Modelling Genetic Variation
in the Brain
Thomas Nichols, PhD
Department of Statistics,
Warwick Manufacturing Group
University of Warwick
joint with
Becky Inkster
Institute of Psychiatry
King’s College London
(GSK3β & WNT
pathway VBM)
Maria Vounou,
Giovanni Montana
Statistics Section, Dept.
of Mathematics
Imperial College
(Sparse Reduced Rank
Regression)
Outline
•  Background
–  Structural brain imaging & VBM
–  Genetics
–  “Imaging Genetics”
•  Candidate SNP VBM
•  Multivariate SNP analyses
Neuroimaging Background:
Structural Brain Image Data
•  Morphometry
–  Quantification of shape/volume of
brain structures
•  Traditional Morphometric Analysis
–  Laborious hand-tracing of structures
–  Accurate, but imperfect inter-rater
reliability
•  Voxel Based Morphometry
–  Automated morphometry method
Voxel Based Morphometry
Original MRI → Segment → Warp to Atlas Space → Modulate → Smooth
Subject Space
T1-weighted MRI
Subject Space
Gray Matter
Voxel Based Morphometry
Original MRI → Segment → Warp to Atlas Space → Modulate → Smooth
Modulation
Gives units of
subject GM
volume in
atlas space
Atlas Space
Atlas Space
Atlas Space
Allows
analysis in
common
space while
retaining
individual
differences
T1-weighted MRI
Gray Matter
Modulated
Gray Matter
Voxel Based Morphometry
Original MRI → Segment → Warp to Atlas Space → Modulate → Smooth
Smoothing
•  Accounts for imperfect
registration of individuals to
atlas
–  Even identical twins have
different cortical foldings
–  Exact match impossible
•  Discards fine spatial details in
exchange for reduced noise
Atlas Space
Atlas Space
–  Generally searching for
moderate scale differences
Done!
•  3D image is n=1
–  A single (100,000-dimensional)
phenotypic measurement on 1
individual
Modulated
Gray Matter
Smoothed,
Modulated GM
Genetics Background
•  Genotype
–  The genetic constitution of an organism or cell
–  46 chromosomes in humans
–  23 pairs of homologous chromosomes
•  One each from each parent
•  Gene
–  A series of basepairs (DNA bits) which code
for a trait
–  Four different possible basepairs, the
nucleotides
•  Adenine, Thymine, Cytosine, & Guanine
Genetics Background
•  Single Nucleotide Polymorphisms (SNP)
–  Locations where single base-pair differences bases
have been found in the population
•  SNP Example
–  If some of the population has sequence…
AATGTGATAGCTT
–  And if remaining has…
AATGTGACAGCTT
–  We have found a SNP!
SNP
•  SNP data
–  Homologous chromosomes
–  For each SNP, for each individual: 0, 1 or 2 count
8
Genetics
Background
•  Millions of SNPs
•  Thanks to correlation
(linkage disequilibrium),
only need ≈500k to “tag”
all variation
5
18
16
x 10
2.5
x 10
3,079,843,747 Base Pairs †
2
1.5
1
0.5
0
Number of SNPs per Chromosome
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X Y
Number of Genes per Chromosome
4500
20,296,765 SNPs *
4000
14
3500
12
3000
10
2500
8
2000
6
1500
4
1000
2
500
0
Number of basepairs per Chromosome
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X Y
0
32,185 Genes †
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X Y
†
Genetics Background
•  SNPs vs. genes
–  Each gene often has several variants
–  1 or more (but not many) SNPs typically
needed to identify a gene
–  SNPs may not lie directly on coding portion of
gene
•  Due to linkage disequilibrium (correlation), close is
good enough
•  Non-coding, regulatory region may be causal
Location on
chromosome
Exon
Exon
Exon
Exon
Exon
SNPs
Exon
Imaging Genetics
•  Motivation
–  Brain structure heritable
–  Objective, reproducible phenotype
•  Important in psychiatry
Brain Phenotype
h2
Whole brain volume
0.78
Total gray matter volume
0.88
Total white matter volume
0.85
Glahn, Thompson, Blangero. Hum Brain Mapp 28:488-501, 2007
Thickness of Cortical GM (r2)
–  Current best measures are coarse,
with weak reproducibility
»  e.g. HAM-D (depression), MMSE
(cognition, AD)
–  Sensitive
•  Brain anatomy/function closer to
disease process than other measures
–  Use to collaborate other findings
Thompson et al, Nature Neuro, 4(12):1253-1258,. 2001
Heritability of GM Thickness
(h2 & corrected P-value)
•  E.g. Large WGA finds modest
significance
Use brain imaging to build confidence
in finding
Thompson & Toga, Annals of Medicine 34(7-8):523-36, 2002
Imaging Genetics Menu
Imaging
Candidate ROI
Many ROI
Voxelwise
Genetics
Candidate SNP
Candidate Gene
Genome-wide SNP
Genome-wide Gene
[Filippini et al. 2009]
29,812 voxels
1 SNP
[Joyner et al. 2009]
4 ROIs, 11 SNPs
[Potkin et al. 2009]
1 BOLD ROI
317, 503 SNPs
[Stein et al. 2010]
31,622 voxels
448,293 SNPs
[Hibar et al. 2011]
31,622 voxels
18,044 SNPs
(Jason Stein/Andy Saykin/Bertrand Thirion)
Outline
•  Background
–  Structural brain imaging & VBM
–  Genetics
–  “Imaging Genetics”
•  Candidate SNP VBM
•  Multivariate SNP analyses
GSK3β Background
•  High heritability of depression (Kendler et al.
2006; Sullivan et al., 2000).
•  Meta-analytical evidence from MRI studies for a
role of hippocampal integrity in depression
(Campbell et al., 2004).
•  There is strong genetic regulation of
neurodevelopment (reviewed by Wilson and
Rubenstein, 2000; O Leary et al., 2002).
•  The Wnt signaling pathway is one network of
proteins that play a role in embryogenesis
•  GSK3β plays a key role in Wnt pathway
Wnt Signaling Pathways
regulates the development of the hippocampus
GSK HiTDIP Study
•  Major Depressive Disorder (MDD) Association Study
–  “High Throughput Human Disease Specific Targets”
–  7,000 SNPs covering 2,000 genes with tractable targets
–  1000 cases, 1000 controls
•  Imaging Subset
–  200 cases, 200 controls (of 1000 & 1000) scanned with
anatomical MRI protocol
–  Optimized VBM with SPM5 s segmentation tool
–  324 images passed QC
•  366 subjects data delivered
•  42 subjects set aside
(clinical exclusion, pathologies or failed segmentation)
•  Glycogen synthase kinase 3β (GSK3β)
–  Plays key role in WNT pathway, influential in development
Modelling Candidate SNPs
•  Mass Univariate Modelling
–  Fit same univariate linear model at each voxel
•  Quantitative Trait Multiple Regression
–  Linear model fit at each voxel
•  Regressors
–  Genetic
–  Group (Case/Control)
–  Demographic / nuisance variables
0
1
SNP Count
2
Xj
Gray Matter Volume
•  Dominant
Gray Matter Volume
Y
Y
Gray Matter Volume
•  Recessive
Gray Matter Volume
SNP Models for Gray Matter Data
Y
0
1
SNP Count
2
0
1
SNP Count
2
Y
•  Additive
•  Genotypic
0
1
SNP Count
2
Xj
Xj
Xj
Mass Univariate Modelling
Genetic Effects
•  Concerns about leverage/influence
–  100’s not 1000’s of subjects
–  Rare SNP can make a few
subjects very influential
•  An ever-greater problem
as sample size shrinks
Y
Gray Matter Volume
•  100 subjects + 10% MAF
→ 1 subject with rare
genotype expected!
0
1
Allele Count
2
Xj
Mass Univariate Modelling
Genetic Effects
•  Ad hoc solution
–  If expected rare genotype frequency <10%
merge genotypes
•  If MAF > 0.31 (=√0.1)
–  2DF Genotypic model
•  Additive + Nonadditive Parameterization
–  Additive
[ -1
0 +1 ]
–  Nonadditive [ -1/2 +1 -1/2 ]
(orthogonalize w.r.t. additive regressor * )
•  If MAF < 0.31
–  Use dominant/recessive model
tested
not tested
Mass Univariate Modelling
Nuisance Effects
•  Age & Gender
–  Substantial normal variation in GM w/ Age
•  Total Gray matter
–  Accounts for differences in head size
–  Discounts global changes to find localized changes
•  Scanner (Pre/Post Upgrade)
–  Upgrade 2/3-through study altered image contrast
•  Medication (Yes/No, for cases only)
–  Neurotrophic effects reported for some Rx
Model Diagnosis for Imaging
•  Why bother?
–  Largish n, continuous data, Central Limit
Theorem should carry us
–  Type I Error generally OK due to robustness
of t-test/ANOVA-like models
Failed GM
segmentation
due to data
formatting
error
•  Sensitivity!
–  Decreased sensitivity due to inflated error
variance σ
–  Suboptimal sensitivity due to non-normality
•  How!?
–  100,000 voxels, 400 subjects
–  100,000 QQ plots to look at all 40 million
data points?
Warping
artefacts seen
in modulated
GM
Model Diagnosis for Imaging
Model Summaries
•  Model summaries
–  Images of diagnostic stats
•  Scan summaries
–  Vectors of ad hoc measures
•  Dynamic graphical tool
–  Explore many summaries
simultaneously
–  Easily jump from
summary image to plots, from
plots to residual images
•  End Result
–  Swiftly localize and
understand problems
Luo & Nichols NeuroImage 19:1014–1032, 2003
Statistic
Assesses
Null Distn
Cook-Weisberg Var(εi) = σ2
Chi-Squared
Shapiro-Wilk
ε ~ Normal
(tabulated)
Outlier Count
Artifacts
Binomial
Std. Deviation
Artifacts
Scan Summaries
Summary
Interpretation
Global intensity
Whole-brain signals
or artifacts
Outlier Count
Artifacts
Any preprocessing
parameters
e.g. head size
Experimental
predictors
Suggests cause of
artifacts
For investigating
mismodelled signal
in residuals
http://go.warwick.ac.uk/tenichols/software
Model Diagnosis w/ SPMd
Model Summaries
Scan Summaries
Model Detail
Scan Detail
Outline / Motivation
•  Data
–  Intro to Voxel Based Morphometry data
•  Model
–  Quantitative trait regression w/ Mass Univariate Model
•  Diagnosis
–  100,000 Q-Q plots anyone?
•  Inference
–  Cluster size under nonstationarity
–  Candidate screening procedure
•  Results
–  GSK3β in MDD
•  Future Directions
Inference On Images:
Voxel-wise vs. Cluster-wise
•  Voxel-wise
–  Reject Ho, point-by-point, by statistic magnitude
•  Cluster-wise
–  Define contiguous blobs with arbitrary threshold uclus
–  Reject Ho for each cluster larger than kα
uclus
space
Cluster not
significant
kα
kα
statistic
image
Cluster
significant
Cluster Inference & Stationarity
•  Cluster-wise preferred over voxel-wise
VBM:
Image of
FWHM
Noise
Smoothness
–  Generally more sensitive
Friston et al, NeuroImage 4:223-235, 1996
–  Spatially-extended signals typical
•  Problem w/ VBM
–  Standard cluster methods assume
stationarity, constant smoothness
–  Assuming stationarity, false positive clusters
will be found in extra-smooth regions
–  VBM noise very non-stationary
•  Nonstationary cluster inference
–  Must un-warp nonstationarity
–  Reported but not implemented
•  Hayasaka et al, NeuroImage 22:676– 687, 2004
–  Now available as SPM toolbox
•  http://fmri.wfubmc.edu/cms/software#NS
Nonstationary
noise…
…warped to
stationarity
Inference in Imaging Genetics:
Creeping Multiple Testing Problem
•  Even just with candidate analyses,
Can end up searching over…
–  Genes
–  SNPs within a gene
–  Space (voxels or clusters)
–  Different contrasts on GLM
•  Main effect? By clinical subgroup? Interactions?
•  Can quickly lose confidence in results
–  E.g. 0.005 FWE-corrected is great…
…Unless it s the 25th statistic image you ve seen
Inference in Imaging Genetics
Multiple Testing Strategy
•  Define strict primary outcome
–  For given gene, use single SNP
•  Best (large) association study significance, otw
•  Best nonsynonymous exonic available, otw
•  Best 5 intronic available
–  For each SNP, only consider main effect of gene
•  If fitting gene x group interaction, test for average effect
–
–
Any association is more likely than a disease-specific association
Even if disease-specification association, opposing sign of effect unlikely w/ VBM
–  1-number summary per gene
•  Minimum nonstationary cluster FWE-corrected P-value for association (1 DF
F-stat)
–  Bonferroni correction for number of genes
•  Primary outcomes then have strong FWE control
–  Over brain, over genes
–  (1-α)100% confidence of no false positives anywhere
•  Secondary outcomes
–  Interactions, sub-group results
–  Use same FWE-inferences, but mark as post-hoc
Results: Model Diagnosis
Outlier Detection with Shapiro -Wilk
-log10 P Shapiro-Wilk
R
Two outliers
Mean Smoothed Mod. GM
Results: Model Diagnosis
Characterising Outliers with Standardized Residual Images
R
R
Subject 193
Subject 194
Outlier
Subject 195
Note: Compare standardized residuals to +/-6.128
Subject 194 raw T1
(Bonferroni for 324 images, each with 173,823 voxels, at
each a 2-sided test)
Severe enlargement of inferior
horn of lateral ventricle
Results:
Outlier
Exploration
Subject 194
Outlier
Inferior Horn of
Lateral Ventricle
In most of us, this is
a pencil-lead-thick
fluid-filled space
In this subject it was
a pencil-thick
Clinical collaborator
verified it as
abnormal & subject
was removed
Randomly
Selected
Control
GSK3β and Structural Differences
R
L
2 SNPs in strong
linkage disequilibrium
showed significant
associations with GM
differences in MDD
patients:
rs6438552
rs12630592
Brain regions where
SNP clusters show colocalization.
GSK3β-Gray Matter association in bilateral superior temporal
gyrus (STG) and right hippocampus
Inkster, B., Nichols, T. E., Saemann, P. G., Auer, D. P., Holsboer, F., Muglia, P., & Matthews, P. M. (2009).
Association of GSK3 Polymorphisms With Brain Structural Changes in Major Depressive Disorder.
Archives of General Psychiatry, 66(7), 721-728.
AA genotype group associated with
decreased GM concentration in right STG
P = 0.0004
(corrected for whole brain search
and multiple SNP testing)
rs6438552 is a putative functional SNP
i.e. it regulates the selection of splice acceptor sites in vitro.
Inkster, B., Nichols, T. E., Saemann, P. G., Auer, D. P., Holsboer, F., Muglia, P., & Matthews, P. M. (2009).
Association of GSK3 Polymorphisms With Brain Structural Changes in Major Depressive Disorder.
Archives of General Psychiatry, 66(7), 721-728.
Wnt Signaling Pathways
WNT3A
FZD3
KRM1
DVL2
CTNNB1
AXIN2
TCF4
LEF1
SMAD1
PPARgC1a
EMX2
ZEB2
regulates the development of the hippocampus
WNT pathway genes
R
RR
ZEB2
FZD3
DVL2
AXIN2
GSK3β
SMAD1
PPARGCA1
EMX2
Inkster, B., Nichols, T. E., Saemann, P. G., Auer, D. P., Holsboer, F., Muglia, P., & Matthews, P. M. (2010).
Pathway-based approaches to imaging genetics association studies: WNT signaling, GSK3beta substrates and major depression.
NeuroImage, 53(3), 908-917.
Outline
•  Background
–  Structural brain imaging & VBM
–  Genetics
–  “Imaging Genetics”
•  Candidate SNP VBM
•  Multivariate SNP analyses
•  Voxel/Region QTL
–  Whole genome
association
–  Must have right ROI
100,000
voxels
•  Candidate SNP
–  Full image result
–  Must have right SNP
500,000 SNPs
≈ 1010
tests!
500,000 SNPs
≈ 106
tests
500,000 SNPs
100,000
voxels
•  Full cross analysis
–  Massive multiple testing
problem!
100,000
voxels
Possible MassUnivariate Analyses
≈ 105
tests
Multivariate Regression
Genotypes
Images
Y
=
N × NV
Regression
Coefficients
X
+
N × NG
•  Silly…
–  If N > NG, fit equivalent
to NV univariate models
fit independently
–  Much redundancy in C
•  rank{C} ≤ min(NV, NG)
≪ NV ∙ NG
Error
E
N × NV
C
NG × NV
N # subjects
NV # voxels/ROIs
NG # genes/SNPs
Reduced Rank Regression
Images
Y
Genotypes
=
N × NV
•  Fix rank r
•  Approximate
Image
Coefficients
X
A
r × NV
N × NG
Error
+
E
N × NV
Genotype
Coefficients B
C≈BA
B & A each rank r
N×r
N # subjects
NV # voxels/ROIs
NG # genes/SNPs
Sparse Reduced Rank
Regression
Images
Y
Genotypes
=
N × NV
•  Fix rank r
•  Approximate
X
Sparse Image
Coefficients
+
A
N × NG
r × NV
Error
E
N × NV
Sparse
Genotype B
Coefficients
C≈BA
B & A each rank r
NG × r
•  Enforce sparsity
Vounou, M., Nichols, T. E., & Montana, G. (2010). Discovering genetic associations with high-dimensional
neuroimaging phenotypes: A sparse reduced-rank regression approach. NeuroImage, 53(3), 1147-59.
N # subjects
NV # voxels/ROIs
NG # genes/SNPs
Sparse Reduced Rank
Regression - Estimation
•  RRR
–  Y = X A B + E
–  For fixed rank r, find A & B that minimize
M = tr { (Y−XBA) Γ (Y−XBA)’ }
for some NV × NV matrix Γ, e.g. Γ = I
•  SRRR
–  For rank 1, find a & b that minimize
M = tr { (Y−Xba’) Γ (Y−Xba’)’ }
+ λa||a||1 + λb||b||1
–  Then subtract Xba’ from the data, and repeat
–  Need to specify final rank r, λa & λb
•  Can set λa & λbin terms of #|a|>0 & #|b|>0
Simulation: Phenotype & SNPs
•  Simulated MRI data
–  ADNI T1 images through SPM5 VBM pipeline
–  NV = 111 ROIs, placed on VBM data from 189 MCI
ADNI subjects
•  GSK CIC Atlas, based on Harvard-Oxford atlas
–  Estimate covariance Σ after adjusting for age &
gender
–  Simulate ROI data (for arbitrary N) with covariance Σ
•  Evaluate with realistic genetic population w/
FREGENE
–  Simulates sequence-level data in large population
–  Provides 10K individuals, 20Mb chromosome (~180K
SNPs)
•  Chadeau-Hyam, et al. BMC Bioinformatics, 9:364, 2009
Simulation: Phenotype & SNPs
•  FREGENE SNP simulation
–
–
–
–
Population of 10,000 evolved over 200,000 generations
20Mb simulated
37,748 SNPs with MAF>0.05
Select k=10 causative SNPs
•  From all possible having MAF=0.2
•  Used to induce phenotypic effect
–  But then dropped from consideration
•  Represents realistic setting, where causative SNP is not seen, but effect
captured through local LD
–  From population of 10,000, repeatedly sample cohorts of size N
•  Simulated association in MRI data
–  Add genetic effect to Frontal and Temporal ROIs with causative
SNPs
•  γ = 0.06, 0.08, or 0.1 reduction in mean GM in affected ROI
•  Calibrated to Filipini et al. (2009)
–  10% reduction in GM ApoE ε4/ε4 subjects relative to subjects with no ε4 alleles
Out of Africa (OoA)
split & bottleneck
Founding population
in Africa
Expansion
Expansion
Expansion
Chadeau-Hyam, et al. BMC Bioinformatics, 9:364, 2009
Asian &
European split
FREGENE: Evolutionary
model of world population
•  Linkage disequilibrium (LD)
–  SNPs not independent
–  Highly structured,
heterogeneous
dependence
•  Population sub-structure
–  Ethnic differences &
migration patterns
induce systematic
variation
•  Multivariate analysis
–  Want realistic multivariate
structure in our simulations
The Wellcome Trust Case Control Consortium, Nature 447, 661-678, 2007.
Why try so hard?
Why not rand{0,1,2}500,000 ?
Realistic Phenotype
•  All pairwise GM
correlations
among NV = 111
ROIs
Realistic Genotypes
•  Correlation of first
1000 simulated
SNPs
Simulation Setting:
Horse shoes & Imaging Genetics
•  “True positive” with
missing causative
SNP
–  Declare true positive
if LD coefficient close
enough
•  LD-linked SNPs
–  Of 1990 SNPs
–  51 linked (r>0.8) to
one or more the 10
causative SNPs
SRRR Simulation Results
•  Power to detect 1 or more SNPs (NG=1990)
•  For ranks r = 1,2,3 dominates Mass Uni.
–  Better for higher r
SRRR Simulation Results
•  Power to detect 1 or more SNPs (NG=1990)
•  For ranks r = 1,2,3 dominates Mass Uni.
–  Better for higher r; here r = 3, high eff. size.
SRRR Simulation Results
•  Power to detect 1 or more ROIs
•  Less difference
–  Power can be manipulated by varying λ by
rank
SRRR: Multivariate vs. MassUnivariate
•  Does this NG=1990
result generalize?
•  For up to 40k SNPs
–  r = 3, med. effect
size, N=1000
–  Power 2-5 greater
–  Absolute power still
tiny
SRRR Simulation Results
•  Power to detect 1 or more SNPs (NG=1990)
•  For ranks r = 1,2,3 dominates Mass Uni.
–  Better for higher r; here r = 3
Sparse Reduced Rank Regression
for SNP – MRI Association
•  Detailed simulation of imaging & genetic
correlations structure
–  Suggests multivariate approach will outperform mass-univariate
–  Power tiny, in any event
•  Much work to do
–  Haven’t addressed how to optimize phenotype
–  Haven’t tried to estimate penalty parameters
λa, λb or r
•  Currently investigating stability selection
–  See #316 Le Floch et al
Conclusions
•  VBM
–  Powerful, automated anatomical analysis
–  Need careful raw data, preprocessing & model QC
•  Imaging Genetics
–  Mash-up of two large data, massive multiple testing
problems
•  Candidate SNP VBM
–  Given a SNP, just like a traditional imaging analysis
–  Multiple SNPs possible too, but need combining
methods
•  Multivariate Sparse Reduced Rank Regression
–  Promising, but little power unless have 1,000’s of
subjects