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18. An article which discusses the limitations of a small scale study

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17. An article which uses purposive or opportunistic sampling

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17. An article which uses purposive or opportunistic sampling

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12.Longitudinal associations between childhood obesity and academic achievement:Systematic review with focus group data

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12.Longitudinal associations between childhood obesity and academic achievement:Systematic review with focus group data

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https://link.springer.com/article/10.1007%2Fs13679-017-0272-9

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Abstract

Purpose

The purposes of this study were to review the evidence on longitudinal associations between child and adolescent obesity and academic achievement and to provide perceptions of adolescents with obesity and their parents on this topic.

Recent Findings

Synthesis of 31 studies (from 17 cohorts) suggested that relationships between obesity and academic achievement are not well established, except for adolescent girls' maths attainment, potentially mediated by both weight-related bullying and executive cognitive functions. Focus groups with adolescent girls with obesity confirmed experiences of psychosocial distress at school particularly during Physical Education. Adolescents perceived that obesity was not related to academic achievement directly, but by their attitude to school.

Summary

Interventions are warranted to promote psychosocial wellbeing and cognitive abilities linked to academic achievement in adolescent girls with obesity. Physical Education should be a positive experience for children and adolescents with obesity.

Introduction

The prevalence of childhood obesity is increasing in many countries with adverse societal impact [ 1]. Nevertheless, to date, successful efforts to tackle the problem of childhood obesity have been insufficient [ 1]. Further economic evaluation is required, including an improved understanding of the link between academic achievement and childhood obesity 'strengthening the economic arguments for interventions' [ 1]. A high value is placed on academic achievement by policymakers, schools, and families [ 2]. If obesity in childhood and adolescence is related to academic achievement, this would provide increased support for interventions aimed at preventing and treating obesity in young people.

'Direct' mechanisms for how childhood obesity could be associated to poorer academic achievement have been postulated [ 3, 4] which are associated with children's cognitive ability [ 5, 6**]. 'Indirect' mechanisms could include obesity-related adverse physical and mental health leading to increased school absenteeism [ 7], obesity-related psychosocial distress (e.g. isolation, bullying) [ 8, 9], stigmatisation by peers and teachers [ 10, 11], poor sleep due to obesity-related disordered breathing [ 12, 13], cardio-metabolic co-morbidities [ 14, 15, 16], nutritional intake [ 17] and low levels of physical activity or fitness [ 18].

Two literature reviews have examined the association between overweight/obesity and academic achievement [ 19, 20]. Both concluded, from cross-sectional data, that children and adolescents with overweight and obesity typically perform less well in school compared to normal-weight peers. Caird et al. [ 19] noted a lack of longitudinal studies addressing potential mediators of the relationship between childhood obesity and academic achievement. Since the publication by Caird et al. [ 19] further evidence on the longitudinal relationship between obesity and academic achievement has emerged. Further, limited research has been undertaken to gain insight into the views of adolescents with obesity and their parents on the link between obesity and academic achievement. Adolescent girls with obesity tend to experience weight-related bullying in school more often which might make them more vulnerable to lower academic achievement [ 21]. Therefore, the aims of this study were to systematically review and critique the evidence on the longitudinal associations between childhood obesity and academic achievement and to complement the review findings with qualitative data on the perspectives of adolescent girls specifically, and their parents. The following research questions were addressed:

1. 1.
Is there evidence of a longitudinal association between childhood or adolescent obesity and academic achievement?

2. 2.
Does a change in obesity status over time influence the association with academic achievement?

3. 3.
What factors moderate or mediate the association between childhood or adolescent obesity and academic achievement?

4. 4.
How do adolescent girls with obesity and their parents perceive the link between obesity and academic achievement?

PART 1: Methods and Results of the Systematic Review

Methods

In January 2017, a literature search was performed in Medline, Embase, PsycINFO, Education Resource Information Centre and SportDiscus; this was restricted to English language but not publication year. The search strategy was adapted for each database and is shown for Medline (ovid) in Table 1. Reference lists and forward citations of included studies were screened. Titles and abstracts were screened independently for eligibility (AM, JNB). Full-text articles were independently screened for inclusion by AM, JNB and SM (DHS arbitrated) using the following inclusion criteria:

1. (i).
Study design: observational prospective cohort studies

2. (ii).
Participants: healthy 3-18-year olds (i.e. absence of conditions associated with overweight/obesity and impaired school performance)

3. (iii).
Exposure: measures or estimates (i.e. self-reported) of body weight status as either body mass index (BMI; values, percentiles or z scores), body fat or waist circumference. Eligible exposures were also the BMI-derived weight status classifications overweight, obesity and overweight and obesity combined

4. (iv).
Outcome: measures or estimates of academic achievement limited to standardised test scores, teacher- or self-reported grades of specific school subjects or average attainment during compulsory education

Table 1 Search strategy for ovid Medline

Experimental studies were excluded as these are systematically reviewed elsewhere [ 22]; an update of the evidence from experimental studies is currently in progress.

Data were independently extracted by two reviewers and cross-checked using a tested and pre-defined data extraction template. Included studies were independently scored for methodological quality by AM and SM, disagreements were resolved through discussion. Quality assessment was scored using criteria for observational longitudinal research [ 23]. Quality domains included sampling and recruitment, participant characteristics, attrition, data collection methods, and data (total 19 items). Quality criteria were scored as positive, negative or 'unclear'. If a study provided no or insufficient information, we scored the criterion at issue as 'unclear'. Where the study referred to additional publications, we retrieved the publication to score the quality criterion. Studies were considered of high methodological quality when the percentage of items that scored positively was ≥70% [ 24].

A narrative evidence synthesis was performed for which individual study findings were synthesised graphically by school subjects, type of obesity measure and sex. The primary analysis considered the study quality in that studies with an overall quality score of <70%, and studies with a quality score ≥70% but with self-reported weight and height, and/or academic achievement were removed from the evidence synthesis. Where both objectively and subjectively assessed academic achievement was available, findings of objectively obtained scores were considered for the evidence synthesis. Study and population characteristics and assessment tools for academic achievement varied substantially between included studies. Although several studies utilised similar cohort studies for their analyses, combining effect sizes of the same study population would result in overestimation of the effect size [ 25]. Therefore, no meta-analysis was performed.

Results

The systematic literature search results were summarised in Fig. 1. Eighty potentially relevant full-text articles were identified, of which 30 articles (31 studies) were included in this review.

Literature search and study selection flow diagram

A detailed description of population characteristics, exposure, outcomes, confounding/mediating variables and main results of the included studies is provided in Table 2. The included studies comprised 17 distinct cohort datasets, of which 11/31 studies performed secondary analysis of the Early Childhood Longitudinal Study-Kindergarten Class (ECLS-K) in the USA [ 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. Seven studies used five other datasets from the USA [ 35, 37, 38, 39, 40, 41, 42], and eight studies utilised existing cohort datasets from Canada [ 43, 44], the UK [ 45, 46], The Netherlands [ 47, 48], Australia [ 49], and Taiwan [ 50]. Five studies from Australia, Germany, Taiwan, Thailand, and Peru used primary data for their prospective analyses [ 51, 52, 53, 54]. The number of participants ranged from 405 to 21,260 (median 3362 participants) and the follow-up periods ranged from 1 to 9 years (median 3 years). Twenty-four studies assessed the association between overweight and/or obesity and academic achievement whereas two studies related percentage body fat [ 46, 53], one study BMI z scores [ 32] and two studies BMI [ 40, 54], to children's academic achievement. Measures of academic achievement were obtained from school records or standardised tests by 25 studies and five studies relied on teacher or self-reported academic attainment [ 37, 42, 44, 52, 55].

Table 2 Characteristics of included studies reporting on the longitudinal relationship between child and adolescent obesity and academic achievement

Sixteen studies assessed the association between overweight or obesity at one time point and academic achievement later in life (Fig. 2a). Of those, eight studies were excluded from the primary analysis due to methodological shortcomings (Fig. 2b). Fourteen studies using data from eight cohorts assessed the association between change in obesity status and academic achievement (Fig. 2b). However, only four studies (two cohorts) were of high methodological quality and so included in the primary analysis (Fig. 2 a). Change in obesity status was classified as persistent obesity, development of obesity, and 'growing out' of obesity (change from obesity to overweight or healthy weight).

Evidence synthesis on the longitudinal association between child and adolescent overweight and obesity and academic achievement. a Primary analysis with 15 high-quality studies. b 24 high- and low-quality studies. Each study may be represented in multiple outcomes and weight categories

All included studies adjusted for a number of confounding variables known to be associated with both obesity and academic achievement such as measures of socioeconomic status (e.g. parental education, family income). The quality scoring of each individual study ranged from 53% [ 26] to 95% [ 30, 45]. The level of agreement for the quality scoring between the reviewers (AM, SM) was 96% ( k = 0.91). In 24/30 studies, methodological quality was scored high (≥70%).

Association Between Overweight or Obesity and Academic Achievement

Figures 2a, and b shows the graphically synthesised data of the included studies on the association between child and adolescent overweight and obesity and academic achievement, namely reading/language, maths, science and a combined average of school subjects. Table 1 also summarises the results for each of the included studies.

Overweight and Academic Achievement

Five high-quality studies (four articles) assessed the link between overweight and maths achievement with one study [ 34] suggesting significantly lower maths scores in adolescents with overweight at 14 years compared to healthy weight peers ( n = 5072). Four studies reported that there was no evidence of an association between overweight and maths achievement [ 31, 35, 49].

In terms of reading achievement, six high-quality studies (five articles) consistently concluded that there is no evidence of a significant negative association between overweight and reading achievement [ 31, 35, 41, 45, 49].

Overweight-related associations with science achievement was assessed in one study ( n = 18,820) which indicated no significant relationship between overweight at age 7 years and science achievement at 14 years [ 31].

Eight studies assessed the association between overweight and average achievement score of multiple school subjects [ 37, 38, 42, 44, 47, 48, 50, 55]. All but one study [ 47] showed methodological limitations and so were removed from the primary analysis (Fig. 2a). The study indicated that there is no significant association between children aged 8 years with overweight and average achievement at 12 years ( n = 1543).

Obesity and Academic Achievement

Ten articles assessed the association between obesity at baseline and academic achievement at follow-up [ 27, 31, 35, 36, 38, 39, 42, 45, 47, 49].

Based on three high-quality studies using data from three distinct cohorts ( n = 8641), there was consistent evidence for significantly lower maths scores at age 13-14 years in girls with obesity compared to healthy-weight peers [ 35, 45, 49]. This finding was not evident in preschool-aged girls with obesity [ 35]. Two studies, provided data jointly for girls and boys from the ECLS-K cohort, and suggested a significantly negative association between obesity at age 5-7 years and maths achievement at age 14 years [ 31, 34]. For boys with obesity, three of the four cohort studies did not find a significant association [ 35, 45].

Six high-quality studies (four articles) provided evidence on reading achievement in children with obesity compared to children with healthy weight (Fig. 2a). One study considered boys and girls as one study sample and found that students with obesity at age 7 years performed less well in reading and language achievement at age 14 compared to peers with healthy weight [ 31]. Where studies distinguished between reading achievement of boys and girls, 4/5 studies suggested no association in both sexes (Table 2).

There was no evidence of an association between obesity and science achievement when girls and boys were analysed as one study sample [ 31], whereas stratified analysis for gender suggested lower science scores in girls with obesity but not in boys at age 13 and 16 years [ 45].

Associations Between Change in Obesity Status and Academic Achievement

Persistent Obesity and Academic Achievement

Using the same cohort (ECLS-K), two studies reported significantly lower maths achievement scores in girls and boys aged 9 and 11 with persistent obesity [ 29, 30].

Data from two high-quality studies (two cohorts) on reading achievement were conflicting for girls, with one study indicating a significantly negative association at age 13 and 16 years [ 45] and one study indicating no significant association at age 9 years [ 30]. The latter study also indicated no evidence of an association between boys with persistent obesity and reading achievement (Table 2).

No high-quality data were available for science scores and combined average scores of multiple school subjects (Fig. 2 b).

Development of Obesity and Academic Achievement

There was no evidence of an association between developing obesity and reading [ 30, 45] and science [ 45] achievement in boys and girls.

Developing obesity was not significantly associated with lower maths scores in male and female adolescents aged 13 and 16 years [ 45] but was in younger children aged 9 and 11 years [ 29, 30]. When combining data of boys and girls and using a longer follow-up duration up to age 14 years, Kranjac (2015) confirmed a negative association between adolescents who developed obesity over time and maths achievement, compared to peers that maintained a healthy weight [ 34].

Data from high-quality studies were lacking for combined average school subject achievement.

'Growing Out' of Obesity and Academic Achievement

Based on a single high-quality study [ 45], there was no evidence of an association between growing out of obesity and reading/language, maths and science achievement (Table 2). No data were available for combined school subject achievement.

Moderating or Mediating Factors

The present systematic literature review and data synthesis of high-quality studies indicated that the association between childhood obesity and academic achievement varied by sex, age and school subject. A significantly negative association between obesity and maths achievemement was evident in adolescent girls, while the evidence suggested no association with math achievement in younger girls with obesity and in boys with obesity in general.

Out of the 30 included studies, six studies performed mediation analyses to identify the role of plausible factors that potentially mediate the relationship between child/adolescent obesity and academic achievement. The following mediating factors were assessed:

Cognitive Abilities

Manes (2015) concluded that childhood obesity at age 9 years indirectly predicted academic achievement at age 15 years through the cognitive processes involving executive functioning and concentration [ 40]. In contrast, Booth et al. (2014) suggested no mediating role of full-scale intelligence quotient between obesity at age 11 years and academic achievement at age 16 years [ 45].

Age of Menarche

Booth et al. also found no mediating effect of age of menarche [ 45].

Physical Health

One study indicated that self-reported health problems influencing performance at school did not mediate the association between obesity age 8 years and academic achievement age 12 years [ 47].

Internalising Behaviour (Including Anxiety, Self-Esteem and Depressive Symptoms)

Teacher-rated internalising behaviour was found to have a significant mediating effect between persistent obesity from preschool age and maths achievement in boys and girls aged 9 and 11 years [ 29]. However, Manes (2015) reported that internalising symptoms assessed using a standardised inventory did not mediate the association between obesity and maths and reading achievement [ 40]. This finding was supported by four other studies, which assessed the mediating role of depression [ 38, 45, 47, 55].

Self-Efficacy

Two studies consistently concluded that the association between child and adolescent obesity and academic achievement at age 14 is not mediated by general self-efficacy [ 34, 55].

Psychosocial Factors

Psychosocial distress in the form of being bullied [ 47] and teacher-rated interpersonal skills [ 29] had significant mediating effects between weight status at age 7-8 and academic achievement in girls age 11-12 years.

In summary, the current evidence suggested that cognitive processes involving executive functions and psychosocial factors might mediate the association between obesity and academic achievement.

PART 2: Methods and Results of the Qualitative Research

This preliminary exploratory qualitative study was complementary to the quantitative data from the systematic review providing more in-depth insight of a few adolescent girls, and providing them with a voice that allows greater understanding of the relationship between obesity and academic achievement [ 35, 45, 49].

Methods

Following completion of a weight management programme (Get Going NHS Lothian: www.nhslothian.scot.nhs.uk/getgoing/) in Scotland, four obese adolescent girls (aged 12-15 years, mean body mass index 99.6th percentile relative to 1990 UK reference) and one of their respective parent/guardians (i.e. 4 dyads in total) participated in separate focus groups of 60 to 90 min (i.e. two focus groups). Focus groups were led by AM, included semi-structured open-ended questions and were audio recorded and subsequently transcribed verbatim using NVivo10 [ 56]. Data from adolescents and parents/guardians were analysed separately following an inductive thematic analysis [ 57]. The text was coded and similar codes were clustered into hierarchical themes. Trustworthiness of the analysis was enhanced through independent coding (AM, AN) and member checking. This study was approved by the University of Edinburgh and the National Health Service South East Scotland Research Ethics Committee.

Results

From the focus groups, it was evident that, despite negative body weight-related experiences in school, the girls had a generally positive attitude to education and school. They said that they do very well in school and usually outperform healthy-weight classmates. The only subject the girls mentioned that they were not good at was Physical Education (PE) and this was perceived to be related to the girls' body weight. Parents/guardians and adolescents perceived that academic achievement is not necessarily related to body weight per se; academic achievement depends on pupils' attitude towards learning and efforts put into school work. However, both adolescents and parents/guardians believed that the reason for the girls' good academic achievement is that they were less distracted from school work due to the lack of friends and absence of good peer relationships. The girls said that having friends in class can lead to girls spending more time talking to their friends rather than concentrating and taking school seriously.

Additional themes emerged on female adolescents' experiences in the school environment that were directly influenced by their body weight status. Both adolescent girls and parents/guardians reported negative psychological consequences of obesity, for example, low self-confidence and unhappiness due to body weight. Social consequences were also highlighted, for example, the girls reported difficulties being accepted and understood by healthy-weight peers in school, and were felt to be lacking friends. From both adolescents' and parents/guardians' responses, it emerged that the girls experienced rejection by peers in school, negative body weight-related comments, classmates laughing and gossiping about them and that at times they feel isolated and ignored. The girls viewed having a higher body weight than other teenagers in school as a disadvantage during PE, and for getting appropriately sized clothes for school (lab coats, school uniform, PE kits). PE was reported to be an environment where the girls were strongly exposed to body weight-related teasing. Some girls also felt ignored and excluded from activities by PE teachers. In contrast, classroom teachers were perceived as non-judgmental towards the increased body weight. Eating healthily in school and receiving support from classmates to do so was described as difficult and sometimes even a reason for being isolated from peers.

Discussion

This paper builds on previous reviews on childhood obesity and its association with academic achievement [ 19, 20], specifically focusing on longitudinal studies. It also provides insight into mediators or moderators, and perceptions of adolescent girls with obesity and their families on the obesity-academic achievement association. In summary, the systematic evidence synthesis showed that obesity is negatively associated with adolescent girls' maths achievement. There is some evidence that this negative association is mediated by psychosocial experiences such as body weight-related bullying and participants' cognitive abilities involving executive functions. However, discrepancies arise for other subjects and age groups where there was less convincing evidence of associations. The role of change in obesity status over time on academic achievement is less well investigated in the majority of existing literature.

Studies in younger children and pre-adolescents generally indicated no association between obesity and academic achievement. This finding is plausible when taking into consideration the developmental trajectories of cognitive abilities related to academic achievement, social functioning and emotional control such as executive function [ 58, 59] (i.e. reasoning, working memory, cognitive flexibility, inhibition). These cognitive abilities begin developing in infancy but develop steeply from the age of 6 years, develop throughout adolescence and reach adult levels at about age 20 [ 59, 60, 61]. It is suggested that puberty influences neural reorganisation in the prefrontal cortex, the brain area linked to executive functions [ 62]. Therefore, obesity-related deficits in academic achievement might manifest only during adolescence [ 58].

Working memory, reasoning, inhibition and cognitive flexibility are strongly associated with maths achievement [ 63] and have been shown to be impaired in children and adolescents with obesity [ 6**]. Although maths and reading share multiple cognitive processes [ 64], for non-verbal maths tasks, different brain regions are implicated [ 65]. The brain regions invoked during non-verbal maths tasks are also areas (prefrontal cortex, hippocampus) which have been suggested to be associated with obesity and energy-balance related behaviours [ 66- 67, 68 **]. This might explain why obesity in childhood and adolescence appeared to be negatively associated with maths but not reading/language achievement. Further research to understand these mechanisms is warranted.

The negative association between obesity and academic achievement in girls, but to a lesser extent in boys, could be attributed to the fact that obese girls face more incidences of body weight-related stigmatisation [ 21] and are more likely to be distressed by teasing/bullying than boys [ 69, 70]. Weight-based teasing was shown to be linked to lower academic achievement [ 70].

Consistent with the published literature [ 71, 72, 73, 74, 75] were the experiences of participants from our focus group study who experienced social rejection, difficulties in making friends and stigmatisation, although not from teachers. Stigmatisation and teasing was particularly evident in PE, and consistent with previous research [ 67, 69], the girls reported that their body weight stopped them performing well in this environment. Although, it appears body weight did influence the school experience for our participants, they did not feel that their body weight directly influenced academic achievement of other school subjects specifically, but instead highlighted the importance of attitude to school. Similar to previous research [ 65], participants reported focusing on education rather than social relationships with peers, perhaps as a coping mechanism to avoid obesity-related bullying and peer rejection.

Strengths and Limitations

Strengths of this work include the use of rigorous systematic review methodology and a focus on longitudinal data to provide a more nuanced insight into the association between obesity and academic achievement. The ability to distinguish between obesity and overweight as exposure variables was also a strength. Adding focus group data provided useful preliminary insight into the perceptions of adolescent girls with obesity and their parents to complement and advance the observational literature.

However, some limitations are notable. Included studies were in English language only; thus, we might have missed relevant studies published in non-English language. Focus group findings are limited in their generalisability given the low number of participants (saturation of themes potentially not reached) and the fact that all those who participated had recently completed a weight management programme (non-participation and non-completion might influence perceptions).

Implications for Research and Practice

The current evidence on the association between obesity and maths attainment is available primarily from North American and European cohorts and entirely from high-income countries. Further research is needed to establish if an obesity-related deficit in academic achievement is evident in children and adolescents from middle-low-income countries. Given the current steep rise in the prevalence of childhood obesity in these countries [ 1], if there is an association between obesity and academic achievement in these contexts, the subsequent economic impact of obesity-related deficits may have even more implications for economic growth of middle-low-income countries and for human capital. Prospective cohort studies indicated that adolescent obesity is negatively associated with years of schooling [ 76, 77], school completion [ 78], enrolment in higher education [ 79, 80], income [ 76, 77, 81] and employment status [ 82]. The economic argument for the implementation of effective childhood obesity prevention and treatment programmes could therefore be substantial.

The current evidence is also limited on the impact of mediating factors. None of the included longitudinal studies evaluated the mediating effects of sleep deprivation, physical activity levels, type of physical activity, diet/nutritional status and co-morbidities, despite the literature suggesting an important role of these factors in the causal pathway between childhood obesity and academic achievement [ 12, 13, 15, 16, 17, 83]. However, most recently (after our literature search was performed), new findings were published on the link between meeting lifestyle-behaviour recommendations at age 11 years and academic achievement at age 12 years ( n = 4253). Researchers concluded that overweight or obesity was not associated with maths or reading achievement expectation but academic achievement was associated with meeting dietary, sleep and screen time recommendations [ 84]. Future longitudinal studies should include mediation analysis of those factors to contribute to understanding the underlying mechanisms of a negative association between adolescent obesity and academic achievement. This in turn will help to identify the most promising intervention strategy for promoting educational outcomes. In addition, the ability to identify a factor as a mediator depends on the reliability and validity of the measure and so researchers should avoid utilising crude measurement methods.

Academic achievement might be influenced by a teacher-bias towards obesity, in that children and adolescents are perceived as having poorer reasoning, social, physical and cooperation skills which impact on academic achievement [ 10, 11, 33, 85*]. However, the empirical evidence is inconsistent [ 42, 44, 48, 52]; thus, further research to provide empirical tests of this perspective is warranted. Nonetheless, there is convincing evidence on limited educational opportunities when PE teachers hold a biased perception about the abilities of children with obesity [ 86, 87, 88]. Our focus group data also indicated that during PE, the girls felt ignored and not supported by the teachers. This finding highlights the importance of promoting positive PE experiences among adolescent girls with obesity and indicate a potential role for physical literacy programmes in schools. Whitehead (2010) [ 89] placed special emphasis on physical literacy being defined by competence-based and interest-based motivation in PE [ 89]. Chen (2015) recently suggested that a physically literate person should be characterised by self-regulated motivation for physical activity [ 90]. For adolescents with obesity, the PE experiences should be educational, including learning the values and benefits of physical activity for health and quality of life for sustained participation in physical activity. Regular aerobic physical activity, as part of the school curriculum or extracurricular, was shown to benefit children's and adolescents' cognition and academic achievement [ 18, 91]. Children and adolescents with obesity seem to be even more responsive to physical activity programmes for improved cognitive abilities and academic achievement [ 22, 92 *]. This demands PE teachers more to fully understand the complexity of motivation processes in order to deliver positive PE experiences for adolescents with obesity. Furthermore, school-wide policy action is needed to address weight-based teasing/bullying.

Conclusion

The educational cost of obesity is primarily evident for adolescent girls' maths achievement potentially mediated by psychosocial distress and lower executive cognitive functions. There is less consistent evidence for other academic subjects though, suggesting differential relationships. The high prevalence of obesity in adolescent girls means that in addition to the threat to physical and mental health, this large population group is at risk of poor educational outcomes. Poor academic achievement might have long-term consequences on later life opportunities and economic implications. Therefore, findings of this review provide developmental and economic arguments for improved efforts in promoting psychosocial well-being and cognitive abilities linked to academic achievement in adolescent girls with obesity.

Acknowledgements

We thank Vasilis Antonopoulos for the assistance in the literature search and data extraction and Michael Pawlukiewicz for generating Fig. 2. We thank all participants for taking part in the focus group study and the National Health Service Lothian for their support.

Author information

Affiliations

1. Centre for Population Health Sciences, University of Edinburgh, Teviot Place, Edinburgh, EH8 9AG, UK

Anne Martin

2. Institute for Education, Community and Society, Moray House School of Education, University of Edinburgh, Edinburgh, UK

Josephine N. Booth & Sarah McGeown

3. Physical Activity for Health Research Centre, Moray House School of Education, University of Edinburgh, Edinburgh, UK

Ailsa Niven & David H. Saunders

4. Physical Education and Health Sciences, Moray House School of Education, Institute for Sport, University of Edinburgh, Edinburgh, UK

John Sproule & David H. Saunders

5. Physical Activity for Health Group, School of Psychology Sciences and Health, University of Strathclyde, Glasgow, UK

John J. Reilly

Corresponding author

Correspondence to Anne Martin.

Ethics declarations

This study was approved by the University of Edinburgh and the National Health Service South East Scotland Research Ethics Committee.

Conflict of Interest

Anne Martin declares that she has no conflict of interest.

Josephine N. Booth declares that she has no conflict of interest.

Sarah McGeown declares that she has no conflict of interest.

Ailsa Niven declares that she has no conflict of interest.

John Sproule declares that he has no conflict of interest.

David H. Saunders has received speaker's honorarium from the Association of Chartered Physiotherapists in Neurology (ACPIN) conference.

John J. Reilly declares that he has no conflict of interest.

Human and Animal Rights and Informed Consent

This article does not contain any studies with animal subjects performed by any of the authors.

This article does include any study with human subjects. Informed written consent was obtained from the study participants.

Funding

There is no funding to be disclosed.

Additional information

This article is part of the Topical Collection on The Obesity Epidemic: Causes and Consequences

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Martin, A., Booth, J.N., McGeown, S. et al. Longitudinal Associations Between Childhood Obesity and Academic Achievement: Systematic Review with Focus Group Data. Curr Obes Rep 6, 297-313 (2017). https://doi.org/10.1007/s13679-017-0272-9
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Children

Adolescents

Obesity

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Systematic review

Longitudinal cohort studies

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Representation theorems for generators of BSDEs with monotonic and convex growth generators

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In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short), whose generators ar…

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7.Classic Grounded Theory: Applications With Qualitative and Quantitative Data

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7.Classic Grounded Theory: Applications With Qualitative and Quantitative Data

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https://dx-doi-org.ezproxy.is.ed.ac.uk/10.4135/9781071802762.n8

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16. An article which discusses the generalisability of the findings

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15. An article which reports descriptive statistics (e.g. mean and standard deviation)

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14.An article which uses thematic analysis

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5.An article reporting narrative research

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Kshitij Dwivedi, Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing, 1 , 2 ,* Michael F. Bonner, Data curation, Formal analysis, Methodology, Writing - review & editing, 3 Radoslaw Martin Cichy, Conceptualization, Funding acquisition, Methodology, Supervision, Writing - original draft, Writing - review & editing, 1 , ‡ and Gemma Roig, Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Writing - original draft, Writing - review & editing 2 , ‡ *

Ulrik R. Beierholm, Editor

Abstract

The human visual cortex enables visual perception through a cascade of hierarchical computations in cortical regions with distinct functionalities. Here, we introduce an AI-driven approach to discover the functional mapping of the visual cortex. We related human brain responses to scene images measured with functional MRI (fMRI) systematically to a diverse set of deep neural networks (DNNs) optimized to perform different scene perception tasks. We found a structured mapping between DNN tasks and brain regions along the ventral and dorsal visual streams. Low-level visual tasks mapped onto early brain regions, 3-dimensional scene perception tasks mapped onto the dorsal stream, and semantic tasks mapped onto the ventral stream. This mapping was of high fidelity, with more than 60% of the explainable variance in nine key regions being explained. Together, our results provide a novel functional mapping of the human visual cortex and demonstrate the power of the computational approach.

Author summary

Human visual perception is a complex cognitive feat known to be mediated by distinct cortical regions of the brain. However, the exact function of these regions remains unknown, and thus it remains unclear how those regions together orchestrate visual perception. Here, we apply an AI-driven brain mapping approach to reveal visual brain function. This approach integrates multiple artificial deep neural networks trained on a diverse set of functions with functional recordings of the whole human brain. Our results reveal a systematic tiling of visual cortex by mapping regions to particular functions of the deep networks. Together this constitutes a comprehensive account of the functions of the distinct cortical regions of the brain that mediate human visual perception.

1. Introduction

The human visual system transforms incoming light into meaningful representations that underlie perception and guide behavior. This transformation is believed to take place through a cascade of hierarchical processes implemented in a set of brain regions along the so-called ventral and dorsal visual streams [ 1]. Each of these regions has been stipulated to fulfill a distinct sub-function in enabling perception [ 2]. However, discovering the exact nature of these functions and providing computational models that implement them has proven challenging. Recently, computational modeling using deep neural networks (DNNs) has emerged as a promising approach to model, and predict neural responses in visual regions [ 3- 7]. These studies have provided a first functional mapping of the visual brain. However, the resulting account of visual cortex functions has remained incomplete. This is so because previous studies either explain the function of a single or few candidate regions by investigating many DNNs or explain many brain regions comparing it to a single DNN trained on one task only (usually object categorization). In contrast, for a systematic and comprehensive picture of human brain function that does justice to the richness of the functions that each of its subcomponents implements, DNNs trained on multiple tasks, i.e., functions, must be related and compared in their predictive power across the whole cortex.

Aiming for this systematic and comprehensive picture for the visual cortex we here relate brain responses across the whole visual brain to a wide set of DNNs, in which each DNN is optimized for a different visual task, and hence, performs a different function.

To reliably reveal the functions of brain regions using DNNs performing different functions, we need to ensure that only function and no other crucial factor differs between the DNNs. The parameters learned by a DNN depend on a few fundamental factors, namely, its architecture, training dataset, learning mechanism, and the function the DNN was optimized for. Therefore, in this study, we select a set of DNNs [ 8] that have an identical encoder architecture and are trained using the same learning mechanism and the same set of training images. Thus, the parameters learned by the encoder of the selected DNNs differ only due to their different functions.

We generate a functional map of the visual cortex by comparing the fMRI responses to scene images [ 9] with the activations of multiple DNNs optimized on different tasks [ 8] related to scene perception, e.g., scene classification, depth estimation, and edge detection. Our key result is that different regions in the brain are better explained by DNNs performing different tasks, suggesting different computational roles in these regions. In particular, we find that early regions of the visual cortex are better explained by DNNs performing low-level vision tasks, such as edge detection. Regions in the dorsal stream are better explained by DNNs performing tasks related to 3-dimensional (3D) scene perception, such as occlusion detection and surface normal prediction. Regions in the ventral stream are best explained by DNNs performing tasks related to semantics, such as scene classification. Importantly, the top-3 best predicting DNNs explain more than 60% of the explainable variance in nine ventral-temporal and dorsal-lateral visual regions, demonstrating the quantitative power and potential of our AI-driven approach for discovering fine-grained functional maps of the human brain.

2. Results

2.1 Functional map of visual cortex using multiple DNNs

Our primary goal is to generate a functional map of the visual brain in terms of the functions each of the regions implements. Our approach is to relate brain responses to activations of DNNs performing different functions. For this, we used an fMRI dataset recorded while human subjects (N = 16) viewed indoor scenes [ 9] and performed a categorization task; and a set of 18 DNNs [ 8] optimized to perform 18 different functions related to visual perception (some of the tasks can be visualized here: https://sites.google.com/view/dnn2brainfunction/home#h.u0nqne179ys2) plus an additional DNN with random weights as a baseline. The different DNNs' functions were associated with indoor scene perception, covering a broad range of tasks from low-level visual tasks, (e.g., edge detection) to 3-dimensional visual perception tasks (e.g., surface normals prediction) to categorical tasks (e.g., scene classification). Each DNN consisted of an encoder-decoder architecture, where the encoder had an identical architecture across tasks, and the decoder varied depending on the task. To ensure that the differences in variance of fMRI responses explained by different DNNs from our set were not due to differences in architecture, we selected the activations from the last two layers of the identical encoder architecture for all DNNs.

The layer selection was based on an analysis finding the most task-specific layers of the encoder (see S1 Text and S2 Fig). Furthermore, all DNNs were optimized using the same set of training images, and the same backpropagation algorithm for learning. Hence, any differences in our findings across DNNs cannot be attributed to the training data statistics, architecture, or learning algorithm, but to the task for which each DNN was optimized.

To compare fMRI responses with DNNs, we first extracted fMRI responses in a spatially delimited portion of the brain for all images in the stimulus set (). This could be either a group of spatially contiguous voxels for searchlight analysis [ 10- 12] or voxels confined to a particular brain region as defined by a brain atlas for a region-of-interest (ROI) analysis. Equivalently, we extracted activations from the encoders of each DNN for the same stimulus set.

We then used Representational Similarity Analysis (RSA) [ 13] to compare brain activations with DNN activations. RSA defines a similarity space as an abstraction of the incommensurable multivariate spaces of the brain and DNN activation patterns. This similarity space is defined by pairwise distances between the activation patterns of the same source space, either fMRI responses from a brain region or DNN activations, where responses can be directly related. For this, we compared all combinations of stimulus-specific activation patterns in each source space (i.e., DNN activations, fMRI activations). Then, the results for each source space were noted in a two-dimensional matrix, called representational dissimilarity matrices (RDMs). The rows and columns of RDMs represent the conditions compared. To relate fMRI and DNNs in this RDM-based similarity space we performed multiple linear regression predicting fMRI RDM from DNN RDMs of the last two encoder layers. We obtained the adjusted coefficient of determination R ² (referred to as R ² in the subsequent text) from the regression to quantify the similarity between the fMRI responses and the DNN (). We performed this analysis for each of the 18 DNNs investigated, which we group into 2D, 3D, or semantic DNNs when those are optimized for 2D, 3D, or semantic tasks, respectively, and an additional DNN with random weights as a baseline. The tasks were categorized into three groups (2D, 3D, and semantic) based on different levels of indoor scene perception and were verified in previous works using transfer performance using one DNN as the initialization to other target tasks [ 8] and representational similarity between DNNs [ 14]. We finally used the obtained DNN rankings based on R ² to identify the DNNs with the highest R ² for fMRI responses in that brain region ( top). To visualize the results, we color-coded the brain region by color indexing the DNN showing the highest R ² in that brain region ( bottom).

To generate a functional map across the whole visual cortex we performed a searchlight analysis [ 11, 12]. In detail, we obtain the R ²-based DNN rankings on the local activation patterns around a given voxel, as described above. We conducted the above analysis for each voxel, resulting in a spatially unbiased functional map.

We observed that different regions of the visual cortex showed the highest similarity with different DNNs. Importantly, the pattern with which different DNNs predicted brain activity best was not random but spatially organized: 2D DNNs (in shades of blue in ; interactive map visualization available here: https://sites.google.com/view/dnn2brainfunction/home#h.ub1chq1k42n6) show a higher similarity with early visual regions, 3D DNNs (in shades of green) show a higher similarity with dorsal regions, while semantic DNNs (in shades of magenta) show a higher similarity with ventral regions and some dorsal regions.

Together, the results of our AI-driven mapping procedure suggest that early visual regions perform functions related to low-level vision, dorsal regions perform functions related to both 3D and semantic perception, and ventral regions perform functions related to semantic perception.

2.2 Nature and predictive power of the functional map

Using the searchlight results from , we identified the DNN that showed the highest R ² for each searchlight. This poses two crucial questions that require further investigation for an in-depth understanding of the functions of brain regions. Firstly, does a single DNN prominently predict a region's response (one DNN-to-one region) or a group of DNNs together predict its response (many DNNs-to-one region)? A one-to-one mapping between DNN and a region would suggest a single functional role while a many-to-one mapping would suggest multiple functional roles of the brain region under investigation. Secondly, given that the DNNs considered in this study predict fMRI responses, how well do they predict on a quantitative scale? A high prediction accuracy would suggest that the functional mapping obtained using our analysis is accurate, while a low prediction accuracy would suggest that DNNs considered in this study are not suitable to find the function of that brain region. Although it is possible to answer the above questions for each voxel, for conciseness we consider 25 regions of interest (ROIs) tiling the visual cortex from a brain atlas [ 15].

To determine how accurately DNNs predict fMRI responses, we calculated the lower and upper bound of the noise ceiling for each ROI. We included ROIs (15 out of 25) with a lower noise ceiling above 0.1 and discarded other ROIs due to low signal-to-noise ratio. We show the locations of the investigated ROIs in the visual cortex in .

For each ROI we used RSA to compare fMRI responses (transformed into fMRI RDMs) with activations of all 18 DNNs plus a randomly initialized DNN as a baseline (transformed into DNN RDMs). This yielded one R ² value for each DNN per region (see S3 Fig). We then selected the top-3 DNNs showing the highest R ² and performed a variance partitioning analysis [ 16]. We used the top-3 DNN RDMs as the independent variable and the ROI RDM as the dependent variable to find out how much variance of ROI responses is explained uniquely by each of these DNNs while considered together with the other two DNNs. Using the variance partitioning analysis (method illustrated in S1 Fig) we were able to infer the amount of unique and shared variance between different predictors (DNN RDMs) by comparing the explained variance (R ²) of a DNN used alone with the explained variance when it was used with other DNNs. Variance partitioning analysis () using the top-3 DNNs revealed the individual DNNs that explained the most variance uniquely for a given ROI along with the unique and shared variance explained by other DNNs. The DNN that detects edges explained significantly higher variance (p<0.05, permutation test, FDR corrected across DNNs) in ROIs in early and mid-level visual regions (V1v, V1d, V2v, V2d, V3v, and hV4) uniquely than the other two DNNs, suggesting a function related to edge detection. Semantic segmentation DNN explained significantly higher unique variance in ventral ROIs VO1 and VO2, suggesting a function related to the perceptual grouping of objects. 3D DNNs (3D Keypoints, 2.5D Segmentation, 3D edges, curvature) were best predicting DNNs for dorsal ROIs V3d and V3b suggesting their role in 3D scene understanding. A combination of 3D and semantic DNNs were best predicting DNNs for other ROIs (PHC1, PHC2, LO1, LO2, and V3a). It is crucial to note that if two DNNs from the same task group are in the top-3 best predicting DNNs for an ROI, the unique variance of ROI RDM explained by DNNs in the same group will generally be lower than by DNN not in the group. We have observed that DNNs in the same task group show a higher correlation with each other as compared to DNNs in other task groups [ 14]. A higher correlation between the DNNs of the same task group leads to an increase in shared variance and reduces the unique variance of the ROI RDM explained by within task group DNNs. For instance, we can observe this in PHC2 (also in PHC1, V3a), where two semantic DNNs explain less unique variance than a 3D DNN. Therefore, in such cases, we restrain from interpreting that one type of DNN is significantly better than others.

Overall, we observed a many-to-one relationship between function and region for multiple regions, i.e., multiple DNNs explained jointly a particular brain region. In early and mid-level regions (V1v, V1d, V2v, V3v) the most predictive functions were related to low-level vision (2D edges, denoising, and 2D segmentation). In dorsal regions V3d and V3b, the most predictive functions were related to 3D scene understanding. In later ventral and dorsal regions (V2d, hV4, VO1, VO2, PHC1, PHC2, LO1, LO2, and V3a) we observed a mixed mapping of 2D, 3D, and semantic functions suggesting multiple functional roles of these ROIs. The predictability of a region's responses by multiple DNNs demonstrates that a visual region in the brain has representations well suited for distinct functions. A plausible conjecture of the above findings is that these regions might be performing a function related to the best predicting DNNs but is not present in the set of DNNs investigated in this study.

To determine the accuracy of the functional mapping of the above ROIs, we calculated the percentage of the explainable variance explained by the top-3 best predicting DNNs. We calculated the explained variance by best predicting DNNs using cross-validation across subjects (N-fold) and conditions (two-fold). As we use multiple models together for multiple linear regression, we need to cross-validate using different sets of RDMs for fitting and evaluating the fit of the regression. Here, we perform cross-validation across subjects by fitting the regression on one-subject-left-out subject-averaged RDMs on half of the images in the stimulus set and evaluating on the left-out single subject RDM on the other half of the images. The above method is a stricter evaluation criterion as compared to the commonly used one without cross-validation (See S5 Fig). We compared the variance explained by the top-3 DNNs with the lower estimate of the noise ceiling which is an estimate of the explainable variance. We found that variance explained in nine ROIs (V1v, V1d, V2v, V3v, VO1, PHC1, LO2, LO1, V3a) is higher than 60% of the lower bound of noise ceiling (, absolute R ² = 0.085 ± 0.046). In absolute terms, the minimum, median, and maximum cross-validated R ² values across the 15 ROIs were 0.014 (PHC2), 0.044 (VO1), and 0.27 (V1v) which are comparable to related studies [ 17] performing evaluation in a similar manner. This shows that the DNNs selected in this study predict fMRI responses well and therefore are suitable for mapping the functions of the investigated ROIs.

In sum, we demonstrated that in many regions of the visual cortex, DNNs trained on different functions predicted activity. This suggests that these ROIs have multiple functional roles. We further showed quantitatively that more than 60% of the explainable variance in nine visual ROIs is explained by the set of DNNs we used, demonstrating that the selected DNNs are well suited to investigate the functional roles of these ROIs.

2.3 Functional map of visual cortex through 2D, 3D, and semantic tasks

In the previous section, we observed a pattern qualitatively suggesting different functional roles of early (2D), dorsal (3D and semantic), and ventral (semantic) regions in the visual cortex. To quantitatively assess this, we investigated the relation of brain responses and DNNs not at the level of single tasks, but task groups (2D, 3D, and semantic), where DNNs belonging to a task group showed a higher correlation with other DNNs in the group than with DNNs in other task groups (see S1 Text).

We averaged the RDMs of DNNs in each task group to obtain aggregate 2D, 3D, and semantic RDMs. Averaging the RDMs based on task groups reduced the number of DNN comparisons from 18 to 3. This allowed us to perform variance partitioning analysis to compare fMRI and DNN RDMs, which would be impractical with 18 single DNNs due to a large number of comparisons and computational complexity. When used in this way, variance partitioning analysis reveals whether and where in the brain one task group explained brain responses significantly better than other task groups.

We first performed a searchlight analysis to identify where in the cortex one task group explains significantly higher variance uniquely than the other task groups. We selected the grouped DNN RDM that explains the highest variance in a given region uniquely to create a functional map of the task groups in the visual cortex (). Here, due to the reduced number of comparisons, we can clearly observe distinctions where one grouped DNN explains fMRI responses better than the other grouped DNNs (p<0.05, permutation test with 10,000 iterations, FDR corrected across DNNs and searchlights). The resulting functional map (; interactive visualization available in this link: https://sites.google.com/view/dnn2brainfunction/home#h.xi402x2hr0p3) is different from the functional map in in two ways. First, in the functional map here we highlight the searchlight where one DNN group explained significantly higher variance uniquely than the other 2 DNN groups. In the functional map of , we highlighted the DNN that explained the highest variance of a searchlight without performing any statistical analysis whether the selected DNN was significantly better than the second best DNN or not due to the higher number of comparisons. Second, here we compared functions using groups of DNNs (3 functions: 2D, 3D and semantic), whereas in the previous analysis we compared functions using single DNNs (18 functions). The comparison using groups of DNNs allows us to put our findings in context with previous neuroimaging findings that are typically reported at this level.

We observed that the 2D DNN RDM explained responses in the early visual cortex, semantic DNN RDM explained responses in the ventral visual stream, and some parts in the right hemisphere of the dorsal visual stream, and 3D DNN RDM explained responses in the left hemisphere of the dorsal visual stream. The above findings quantitatively reinforce our qualitative findings from the previous section that early visual regions perform functions related to low-level vision, dorsal regions perform functions related to both 3D and semantic perception, and ventral regions perform functions related to semantic perception.

While the map of the brain reveals the most likely function of a given region, to find out whether a region can have multiple functional roles we need to visualize the variance explained by other grouped DNN RDMs along with the best predicting DNN RDM. To achieve that, we performed a variance partitioning analysis using 3 grouped DNN RDMs as the independent variable and 15 ROIs in the ventral-temporal and the dorsal-ventral stream as the dependent variable. The results in show the unique and shared variance explained by group-level DNN RDMs (2D, 3D, and semantic) for all the 15 ROIs.

From we observed that the responses in early ROIs (V1v, V1d, V2v, V3v, hV4) are explained significantly higher (p<0.05, permutation test with 10,000 iterations, FDR corrected across DNNs) by 2D DNN RDM uniquely, while responses in later ventral-temporal ROIs (VO1, VO2, PHC1, and PHC2) are explained by semantic DNN RDM uniquely. In dorsal-lateral ROIs (V3a, V3d) responses are explained by 3D RDM uniquely. In LO1, LO2, and V3b 3D and semantic DNN RDMs explained significant variance uniquely while in V2d all 2D, 3D, and semantic DNN RDMs explained significant unique variance. It is crucial to note that for the ROI analysis here we use grouped DNN RDMs as compared to where we selected top-3 single DNNs that showed the highest R ² with a given ROI. The comparison with grouped DNN RDMs provides a holistic view of the functional role of ROIs which might be missed if one of the DNNs that is related to the functional role of a ROI is not in the top-3 DNNs (as analyzed in ). For instance, in the results suggest both 3D and semantic functional roles of V3b which is not evident from where the top 3-DNNs were all optimized on 3D tasks.

Together, we found that the functional role of the early visual cortex is related to low-level visual tasks (2D), the dorsal stream is related to tasks involved in 3-dimensional perception and categorical understanding of the scene (3D and semantic), and in the ventral stream is related to the categorical understanding of the scene (semantic).

2.4 Functional roles of scene-selective regions

In the previous sections, we focused on discovering functions of regions anatomically defined by an atlas. Since the stimulus set used to record fMRI responses consisted of indoor scenes, in this section we investigate functional differences in functionally localized scene-selective regions. We here focus on two major scene-selective ROIs: occipital place area (OPA) and parahippocampal place area (PPA), putting results into context with the early visual cortex (EVC) as an informative contrast region involved in basic visual processing. The analysis followed the general rationale as used before.

We first investigated the functional differences in these regions by performing variance partitioning analysis using top-3 DNNs (see R ² based ranking of all DNNs in S4 Fig) that best explained a given ROIs' responses (). We found that the DNN that detects edges explained significantly higher variance (p<0.05, permutation test, FDR-corrected) in EVC uniquely than the other two DNNs, suggesting a function related to edge detection. 3D DNNs (3D Keypoints, 2.5D Segmentation, 3D edges) were best predicting DNNs for OPA suggesting its role in 3D scene understanding. A combination of semantic (semantic segmentation, scene classification) and 3D (3D keypoints) DNNs were best predicting DNNs for PPA suggesting its role in both semantic and 3D scene understanding.

We then investigated the functional differences by performing variance partitioning analysis using aggregated 2D, 3D, and semantic DNN RDMs obtained by averaging the individual DNN RDMs in each task group (). We found that for EVC and OPA results are highly consistent with top-3 DNN analysis showing a prominent unique variance explained by the 2D DNN RDM in EVC and the 3D DNN RDM in OPA. Interestingly, in PPA we find that the semantic DNN RDM shows the highest unique variance with no significant unique variance explained by the 3D DNN RDM. The insignificant unique variance explained by the 3D DNN RDM is potentially due to averaging the DNN RDMs of all 3D DNNs (high ranked as well as low ranked) which may lead to diminishing the contribution of an individual high ranked 3D DNN RDM (e.g. 3D keypoints that was in top-3 DNNs for PPA). Overall, we find converging evidence that OPA is mainly related to tasks involved in 3-dimensional perception (3D), and PPA is mainly related to semantic (categorical) understanding of the scene.

3. Discussion

In this study, we harvested the potential of discovering functions of the brain from comparison to DNNs by investigating a large set of DNNs optimized to perform a set of diverse visual tasks. We found a systematic mapping between cortical regions and function: different cortical regions were explained by DNNs performing different functions. Importantly, the selected DNNs explained 60% of the explainable variance in nine out of 15 visual ROIs investigated, demonstrating the accuracy of the AI-driven functional mapping obtained using our analysis.

Our study provides a systematic and comprehensive picture of human brain functions using DNNs trained on different tasks. Previous studies [ 3- 7, 17- 24] have compared model performance in explaining brain activity, but were limited to a few preselected regions and models, or had a different goal (comparing task structure) [ 25]. Using the same fMRI dataset as used in this study, a previous study [ 18] showed that representation in scene-selective ROIs consists of both location and category information using scene-parsing DNNs. We go beyond these efforts by comparing fMRI responses across the whole visual brain using a larger set of DNNs, providing a comprehensive account of the function of human visual brain regions.

We obtained the functional mapping of different regions in the visual cortex on both individual (e.g., 2D edges, scene classification, surface normals, etc.) and group (2D, 3D, semantic) levels of visual functions. We discuss the novel insights gained at the level of individual functions that inform about the fine-grained functional role of cortical regions.

First, we consider 2D DNNs, where the denoising DNN explained significant unique variance in V1v, V1d, V2v, V2d, V3v, and hV4. The denoising task requires the DNN to reconstruct an unperturbed input image from slightly perturbed (e.g., adding Gaussian noise in the current case) input image that encourages learning representations robust to slight perturbations and limited invariance. This suggests that these ROIs might be generating a scene representation robust to high frequency noise.

When considering 3D DNNs, the 3D Keypoint and the 2.5d segment were among the top-3 best predicting DNNs in multiple ROIs. The 3D Keypoints DNN explained significant unique variance in V3d, PHC1, PHC2, LO2, LO1, V3a, V3b, OPA, and PPA. The 3D Keypoints task requires the DNN to identify locally important regions of the input image based on object boundary information and surface stability. This suggests that the ROIs in which 3D Keypoints DNN explained significant variance may be identifying locally important regions in a scene. The identification of locally important regions might be relevant to selectively attend to these key regions to achieve a behavioral goal e.g., searching for an object. The 2.5d segment DNN explained significant unique variance in V3d, LO2, LO1, V3b, V3a, and OPA. The 2.5d segment task requires the DNN to segment images into perceptually similar groups based on color and scene geometry (depth and surface normals). This suggests that the ROIs in which 2.5d segment DNN explained significant variance may be grouping regions in the images based on color and geometry cues even without any knowledge of the categorical information. Grouping regions based on geometry could be relevant to behavioral goals such as reaching for objects or identifying obstacles.

Among semantic DNNs, the semantic segmentation DNN explained significant unique variance in VO1, VO2, PHC1, PHC2, V3a, and PPA. The semantic segmentation task requires the DNN to segment objects present in the image based on categories. This suggests that the ROIs in which semantic segmentation DNN explained significant variance may be grouping regions in the image based on categorical information.

Other DNNs (2D edges, scene classification, and object classification) that showed significant unique variance in ROIs provided functional insights mostly consistent with the previous studies [ 26- 30]. Overall, the key DNNs (denoising, 3D keypoints, 2.5D segment, and semantic segmentation) that explained significant variance in multiple ROI responses uniquely promote further investigation by generating novel hypotheses about the functions of these ROIs. Future experiments can test these hypotheses in detail in dedicated experiments.

The functional mapping obtained using grouped DNNs is complementary to that at the individual level and helps us put functional mapping obtained here in context with previous literature. We found that early visual regions (V1v, V1d, V2v) have a functional role related to low-level 2D visual tasks which is consistent with previous literature investigating these regions [ 26- 28]. In dorsal-ventral ROIs (V3a, V3d, LO1, and LO2) we found functional roles related to 3D and semantic tasks converging with evidence from previous studies [ 31- 35]. Similarly, the prominent semantic functional role of later ventral-temporal ROIs (VO1, VO2, PHC1, and PHC2) found in this study converges with findings in previous literature [ 29- 30]. In scene-selective ROIs, we found a semantic functional role for PPA and 3D functional role for OPA respectively. Our study extends the findings of a previous study [ 23] relating OPA and PPA to 3D models by differentiating between OPA and PPA functions through a much broader set of models. To summarize, the functional mapping using individual DNNs optimized to perform different functions revealed new functional insights for higher ROIs in the visual cortex while at the same time functional mapping using grouped DNNs showed highly converging evidence with previous independent studies investigating these ROIs.

Beyond clarifying the functional roles of multiple ROIs, our approach also identifies quantitatively highly accurate prediction models of these ROIs. We found that the DNNs explained 60% of the explainable variance in nine out of 15 ROIs. Our findings, thus, make advances towards finding models that generate new hypotheses about potential functions of brain regions as well as predicting brain responses well [ 21, 36- 38].

A major challenge in meaningfully comparing two or more DNNs is to vary only a single factor of interest while controlling the factors that may lead to updates of DNN parameters. In this study, we address this challenge by selecting a set of DNNs trained on the same set of training images using the same learning algorithm, with the same encoder architecture, while being optimized for different tasks. Our results, thus, complement previous studies that focused on other factors influencing the learning of DNN parameters such as architecture [ 20, 39, 40], and the learning mechanism [ 41- 43]. Our approach accelerates the divide-and-conquer strategy of investigating human brain function by systematically and carefully manipulating the DNNs used to map the brain in their fundamental parameters one by one [ 21, 44- 46]. Our high-throughput exploration of potential computational functions was initially inspired by Marr's computational level of analysis [ 47] which aims at finding out what the goal of the computation carried out by a brain region is. While Marr's approach invites the expectation of a one-to-one mapping between regions and goals, we found evidence for multiple functional roles (3D + semantic) using DNNs in some ROIs (e.g. LO1, LO2, PHC1, PHC2). This indicates a many-to-one mapping [ 48] between functions and brain regions. We believe such a systematic approach that finds the functional roles of multiple brain regions provides a starting point for a further in-depth empirical inquiry into functions of the investigated brain regions.

Our study is related to a group of studies [ 49- 52] applying DNNs in different ways to achieve a similar goal of mapping functions of brain regions using DNNs. Some studies [ 49- 51] applied optimization algorithms (genetic algorithm or activation maximization) to find images that maximally activate a given neuron's or group of neurons' response. Another related study [ 52] proposes Neural Information Flow (NIF) to investigate functions of brain regions where they train a DNN with the objective function to predict brain activity while preserving a one-to-one correspondence between DNN layers and biological neural populations. While sharing the overall goal to discover functions of brain regions, investigating DNN functions allows investigation in terms of which computational goal a given brain region is best aligned with. With new computer vision datasets [ 53] investigating a diverse set of tasks relevant to human behavioral goals [ 54, 55] our approach opens new avenues to investigate brain functions.

A limitation of our study is that our findings are restricted to functions related to scene perception. Thus, the functions we discovered for non-scene regions correspond to their functions when humans are perceiving scenes. In contrast, our study does not characterize the functions of these regions when humans perceive non-scene categories such as objects, faces, or bodies. We limited our study to scene perception because there are only a few image datasets [ 8, 56] that have annotations corresponding to a diverse set of tasks, thus, allowing DNNs to be optimized independently on these tasks. The Taskonomy dataset [ 8] with annotations of over 20 diverse scene perception tasks and pretrained DNNs available on these tasks along with the availability of an fMRI dataset related to scene perception [ 9], therefore, provided a unique opportunity. However, the approach we presented in this study is not limited to scene perception. It can in principle be extended to more complex settings such as video understanding, active visual perception, and even outside the vision modality, given an adequate set of DNNs and brain data. While in this study we considered DNNs that were trained independently, future studies might consider investigating multitask models [ 57, 58] which are trained to perform a wide range of functions using a single DNN. Multitask modeling has the potential to model the entire visual cortex using a single model as compared to several independent models used in this study. Another potential limitation is that our findings are based on a single fMRI and image dataset, so it is not clear how well they would generalize to a broader sample of images. Given the explosive growth of the deep learning field [ 59] and the ever increasing availability of open brain imaging data sets [ 60, 61] we see a furtive ground for the application of our approach in the future.

Beyond providing theoretical insight with high predictive power, our approach can also guide future research. In particular, the observed mapping between cortical region and function can serve as a quantitative baseline and starting point for an in-depth investigation focused on single cortical regions. Finally, the functional hierarchy of the visual cortex from our results can inspire the design of efficient multi-task artificial visual systems that perform multiple functions similar to the human visual cortex.

4. Materials and methods

4.1 fMRI data

We used fMRI data from a previously published study [ 9]. The fMRI data were collected from 16 healthy subjects (8 females, mean age 29.4 years, SD = 4.8). The subjects were scanned on a Siemens 3.0T Prisma scanner using a 64-channel head coil. Structural T1-weighted images were acquired using an MPRAGE protocol (TR = 2,200 ms, TE = 4.67 ms, flip angle = 8°, matrix size = 192 × 256 × 160, voxel size = 0.9 × 0.9 × 1 mm). Functional T2*-weighted images were acquired using a multi-band acquisition sequence (TR = 2,000 ms for main experimental scans and 3,000 ms for localizer scans, TE = 25 ms, flip angle = 70°, multiband factor = 3, matrix size = 96 × 96 × 81, voxel size = 2 × 2 × 2 mm).

During the fMRI scan, subjects performed a category detection task while viewing images of indoor scenes. On each trial, an image was presented on the screen at a visual angle of ~17.1° x 12.9° for 1.5 s followed by a 2.5s interstimulus interval. Subjects had to respond by pressing a button indicating whether the presented image was a bathroom or not while maintaining fixation on a cross. The stimulus set consisted of 50 images of indoor scenes (no bathrooms), and 12 control images (five bathroom images, and seven non-bathroom images). fMRI data were preprocessed using SPM12. For each participant, the functional images were realigned to the first image followed by co-registration to the structural image. Voxelwise responses to 50 experimental conditions (50 indoor images excluding control images) were estimated using a general linear model.

4.2 Deep neural networks

For this study, we selected 18 DNNs trained on the Taskonomy [ 8] dataset optimized on 18 different tasks covering different aspects of indoor scene understanding. The Taskonomy dataset is a large-scale indoor image dataset consisting of annotations for 18 single image tasks, thus, allowing optimization of DNNs on 18 different tasks using the same set of training images. We briefly describe the objective functions and DNN architectures below. For a detailed description, we refer the reader to Zamir et al. [ 8].

4.2.1 Tasks and objective functions of the DNNs

The Taskonomy dataset consists of annotations for tasks that require pixel-level information such as edge detection, surface normal estimation, semantic segmentation, etc. as well as high-level semantic information such as object/scene classification probabilities. The tasks can be broadly categorized into 4 groups: relating to low-level visual information (2D), the three-dimensional layout of the scene (3D), high-level object and scene categorical information (semantic), and low-dimensional geometry information(geometrical). The above task categorization was obtained by analyzing the relationship between the transfer learning performance on a given task using the models pretrained on other tasks as the source tasks. The 2D tasks were edge detection, keypoint detection, 2D segmentation, inpainting, denoising, and colorization; 3D tasks were surface normals, 2.5D segmentation, occlusion edges, depth estimation, curvature estimation, and reshading; semantic tasks were object/scene classification and semantic segmentation, and low-dimensional geometric tasks were room layout estimation and vanishing point. A detailed description of all the tasks and annotations is provided in http://taskonomy.stanford.edu/taskonomy_supp_CVPR2018.pdf. In this study, we did not consider low dimensional geometric tasks as they did not fall into converging clusters according to RSA and transfer learning as in the case of 2D, 3D, and semantics tasks. To perform a given task, DNN's parameters were optimized using an objective function that minimizes the loss between the DNN prediction and corresponding ground truth annotations for that task. All the DNNs' parameters were optimized using the corresponding objective function, on the same set of training images. Due to the use of the same set of training images the learned DNN parameters vary only due to the objective function and not the difference in training dataset statistics. A complete list of objective functions used to optimize for each task is provided in this link ( https://github.com/StanfordVL/taskonomy/tree/master/taskbank). We downloaded the pretrained models using this link ( https://github.com/StanfordVL/taskonomy/tree/master/taskbank), where further details can be found.

4.2.2 Network architectures

The DNN architecture for each task consists of an encoder and a decoder. The encoder architecture is consistent across all the tasks. The encoder architecture is a modified ResNet-50 [ 62] without average pooling and convolutions with stride 2 replaced by convolutions with stride 1. ResNet-50 is a 50-layer DNN with shortcut connections between layers at different depths. Consistency of encoder architecture allows us to use the outputs of the ResNet-50 encoder as the task-specific representation for a particular objective function. For all the analysis in this study, we selected the last two layers of the encoder as the task-specific representation of the DNN. Our selection criteria was based on an analysis (see S1 Text and S2 Fig) that shows task-specific representation is present in those layers as compared to earlier layers. In this way, we ensure that the difference in representations is due to the functions these DNNs were optimized for and not due to the difference in architecture or training dataset. The decoder architecture is task-dependent. For tasks that require pixel-level prediction, the decoder is a 15-layer fully convolutional model consisting of 5 convolutional layers followed by alternating convolution and transposed convolutional layers. For tasks, which require low dimensional output, the decoder consists of 2-3 fully connected layers.

4.3 Representational Similarity Analysis (RSA)

To compare the fMRI responses with DNN activations we first need to map both the modalities in a common representational space and then by comparing the resulting mappings we can quantify the similarity between fMRI and DNNs. We mapped the fMRI responses and DNN activations to corresponding representational dissimilarity matrices (RDMs) by computing pairwise distances between each pair of conditions. We used the variance of upper triangular fMRI RDM (R ²) explained by DNN RDMs as the measure to quantify the similarity between fMRI responses and DNN activations. To calculate R ², we assigned DNN RDMs (RDMs of the last two layers of the encoder) as the independent variables and assigned fMRI RDM as the dependent variable. Then a multiple linear regression was fitted to predict fMRI RDM from the weighted linear combination of DNN RDMs. We evaluated the fit by estimating the variance explained (R ²). We describe how we mapped from fMRI responses and DNN activations to corresponding RDMs in detail below.

Taskonomy DNN RDMs. We selected the last two layers of the Resnet-50 encoder as the task-specific representation of DNNs optimized on each task. For a given DNN layer, we computed the Pearson's distance between the activations for each pair of conditions resulting in a condition x condition RDM for each layer. This resulted in a single RDM corresponding to each DNN layer. We followed the same procedure to create RDMs corresponding to other layers of the network. We averaged the DNN RDMs across task clusters (2D, 3D, and semantic) to create 2D, 3D, and semantic RDMs.

Probabilistic ROI RDMs. We downloaded probabilistic ROIs [ 15] from the link ( http://scholar.princeton.edu/sites/default/files/napl/files/probatlas_v4.zip). We extracted activations of the probabilistic ROIs by applying the ROI masks on the whole brain response pattern for each condition, resulting in ROI-specific responses for each condition for each subject. Then for each ROI, we computed the Pearson's distance between the voxel response patterns for each pair of conditions resulting in a RDM (with rows and columns equal to the number of conditions) independently for each subject. To compare the variance of ROI RDM explained by DNN RDMs with the explainable variance we used independent subject RDMs. For all the other analyses, we averaged the RDMs across the subjects resulting in a single RDM for each ROI due to a higher signal to noise ratio in subject averaged RDMs.

Searchlight RDMs. We used Brainiak toolbox code [ 63] to extract the searchlight blocks for each condition in each subject. The searchlight block was a cube with radius = 1 and edge size = 2. For each searchlight block, we computed the Pearson's distance between the voxel response patterns for each pair of conditions resulting in a RDM of size condition times condition independently for each subject. We then averaged the RDMs across the subjects resulting in a single RDM for each searchlight block.

4.4 Variance partitioning

Using RSA to compare multiple DNNs we do not obtain a complete picture of how each model is contributing to explaining the fMRI responses when considered in conjunction with other DNNs. Therefore, we determined the unique and shared contribution of individual DNN RDMs in explaining the fMRI ROI RDMs when considered with the other DNN RDMs using variance partitioning.

We performed two variance partitioning analyses on probabilistic ROIs: first using the top-3 DNNs that best explained a given ROI's responses and second using RDMs averaged according to task type (2D, 3D, and semantic). For the first analysis, we assigned a fMRI ROI RDM as the dependent variable (referred to as predictand) and assigned RDMs corresponding to the top-3 DNNs as the independent variables (referred to as predictors). For the second analysis, we assigned an fMRI ROI (searchlight) RDM as the dependent variable (referred to as predictand). We then assigned three DNN RDMs (2D, 3D, and semantic) as the independent variables (referred to as predictors).

For both variance partitioning analyses, we performed seven multiple regression analyses: one with all three independent variables as predictors, three with different pairs of two independent variables as the predictors, and three with individual independent variables as the predictors. Then, by comparing the explained variance (R ²) of a model used alone with the explained variance when it was used with other models, we can infer the amount of unique and shared variance between different predictors (see S1 Fig).

4.5 Searchlight analysis

We perform two different searchlight analyses in this study: first to find out if different regions in the brain are better explained by DNNs optimized for different tasks and second to find the pattern by taking the averaged representation DNNs from three task types (2D, 3D, and semantic). In the first searchlight analysis, we applied RSA to compute the variance of each searchlight block RDM explained by 19 DNN RDMs (18 Taskonomy DNNs and one randomly initialized as a baseline) independently. We then selected the DNN that explained the highest variance as the preference for the given searchlight block. In the second searchlight analysis, we applied variance partitioning with 2D, 3D, and semantic DNN RDMs as the independent variables, and each searchlight block RDM as the dependent variable. For each searchlight block, we selected the task type whose RDMs explained the highest variance uniquely as the function for that block. We used the nilearn ( https://nilearn.github.io/index.html) library to plot and visualize the searchlight results.

4.6 Comparison of explained with explainable variance

To relate the variance of fMRI responses explained by a DNN to the total variance to be explained given the noisy nature of the fMRI data, we first calculated the lower and upper bounds of the noise ceiling as a measure of explainable variance and then compared cross-validated explained variance of each ROI by top-3 best predicting DNNs. In detail, the lower noise ceiling was estimated by fitting each individual subject RDMs as predictand with mean subject RDM of other subjects (N-1) as the predictor and calculating the R ². The resulting subject-specific R ² values were averaged across the N subjects. The upper noise ceiling was estimated in a similar fashion while using mean subject RDMs of all the subjects (N) as the predictor. To calculate variance explained by the best predicting DNNs we fit the regression using cross validation in 2N folds (2 folds across conditions, N folds across subjects) where the regression was fit using the subject averaged RDMs of N-1 subjects and the fit was evaluated using R ² on the left out subject and left out conditions. Finally, we then calculated the mean R ² across 2N folds and divided it by the lower bound of the noise ceiling to obtain the ratio of the explainable variance explained by the DNNs.

4.7 Statistical testing

We applied nonparametric statistical tests to assess the statistical significance in a similar manner to a previous related study [ 64]. We assessed the significance of the R ² through a permutation test by permuting the conditions randomly 10,000 times in either the neural ROI/searchlight RDM or the DNN RDM. From the distribution obtained using these permutations, we calculated p-values as one-sided percentiles. We calculated the standard errors of these correlations by randomly resampling the conditions in the RDMs for 10,000 iterations. We used re-sampling without replacement by subsampling 90% (45 out of 50 conditions) of the conditions in the RDMs. We used an equivalent procedure for testing the statistical significance of the correlation difference and unique variance difference between different models.

For ROI analysis, we corrected the p-values for multiple comparisons by applying FDR correction with a threshold equal to 0.05. For searchlight analyses, we applied FDR correction to correct for the number of DNNs compared as well as to correct for the number of searchlights that had a significant noise ceiling.

We applied a two-sided t-test to assess the statistical significance of the cross-validated explained variance across N subjects. We corrected the p-values for multiple comparisons by applying FDR correction.

Supporting information

S1 Fig
Variance partitioning overview.
Given a set of multiple independent variables and dependent variables, multiple linear regression results in R-squared (R ²) that represents the proportion of the variance for a dependent variable that's explained by independent variables in a regression model. To find how 3 DNN RDMs together explain the variance of a given fMRI RDM we perform 7 multiple regression and illustrate unique and shared variance explained by models through a Venn diagram.

(TIFF)

S2 Fig
Selecting task-specific DNN representation to compare with fMRI data.
A) Spearman's correlation of all DNN RDMs at a given layer of the encoder with other DNN RDMs computed at the same layer. We report the mean pairwise correlation of all 18 DNNs at different layers of the encoder. B) Spearman's correlation of all DNN RDMs at a given layer of the encoder with a randomly initialized model with the same architecture computed at the same layer. We report the mean correlation of all 18 DNNs with the randomly initialized DNN at different layers of the encoder. C) Spearman's correlation of all DNN RDMs at a given layer of the encoder with deeper layers (block4 and encoder output) of 2D DNNs. We report the mean correlation of the key layers of all 18 DNNs with deeper layers (block4 and encoder output) of 2D DNNs. D) Spearman's correlation between layers at different depths for DNNs corresponding to different task types. We report the mean correlation between different layers averaged across different DNNs of the same task type. E) Effect of adding all the key layers on unique and shared variance of fMRI RDMs from different ROIs as compared to selecting only task-specific layers for variance partitioning analysis. We report the change in variance explained (variance change) for 7 variance partitions when all key layers were used for analysis as compared to selecting task-specific layers.

(TIFF)

S3 Fig
R² ranking for 18 Taskonomy DNNs and random baseline in anatomical ROIs.
The bar plot shows the absolute total variance of each ROI RDM explained by task-specific layer RDMs of a given DNN. The asterisk denotes the significance of total variance (p<0.05, permutation test with 10,000 iterations, FDR-corrected across DNNs). The error bars show the standard deviation calculated by bootstrapping 90% of the conditions (10,000 iterations).

(TIFF)

S4 Fig
R² ranking for 18 Taskonomy DNNs and random baseline in functionally localized ROIs.
The bar plot shows the absolute total variance of each ROI RDM explained by task-specific layer RDMs of a given DNN. The asterisk denotes the significance of total variance (p<0.05, permutation test with 10,000 iterations, FDR-corrected across DNNs). The error bars show the standard deviation calculated by bootstrapping 90% of the conditions (10,000 iterations).

(TIFF)

S5 Fig
Effect of cross validation on variance explained (R²).
A) Variance of each ROI explained by top-3 best predicting DNNs compared for different cross-validation settings (blue bars: no cross validation; orange bars: cross validation across subjects; green bars: cross validation across subjects and stimuli). The error bars show the 95% confidence interval calculated across N = 16 subjects. All the R ² values are statistically significant (p<0.05, two-sided t-test, FDR-corrected across ROIs) B) Variance of each ROI explained by 1000 randomly generated RDMs compared for different cross-validation settings (blue bars: no cross validation; orange bars: cross validation across subjects; green bars: cross validation across subjects and stimuli). The error bars show the 95% confidence interval calculated across N = 16 subjects.

(TIFF)

S1 Text
Selecting task-specific DNN representations.
(DOCX)

Funding Statement

G.R. thanks the support of the Alfons and Gertrud Kassel Foundation. R.M.C. is supported by DFG grants (CI241/1-1, CI241/3-1) and the ERC Starting Grant (ERC-2018- StG 803370). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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Unveiling functions of the visual cortex using task-specific deep neural networks

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Abstract

Linkage analysis is often followed by association mapping to localize disease variants. In this paper, we evaluate approaches to determine how much of the observed linkage evidence, namely the identity-by-descent (IBD) sharing at the linkage peak, is explained by associated SNPs. We study several methods: Homozygote Sharing Tests (HST), Genotype Identity-by-Descent Sharing Test (GIST), and a permutation approach. We also propose a new approach, HSTMLB, combining HST and the Maximum Likelihood Binomial (MLB) linkage statistic. These methods can identify SNPs partially explaining the linkage peak, but only HST and HSTMLB can identify SNPs that do not fully explain the linkage evidence and be applied to multiple-SNPs. We contrast these methods with the association tests implemented in the software LAMP. In our simulations, GIST is more powerful at finding SNPs that partially explain the linkage peak, while HST and HSTMLB are equally powerful at identifying SNPs that do not fully explain the linkage peak. When applied to the North American Rheumatoid Arthritis Consortium data, HST and HSTMLB identify marker pairs that may fully explain the linkage peak on chromosome 6. In conclusion, HST and HSTMLB provide simple and flexible tools to identify SNPs that explain the IBD sharing at the linkage peak.

Key Words: Genetic linkage, Linkage disequilibrium, Conditional allele sharing

Introduction

There are two common statistical techniques used for gene mapping of complex human diseases: linkage analysis and association analysis. A frequently used gene mapping strategy consists of performing a genome scan to identify regions of linkage followed by association analysis to further localize the disease susceptibility loci. Once associated single nucleotide polymorphisms (SNPs) in a linkage region have been identified, it is of interest to determine if they explain none/some/all of the linkage evidence, namely the identity-by-descent (IBD) sharing at the linkage peak. In this paper, we focus on investigating methods that determine whether the IBD sharing at the linkage peak can be explained by the associated SNPs in a linked region. There are two types of such tests: the first type identifies SNPs that partially explain the linkage peak; the second type further identifies SNPs that do not fully explain the linkage peak. If none of the single associated SNPs fully explains the IBD sharing at the linkage peak, further investigation may reveal additional disease loci.

When a linkage signal is observed, one or multiple disease loci may contribute to the linkage signal. In general, complex diseases are most likely caused by multiple disease loci. For example, in Zavattari et al. [ 1], three loci in the human lymphocyte antigen (HLA) region were identified as risk modifiers in addition to and independent of HLA-DQB1, -DRB1 for type 1 diabetes mellitus in Sardinian population. When multiple disease loci contribute to a linkage signal, no single SNP should fully explain the IBD sharing at the linkage peak. Therefore, it is important for methods to consider multiple SNPs simultaneously.

Several researchers have developed methods to identify SNPs that explain the linkage peak for complex diseases [ 2, 3, 4, 5, 6, 7], among others[]. Horikawa et al. [ 2] identified SNPs associated with the IBD sharing at the linkage peak by looking for an increase in the logarithm of odds (LOD) score in a subset of families defined by the proband carrying the risk genotypes. The significance of the change in LOD score in the subset of Ns families with the risk genotypes is assessed using a permutation approach, by randomly selecting subsets of Ns families, irrespective of proband genotypes. Sun et al. [ 3] defined a test statistic based on the premise that there should be no unexplained IBD oversharing in a sample of affected sibling pairs (ASPs) or nuclear families when conditioning on the affected offspring genotypes of a single causal locus. Li et al. [ 4] developed the Genotype Identity-by-Descent Sharing Test (GIST) to test the correlation between the family nonparametric linkage (NPL) score and a family genotype weight defined on the basis of affected offspring genotypes; the correlation should be zero when the tested SNP does not explain the IBD sharing at the linkage peak. Biernacka and Cordell [ 5] extended the method proposed in Sun et al. [ 3] by further conditioning on parental genotypes, which eliminates the need to estimate allele or haplotype frequencies.

The Homozygote Sharing Tests (HST) proposed in Dupuis and Van Eerdewegh [ 6] compare the IBD sharing from homozygous parents to the IBD sharing from heterozygous parents, where homozygosity status is defined at a SNP or a group of SNPs of interest. There are two test statistics: HST.P identifies SNPs that partially explain the IBD sharing at the linkage peak by looking for more IBD sharing from heterozygous parents than from homozygous parents; HST.F identifies SNPs that do not fully explain the linkage peak by looking for excess IBD sharing from homozygous parents. In this paper, we propose a novel extension combining HST and the Maximum Likelihood Binomial (MLB) statistic proposed in Abel et al. [ 8, 9], called HSTMLB, to identify SNPs that partially explain the IBD sharing at the linkage peak (HSTMLB.P) or SNPs that do not fully explain the IBD sharing at the linkage peak (HSTMLB.F).

Our evaluation focuses on HST, HSTMLB, GIST and the method proposed in Horikawa [ 2], because Sun et al.'s [ 3] approach is sensitive to allele frequencies and no software is available for the method proposed by Biernacka and Cordell [ 5]. In addition, we contrast these approaches with the association tests proposed in Li et al. [ 10], where a likelihood function was constructed using disease-SNP haplotype frequencies and disease penetrances as parameters. Two likelihood ratio tests for association are implemented in the software 'Linkage and Association Modeling in Pedigrees' (LAMP). LAMP.P assesses whether the tested SNP is in linkage disequilibrium (LD) with a disease SNP, while LAMP.F assesses whether the tested SNP is in perfect LD with a disease SNP. Table summarizes the characteristics of the studied methods to identify SNPs that explain the IBD sharing at the linkage peak.

We conduct two simulation studies: one that considers the case of a single disease SNP and that compares all approaches listed above, and one that considers two disease SNPs and offers a comparison of approaches applicable to multiple SNPs (HST and HSTMLB). In addition, we also apply these approaches to a rheumatoid arthritis (RA) real dataset provided by the North American Rheumatoid Arthritis Consortium (NARAC) in the Genetic Analysis Workshop (GAW) 15. RA is a common chronic complex disease with a genetic component that has long been recognized. Approximately 1% of the worldwide population is affected by RA (Orozco et al. [ 11]) which causes inflammation and destruction of the joints. Previous studies have identified linkage signals in genome-wide linkage scans [ 12, 13, 14], with the most consistent linkage evidence found around the HLA region on chromosome 6.

Methods and Results

We first review the HST method to identify SNPs explaining the IBD sharing at the linkage peak in a sample of ASPs with parental genotypes. We then introduce a new extension, HSTMLB.

Homozygote Sharing Tests

Risch [ 15] proposed the Maximum LOD Score (MLS) method to detect increased or higher than expected IBD sharing among ASPs. Under the null hypothesis of no linkage, the probability of an ASP sharing 0, 1 or 2 alleles IBD is 1/4, 1/2 and 1/4 respectively. A trinomial likelihood ratio can be used to test for departure from the null hypothesis. Increased sharing would indicate that the tested SNP is linked to the disease susceptibility loci. The HST uses a likelihood function that decomposes the IBD sharing in MLS by conditioning on parental genotype. Further assuming a multiplicative model of transmission, it can be shown that the parental transmissions are independent at any SNP (Appendix A). Consequently, the general likelihood function for HST can be expressed as the product of two independent binomial variables.

More formally, let X homo and X het be the number of alleles IBD shared by an ASP from a homozygous parent and a heterozygous parent, respectively. Under the assumption of independent parental transmissions, X homo and X het are two independent Bernoulli variables with means α and α, respectively. Therefore, the log-likelihood function of N independent ASPs is

If a SNP is independent of the disease SNP, the IBD sharing probabilities are independent of parental genotypes. Hence, the IBD sharing probabilities from homozygous and heterozygous parents are both equal and are greater than 1/2 due to linkage. If a SNP is in LD with the disease SNP, increased IBD sharing from homozygous parents may be observed and this sharing should be less than the sharing from heterozygous parents. To determine whether a SNP partially explains the IBD sharing at the linkage peak, Dupuis and Van Eerdewegh [ 6] proposed HST.P, a likelihood ratio test of the hypotheses

If a SNP is in perfect LD with the sole disease SNP in the linked region, there should be no excess IBD sharing from parents homozygous for the causal SNP, so α = 1/2 (Appendix B). The fact that α = 1/2 at the disease SNP was also derived in a different way in Robinson et al. [ 16]. Once SNPs that partially explains the linkage peak have been detected, Dupuis and Van Eerdewegh [ 6] proposed HST.F, a likelihood ratio test of the hypotheses H ₀: α = 1/2 vs. H ₁: α > 1/2 to further identify SNPs that do not fully explain the linkage peak. Rejection of the null hypothesis suggests the tested SNP does not fully explain the observed IBD sharing at the linkage peak.

Under the null hypotheses, both HST.P and HST.F follow a χ ² mixture distribution of 0.5 χ ₀ ² + 0.5 χ ₁ ². A theoretical power approximation for HST.F is provided in Appendix C; a similar procedure may be used to obtain a power approximation for HST.P.

The HST.P and HST.F can be extended to multiple-SNP analysis in cases where none of the single SNPs can fully explain the observed IBD sharing at the linkage peak. The only difference between single-SNP analysis and multiple-SNP analysis in HST is the definition of parental homozygosity. For single-SNP analysis, a homozygous parent has two identical alleles at the tested SNP, while for multiple-SNP analysis, a homozygous parent has two identical haplotypes at the tested SNPs. In multiple-SNP analysis, the parental homozygosity determination does not involve haplotype reconstruction, because an individual with two identical haplotypes indicates that the individual is homozygous at all sites included in the haplotype.

When affected sibships with more than two affected siblings are present, the IBD sharing is not independent among all possible sibling pairs, violating one of the assumptions used to derive the HST asymptotic distributions. One approach to handle this issue is to multiply the likelihood of i-th nuclear family by a weight of 2/ S, where Sdenotes the number of affected children in the family (Van Eerdewegh et al. [ 17]). We incorporated this correction factor in our R (R Development Core Team [ 18]) implementation of the HST statistics.

Homozygote Sharing Tests in Maximum Likelihood Binomial

In nuclear families, if a particular locus is unlinked to a disease of interest, the probability that an offspring inherits the maternal allele of grand maternal origin is 1/2 based on Mendelian inheritance. Therefore, the probability that k out of S offspring inherits the grand maternal allele from their mother follows a binomial distribution with probability parameter γ equal to 1/2. The same holds for the grand paternal allele transmitted from a father to his offspring. However, if the locus is linked to the disease, affected offspring will have a higher probability of inheriting the allele that is linked to the causal variant, creating a distortion from the 1/2 transmission by Mendel's first law. The MLB statistic [ 8, 9] tests whether γ is equal to 1/2 by using a binomial likelihood ratio. Rejection of the null hypothesis γ = 1/2 indicates evidence for linkage. Because one does not know a priori which of the grand paternal or grand maternal allele segregates the disease susceptibility allele, the part of the likelihood involving γ in a nuclear family can be written as: f(γ) = γ k(1 - γ) S - k + (1 - γ) k γ S - k.

In the same spirit as the HST statistics, with parental genotypes, the likelihood can be expressed as a function of γ and γ, where these parameters are estimated based on the transmissions from heterozygous and homozygous parents, respectively. We propose a likelihood ratio test, which we denote HSTMLB.P, to test the hypotheses H ₀: 1/2 < γ = γ vs. H ₁: 1/2 ≤ γ < γ to determine whether a SNP partially explains the observed IBD sharing at the linkage peak. Similarly, we suggest a second likelihood ratio test, HSTMLB.F, for the hypotheses H ₀: γ = 1/2 vs. H ₁: γ > 1/2 to determine if a SNP does not fully explain the observed IBD sharing at the linkage peak. Under their null hypotheses and assuming independent parental transmission, the distribution of both HSTMLB statistics asymptotically follows a χ ² mixture distribution of 0.5 χ ₀ ² + 0.5 χ ₁ ². The software Genehunter [ 19] was modified to compute the HSTMLB statistics.

Simulation Study - A Single Di-Allelic Disease Locus

A simulation study using several models for complex disease was conducted to compare and contrast the HST and HSTMLB with other methods. We simulated a single disease locus consisting of risk allele D with allele frequency p and normal allele d. Table characterizes the four simulated disease models. The population prevalence of disease was fixed at 5%.

We generated 200 nuclear families each with one ASP. For each replicate, we simulated the disease SNP, a SNP in LD ( r ² = 0.1, 0.2, ..., 0.9) with the disease SNP, a SNP in linkage equilibrium (LE) with the disease SNP, and a fully informative microsatellite marker used to determine IBD sharing. All SNPs had identical allele frequency p, and there was no recombination between the simulated markers and hence, the observed IBD sharing is constant across the simulated markers. For every disease model in table , we used 9,000 replicates to assess type I error rates and 1,000 replicates to evaluate power. We compared HST.P, HSTMLB.P, GIST, and Horikawa (denoted Hori in tables and figures) for identifying SNPs that partially explain the observed IBD sharing and contrasted them with LAMP.P. When attempting to identify SNPs that do not fully explain the observed IBD sharing, HST.F and HSTMLB.F were contrasted with LAMP.F. Note that the fully informative marker was used as the flanking marker for LAMP and to determine IBD sharing for the other approaches.

Results: Identifying SNPs That Partially Explain the Linkage Peak

Table presents the type I error rate estimates at a 5% significance level. In general, GIST has slightly conservative type I error rates, while the other methods to identify SNPs that partially explain the IBD sharing have type I error rates close to 5%. LAMP.P has slightly inflated type I error rates in testing whether there is LD between the tested SNP and the disease SNP. Figure presents the power results at a 5% significance level for dominant and recessive disease models. The power increases as the LD between the disease SNP and the tested SNP increases and as the risk allele frequency decreases. GIST is more powerful than other methods to identify SNPs that partially explain the IBD sharing, except when the disease model is dominant where HST.P and HSTMLB.P have better power (results for other disease models are not shown).

Results: Identifying SNPs That Do Not Fully Explain the Linkage Peak

When looking for SNPs that fully explain the IBD sharing, the null hypothesis is that the tested SNP is the only disease locus or is a perfect proxy for the sole disease SNP in the linkage region and thus, fully explains the observed IBD sharing at the linkage peak. Hence, rejecting the null hypothesis indicates that the tested SNP does not fully explain the IBD sharing. Table presents the type I error rates at a 5% significance level. The type I error rates are close to 5% for both HST.F and HSTMLB.F to identify SNPs that do not fully explain the IBD sharing. In contrast, LAMP.F has slightly conservative type I error rates for testing whether a SNP is in perfect LD with the disease SNP. Figure presents the power results at a 5% significance level for multiplicative and recessive models (results for other disease models are not shown). As expected, the power decreases when the LD between the disease SNP and the tested SNP increases. HST.F and HSTMLB.F have similar power, and the power of HST.F was accurately estimated (within 5% difference) using the theoretical approximation (Appendix C). In contrast, LAMP.F has better power to identify SNPs not in perfect LD with the disease SNP under the recessive model.

Table 4

For ASP design, our simulations show that HST (HST.P and HST.F) and HSTMLB (HSTMLB.P and HSTMLB.F) have equivalent power and type I error. Because HSTMLB does not assume independence of sibling pairs within a sibship, it may be applied to sibships with three or more siblings. However, HST uses weighted likelihood to account for the dependence of IBD sharing among sibling pairs. To compare these two approaches on larger sibships, we performed a simulation with p = 0.2 using models described in table to generate 150 sibships of size three or 50 sibships of size four. We found that HST has slightly conservative type I error rates for larger sibships, while HSTMLB has type I error rates close to 5%, and that HSTMLB is slightly more powerful than HST (results not shown).

Simulation Study - Two Di-Allelic Disease Loci

When none of the single SNPs fully explain the IBD sharing at the linkage peak, one may further analyze two-SNP combinations. Among the methods investigated, only HST and HSTMLB can be extended to multiple-SNP analysis to identify SNP combinations that explain the linkage peak. We compared these two methods for single- and two-SNP analyses using simulations in which three SNPs and a fully informative marker to determine IBD sharing were simulated. All three SNPs were set to have equal allele frequencies, and there was no recombination between the simulated markers. To evaluate type I error rates of HST.P and HSTMLB.P in two-SNP analysis, one disease SNP and two mutually independent SNPs were simulated, and the two independent SNPs were used to estimate the type I error rates. The same population prevalence (5%), disease models and penetrances as in table were used. A second three-SNP scenario was simulated where SNPs 1 and 3 were independent and equally contributing disease SNPs, while SNP 2 was in varying degrees of LD ( r ² = 0, 0.1, 0.2, ..., 1.0) with SNP 1 but in LE with SNP 3. This scenario was used to assess the power of HST.P and HSTMLB.P and to evaluate the type I error rates and power of HST.F and HSTMLB.F. We used the same population prevalence (5%), disease models and penetrances as in table , but with penetrances conditional on the risk haplotype of the two disease SNPs. Let D ₁ and D ₂ be the risk alleles of the two disease SNPs, h = D ₁ D ₂ be the haplotype with increased disease risk and all other haplotypes confer the same disease risk. We used subscripts to indicate which SNPs were analyzed; for example, HST.F ₁₃ means that SNPs 1 and 3 were used to determine parental homozygosity status when computed the HST.F statistic. We simulated 9,000 replicates to estimate type I error rates and 1,000 replicates to estimate power at a 5% significance level for HST and HSTMLB.

Results: Identifying SNPs That Partially Explain Linkage Peak

The type I error rates at a 5% significance level of two-SNP analysis for HST.P and HSTMLB.P are close to 0.05 (table ). Figure presents the power of HST.P and HSTMLB.P to identify SNPs that partially explain the observed IBD sharing at the linkage peak for single- and two-SNP analyses for additive and recessive models (results for other disease models are not shown). HST.P and HSTMLB.P have equivalent power; therefore, we only present and discuss results from HST.P in what follows. In single-SNP analysis, the simulated power of HST.P to detect the true disease loci is greater than the power to detect the SNP in LD with one of the disease loci, as expected. In two-SNP analysis, HST.P ₁₃ has the best power, while the power of HST.P ₂₃ increases as the LD between SNPs 2 and 1 increases.

Table 5

Results: Identifying SNPs That Do Not Fully Explain the Linkage Peak

Table presents the type I error rates for HST.F ₁₃ and HSTMLB.F ₁₃ to identify SNP pairs that may fully explain the linkage peak at a 5% significance level. HST.F ₁₃ and HSTMLB.F ₁₃ have type I error rates close to 5% in all scenarios. Figure presents the power of HST.F and HSTMLB.F to reject SNP pairs that do not fully explain the IBD sharing for additive and recessive models (results for other disease models are not shown). Again we use HST.F to report the results because HST.F and HSTMLB.F have equivalent power. In general, single-SNP analysis has better power than two-SNP analysis to reject SNPs that do not fully explain the IBD sharing at the linkage peak due to the reduction in the number of homozygous parents.

Table 6

Application to GAW15 NARAC Data

We compare HST and HSTMLB with other methods included in our simulation studies for identifying SNPs that explain the observed IBD sharing at the linkage peak in the GAW15 NARAC data. The dataset contains 746 multiplex families. The full sample is used in the initial linkage and association analyses. Further analyses to identify SNPs explaining the IBD sharing at the linkage peak are restricted to 280 families with parental genotypes available so that most of the methods could be applied to the same subset of individuals for comparison purpose. Among these 280 families, there are 212 families with one genotyped parent and 68 families with two genotyped parents, and the number of affected sibships ranges from two to four. We focus our analysis on 5,407 SNPs on the 22 autosomal chromosomes.

LD among SNPs can lead to false positive linkage signals in ASP analysis when parental genotype information is incomplete [ 20]. Because LD exists in the set of genotyped SNPs [ 14], we use SNPLINK [ 21] to remove SNPs in LD with D′ greater than 0.5, and select a subset of 4,400 SNPs for linkage analysis. The linkage scan using Merlin [ 22] reveals a region of strong linkage with a maximum LOD score [ 23] of 15.95 at rs11908 at 32,991,663 base pairs (bp) on chromosome 6 close to major histocompatibility complex (MHC). We apply the family-based association test (FBAT) in the presence of linkage [ 24] to the HLA-DRB1 locus and to 14 SNPs falling within the 2-LOD score support region. The HLA-DRB1 genotype is recoded as a di-allelic marker with the highest risk alleles 0401, 0404, 0405, 0408 and 0409 forming allele 1 and the other HLA-DRB1 alleles recoded as allele 2. A total of 5,003 SNPs on chromosomes other than chromosome 6 are used to estimate type I error rates of methods for identifying SNPs that explain the IBD sharing at the linkage peak. FBAT identifies 2 associated markers (the recoded HLA-DRB1 locus and rs1003979) from the 15 markers. We apply HST, HSTMLB, Horikawa and GIST to the 2 associated markers to determine if they explain the IBD sharing at the linkage peak. The results are contrasted with LAMP. Because neither SNP can fully explain the IBD sharing at the linkage peak, we further use FBAT to identify associated marker pairs out of the 105 marker pairs formed by the 15 markers under the linkage peak and then apply two-marker HST and HSTMLB analyses to the associated marker pairs. For GIST and Horikawa, the risk allele is defined as the overtransmitted allele from heterozygous parents. After removing SNPs in LD, all SNPs residing on the same chromosome as the tested SNP are used as flanking markers in the LAMP analyses.

Single-Marker Analysis: Identifying Markers That Partially Explain the Linkage Peak

We estimate the type I error rates at a significance level of 5% for all methods except LAMP.P, using 5,003 SNPs on chromosomes other than 6 and the IBD sharing at the linkage peak on chromosome 6. For HST.P, HSTMLB.P, Horikawa and GIST, the type I error rate estimates are 4.4, 4.7, 5.2 and 5.1%, respectively. The type I error rate for LAMP.P cannot be evaluated using markers on chromosome 6 as it is not known a priori which SNPs are in LE with the disease susceptibility loci on that chromosome. Using SNPs on chromosomes other than chromosome 6 leads to an inflated positive rate of 17.9% for LAMP.P. The inflation may be due to the presence of other disease susceptibility loci in the rest of the genome and, to a smaller extent, to the inadequate modeling of familial aggregation as seen in our simulation results. Table presents the results of identifying markers that partially explain the IBD sharing at the linkage peak on chromosome 6 for the two associated markers in 2-LOD score support region. Only the recoded HLA-DRB1 locus is identified as partially explaining the linkage peak by all methods but HSTMLB.P (HST.P p value = 0.046; HSTMLB.P p value = 0.14, Horikawa p value = 4 × 10 ⁻⁵ and GIST p value <10 ⁻⁶). Note that when using the original multi-allelic HLA-DRB1, both HST.P and HSTMLB.P detect HLA-DRB1 with p values of 0.0341 and 0.0108, respectively. By contrast, the recoded HLA-DRB1 is also identified by LAMP.P as a marker in LD with a disease locus (p value = 1.6 × 10 ⁻²⁴).

Single-Marker Analysis: Identifying Markers That Do Not Fully Explain the Linkage Peak

On chromosomes other than chromosome 6, HST.F and HSTMLB.F have 99.96 and 100% power, respectively, to reject SNPs that cannot alone fully explain the IBD sharing at the linkage peak at a 5% significance level. In contrast, LAMP.F has 87.2% power to identify SNPs that are in partial LD but not in perfect LD with a disease SNP. The recoded HLA-DRB1 marker is rejected by HST.F (p value = 3.7 × 10 ⁻³) and HSTMLB.F (p value = 2.8 × 10 ⁻⁴) as the sole source of the observed IBD sharing at the linkage peak. Similarly, LAMP.F rejects that the recoded HLA-DRB1 marker is in perfect LD with a disease SNP on chromosome 6 with a p value of 1.8 × 10 ⁻⁶. This suggests that further multiple-marker analysis is needed to determine the source of the linkage signal on chromosome 6.

Two-Marker Analysis

FBAT identifies 25 associated marker pairs from the 105 marker pairs formed by the recoded HLA-DRB1 marker and 14 SNPs in the 2-LOD score support region (results not shown). We further apply two-marker HST.P and HSTMLB.P analyses to the 25 associated marker pairs to test if they partially explain the observed IBD sharing at the linkage peak. Table presents the results of 5 marker pairs that at least partially explain the linkage peak identified either by HST or HSTMLB. Both HST and HSTMLB cannot reject 3 marker pairs that may fully explain the IBD sharing at the linkage peak at a significance level of 10%: HLA-DRB1-rs169679, HLA-DRB1-rs11908 and HLA-DRB1-rs1011094. The number of homozygote parents ( N homo) for the 3 marker pairs is about 90, which might suggest that HST.F and HSTMLB.F may not have good power to reject the null hypothesis that the tested marker pair fully explains the observed IBD sharing at the linkage peak. However, the estimates of α and γ for HLA-DRB1-rs169679 are close to what is expected (0.5) under the null hypothesis. In addition, we assess the empirical power of HST.F to identify SNP pairs that do not fully explain the IBD sharing at the linkage peak by permuting the parental homozygosity status to form an empirical null distribution based on 10,000 replicates. The power estimates (HST.F power in table ) are all greater than 90%, so there is good power to reject SNP pairs that do not fully explain the observed IBD sharing at the linkage peak.

Discussion

We have investigated HST and HSTMLB and compared these methods to others to identify SNPs that explain the IBD sharing at the linkage peak. Based on our simulations, we found that examining the IBD sharing probability from homozygous parents provides a simple and powerful way to identify SNPs that do not fully explain the linkage peak (HST.F and HSTMLB.F).

In the first simulation study using an ASP sample, we simulated a single disease SNP and considered various disease models. We found that in most scenarios, GIST was the most powerful to identify SNPs that partially explain the IBD sharing at the linkage peak. On the other hand, when searching for SNPs that do not fully explain the IBD sharing at the linkage peak, HST.F and HSTMLB.F were equally powerful. In contrast, LAMP.P showed great power to detect SNPs in LD with the disease SNP, while LAMP.F showed good power to identify SNPs not in perfect LD with the disease SNP.

In the second simulation study using an ASP sample, we generated two disease SNPs with equal effect on disease susceptibility and evaluated HST and HSTMLB, the only two approaches easily generalizable to multiple-SNP analysis. Our results showed that these two methods are equivalent in terms of power and type I error.

In a sample of ASPs with parental genotypes, we derived the theoretical approximations of HST.P and HST.F, which we used to compute their noncentral χ ² parameters and theoretical power. The theoretical power approximation was precise and close to the empirical power estimated from simulations with the maximum difference less than 5% among all scenarios (results not shown).

In the NARAC dataset, we applied all studied methods to attempt to identify SNPs that explained the IBD sharing at a previously reported linkage peak on chromosome 6. Among the recoded HLA-DRB1 locus and the 14 SNPs in the 2-LOD score support region, HST and HSTMLB could not reject three and four associated marker pairs that may fully explain the IBD sharing observed at the linkage peak at a 5% significance level, respectively. Most of these marker pairs are composed of the recoded HLA-DRB1 locus. We confirmed the consistent evidence that the HLA-DRB1 locus contributes to the RA susceptibility and yet, the HLA-DRB1 locus is not the sole source of the strong linkage signal.

Note that for HST.F and HSTMLB.F, the null hypothesis is H ₀: markers fully explain the IBD sharing at the linkage peak. Failure to reject the null hypothesis may occur when there is insufficient power, which may be due to a low number of homozygous parents at the markers of interest. While this should be less frequent in single associated marker analysis, this is more likely to happen in multiple associated marker analysis, where it will be even rarer for parents to be homozygous at all sites tested. A more liberal significance level than the typical 5% may need to be used in order to avoid falsely identified markers as fully explaining the IBD sharing at the linkage peak.

In conclusion, the HST has the following nice properties: (1) simplicity in theory and computation; (2) ability to test whether associated SNPs explain none/some/all of the IBD sharing at the linkage peak; (3) powerful extensions such as HSTMLB.F; (4) flexibility to work with multiple-SNP combinations or multi-allelic markers. In addition, the idea of decomposing IBD sharing into sharing from heterozygous and homozygous parents can be incorporated into other linkage methods for identifying SNPs that explain the IBD sharing at the linkage peak, such as in HSTMLB and also in the application to quantitative traits. However, a limitation of the HST and HSTMLB is the requirement that parental genotypes be available. Additionally, the power of HST.F and HSTMLB.F rely on the number of homozygote parents, and the generalization to multiple-SNP analysis may reduce power because the number of parents homozygous at all markers decreases with the inclusion of additional markers. In conclusion, among the approaches investigated in this paper, we suggest using HST-based approaches to identify SNPs that explain the IBD sharing at the linkage peak when parental genotypes are available, whereas GIST is preferable when parental genotypes are missing.

Acknowledgements

The computing was conducted on the Linux Cluster for Genetic Analysis (LinGA-I) funded by the NIH (National Center for Research Resources) Shared Instrumentation grant (1S10RR163736-01A1) and the Linux Cluster for Genetic Analysis (LinGA-II) funded by the Robert Dawson Evans Endowment of the Department of Medicine at Boston University School of Medicine and Boston Medical Center. This work was supported by the National Institutes of Health/National Heart, Lung, and Blood Institute Contract N01-HC-25195. We are very grateful for Dr. Amos and Dr. Gregersen's consent to analyze the NARAC data.

Appendix A

Proof of independence of parental transmissions to ASP under a multiplicative model, assuming no imprinting in a linked region. Equivalently, for m = 0, 1 and d = 0, 1, show that

where I M and I D are the number of alleles shared IBD by an ASP from mother and father, respectively.

Proof: For a di-allelic disease locus with risk allele t (frequency p) and normal allele T (frequency q = 1 − p), assuming no imprinting, we define f as the probability that an individual is affected given that he has i copies of risk allele, i = 0, 1, 2. The genotype relative risks are denoted by

and

For a multiplicative model, η = γ ². The population prevalence can be written as

The additive ( V) and dominant ( V) variance components assuming no imprinting [ 25] can be written as:

Therefore, parental transmissions are independent at a marker with recombination fraction θ away from the disease locus for multiplicative models. When θ = 0, this reduces to showing independence of parental transmissions at the disease locus.

Appendix B

Proof of there is no excess IBD sharing from parents homozygous at the disease locus assuming no imprinting. That is, α = 1/2 at the disease locus.

Table B-1

For tt × Tt, a child has the probability 1/2 to be either tt or Tt and thus has the probability

Therefore,

Appendix C

Theoretical approximation for the expected HST.F.

We show the approximation procedure for the expected HST.F; a similar procedure applies to HST.P. Under the alternative hypothesis that the tested SNP is in LD with the disease locus, the distribution of HST.F can be approximated by a distribution, where λ = E(HST.F) 1 is the noncentrality parameter, provided that E(HST.F) is sufficiently large. With the approximated expected HST.F, we can compute the noncentrality parameter and then the theoretical power.

We only show the approximation to the other terms can be obtained similarly.

Let

1.

where these three probabilities are derived below:

and

2.

References

6. Dupuis J, Van Eerdewegh P. Identification of polymorphisms that explain a linkage peak: conditioning on parental genotypes. Genet Epidemiol. 2003;25:247. [ Google Scholar]

18. R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org 2007.

# data[].attributes.attachment_link.title
Evaluation of Approaches to Identify Associated SNPs That Explain the Linkage Evidence in Nuclear Families with Affected Siblings

# data[].attributes.attachment_link.description
Linkage analysis is often followed by association mapping to localize disease variants. In this paper, we evaluate approaches to determine how much of the observed linkage evidence, namely the identity-by-descent (IBD) sharing at the linkage peak, is explained by associated SNPs. We study several methods: Homozygote Sharing Tests (HST), Genotype Identity-by-Descent Sharing Test (GIST), and a permutation approach.

# data[].attributes.headline
1. An article which describes the aim of the research reported as to ‘explore’

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1. An article which describes the aim of the research reported as to ‘explore’