\nrepresentations of MDS representations of these domains\nusing the Bayesian framework would potentially offer deep Finally, there are alternative representational models,\nwhich do not assume stimuli are represented by values on\ndimensions, that can compete with or complement MDS\nmodels. These alternatives include feature-based represen-\ntations (Tversky 1977), such as those found by additive\nclustering and related methods (Shepard and Arabie 1979)\nand special cases such as tree-based models (Corter 1996;\nShepard 1980). One attraction of the Warp-III approach\nwe used is that it could estimate Bayes factors between\nfundamentally different sorts of representations—such as Comput Brain Behav (2020) 3:322–340 337 individual differences, such as INDSCAL (Carroll and\nChang 1970; Carroll 1972). These would be easy to imple-\nment within our generative modeling framework. A model\nlike INDSCAL, which assumes individuals weight the latent\nstimulus dimensions differently, relies on the appropriate\nnumber of dimensions being inferred, and evidence that the\nstimulus domain is separable. In this way, the potential of\nour method to make these inferences is especially important. As a final example, the rectangle and line segment stim-\nuli are used by Kruschke (1993) to study category learning,\nbut the similarity data and category learning data are ana-\nlyzed independently. Other Representations In effect, the similarity data are used to\ngenerate the MDS representation, and that representation is\nthen assumed to provide the fixed basis for category learn-\ning. An alternative approach would be to infer the MDS\nrepresentation jointly from both the similarity judgments\nand the category learning choices. This sort of flexibility\nraises the possibility of tackling more complicated cogni-\ntive phenomena, such as the ability to adapt representations\nin response to changes in the external environment, or the\ncurrent context or goals. comparing dimensional and featural representations—since\nit operates directly on posterior samples for each model\napplied independently to the data. Even further, Navarro and\nLee (2003) proposed a hybrid model of stimulus represen-\ntation that combined both dimensions and features, and it\nwould be conceptually elegant to choose between all of the\ncandidate models, with various combinations of dimensions\nand features, using our methods. Navarro and Lee (2003)\nused an approximate analytic approach for this purpose,\nwhich would be significantly improved by an approach\nbased on Bayes factors. MDS Cognitive Process Models Our modeling approach is generative, but is based\non an extremely simple cognitive model. In essence,\nwe assume that all participants have the same MDS\nrepresentation, and produce dissimilarity judgments for\npairs of stimuli that directly reflect the distances between\nthose stimuli in the representation. It is likely that much\nbetter generative models can be developed by considering\nmore realistic processing assumptions, and especially by\nincluding individual differences. Appendix 1. The ordering heuristic Figure 13 provides a concrete example to motivate the\nneed for the ordering heuristic. It is clear this is an\ninferior representation to the one presented in Fig. 8. In\nFig. 13, the first- and second-line stimuli, which are the two\nshortest, are located at almost the same point, rather than\nbeing appropriately spaced to reflect their psychological\ndissimilarity. Consistent with this intuition, the posterior\ndensity is worse for the representation in Fig. 13 than the\nrepresentation in Fig. 8. We used this ordering heuristic for the colors and colored\nshapes applications. For the line-length application, we used\nthe heuristic as described but then, in an additional step,\nswitched the first stimulus with the second stimulus. This\nswitch helped prevent the posterior for the ninth stimulus,\ncorresponding to the longest line, push against the upper\nbound of 1. For the rectangles with interior line segments\nand Shepard circles applications, we used the heuristic as\na starting point, but we then reordered some of the stimuli\nmanually since it seemed to help with convergence. This suboptimality is caused by the naive application of\nthe constraints identified in Fig. 2 for a one-dimensional\nrepresentation. The first stimulus is fixed at the origin, and\nthe second stimulus is constrained to be positive. It is clear\nfrom Fig. 13 that the second stimulus is indeed inferred to be\npositive, but is extremely close to zero, with the remaining\nlonger line stimuli “flipping” to negative values in the MDS\nspace. This configuration still satisfies the proximity data\nreasonably well, because the required distance between the\nfirst two stimuli is small, and the distances from the first\nand second stimuli to all of the others is approximately\nconserved. Thus, it is the choice of the two similar stimuli\nas those that are constrained that leads to this potential for a\nlocal maximum and suboptimal representation. Conclusion The hierarchical, latent mixture, and common\ncause model structures advocated by Lee (2018) could allow\nfor rich accounts of individual differences in judgment\nprocesses or stimulus representations, and allow for mod-\nels that extend beyond the judgment of similarity to other\ncognitive capabilities like categorization and inference. As\none example, Ennis (1992) considers extended assump-\ntions about MDS representations that allow for the noisy\nrepresentation of perceptual stimuli, which could be incor-\nporated by adding hierarchical structure to the coordinate\nlocations. As another example, there are extensions of the\nbasic MDS model we considered that allow for structured Open Access This article is licensed under a Creative Commons\nAttribution 4.0 International License, which permits use, sharing,\nadaptation, distribution and reproduction in any medium or format, as\nlong as you give appropriate credit to the original author(s) and the\nsource, provide a link to the Creative Commons licence, and indicate\nif changes were made. The images or other third party material in this\narticle are included in the article’s Creative Commons licence, unless\nindicated otherwise in a credit line to the material. If material is not\nincluded in the article’s Creative Commons licence and your intended\nuse is not permitted by statutory regulation or exceeds the permitted Comput Brain Behav (2020) 3:322–340 338 stimuli. Our heuristic for doing this is based on the across\nparticipants averaged pairwise dissimilarity ratings. The\nfirst two stimuli are chosen to be the ones with the largest\naveraged pairwise dissimilarity. The remaining stimuli are\nchosen, one at a time, by considering the minimum averaged\npairwise dissimilarity to the already selected stimuli. Specifically, the next stimulus is always chosen to be the\none with the maximum value for the minimum averaged\npairwise dissimilarity to the already selected stimuli. use, you will need to obtain permission directly from the copyright\nholder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Appendix 2. Transformation ordered vector\n(0–1 bounded) The constrained vector x, 0 ≤x1 ≤x2 ≤. . . ≤xK ≤1,\ncan be transformed to an unconstrained vector y ∈ℜK as\nfollows: yk =\n\u0016\n\u0003−1 (xk)\nif k = 1,\n\u0003−1 \u0014\nxk−xk−1\n1−xk−1\n\u0015\nif 1 < k ≤K, Accordingly, we developed an ordering heuristic to try\nand assign the constraints for the various dimensionalities\nand metrics to stimuli that are sufficiently dissimilar. Because higher dimensionalities place constraints on more\nthan two stimuli, the general approach is to order all of the where \u0003−1(·) denotes the inverse of the normal CDF. The\ninverse transformation is given by: where \u0003−1(·) denotes the inverse of the normal CDF. The\ninverse transformation is given by: xk =\n\u0017 \u0003 (yk)\nif k = 1,\nxk−1 + (1 −xk−1) \u0003 (yk) if 1 < k ≤K, if k = 1, 0\nFig. 13 A suboptimal one-dimensional representation of the line-\nlength similarity data from Cohen et al. (2001), motivating the\nneed for the ordering heuristic. The black lines show the stimuli\nat their inferred locations in the representation, and the blu\nhistograms show the marginal posterior distributions for thes\nlocations at their inferred locations in the representation, and the blue\nhistograms show the marginal posterior distributions for these\nlocations Fig. 13 A suboptimal one-dimensional representation of the line-\nlength similarity data from Cohen et al. (2001), motivating the\nneed for the ordering heuristic. The black lines show the stimuli 339 Comput Brain Behav (2020) 3:322–340 where \u0003(·) denotes the normal CDF. Note that xk is a func-\ntion of y1, y2, . . . , yk (the dependence on y1, y2, . . . , yk−1\nis “hidden” in xk−1). Crucially, xk does not depend\non yk+1, yk+2, . . . , yK. Consequently, the Jacobian matrix\nJ of the transformation is lower triangular so that its\ndeterminant |J | is obtained by multiplying its diagonal\nentries. 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