Spatial point process modelling of coordinate-based meta-analysis data
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Pantelis Samartsidis , Tor Wager , Lisa Feldman Barrett , Timothy D. Johnson , and Thomas Nichols
1University
of Warwick
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3
University of Colorado at Boulder
Introduction
Northeastern University
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University of Michigan
Results
Neuroimaging meta-analysis is a topic of growing interest, in part due to the small sample
sizes in individuals studies [Carp,2012], concerns of prevalence of Type II errors [Wager
et al, 2009] and low reliability [Raemaekers et al, 2007]. Since full statistical images are
generally not shared, only Coordinate-Based Meta-Analysis (CBMA) based on XYZ peak
locations is feasible. However CBMA methods are not based on a generative statistical
model and generally cannot account for study-specific characteristics.
In this work, we present a novel method, based on Bayesian point processes that is interpretable and can account for the effect of study specific characteristics. We apply our
method in a meta-analysis of fMRI studies.
• Mean posterior log-intensities for emotions can be found in Column 1. Column 2 are the corresponding
quantities for executive control. Column 3 is the standardised mean posterior difference
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Methods
Notation
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i
• Foci xi = {xi j }nj=1
from study i = 1, 2, . . . , I
K
• Covariates{zi k }k=1:
a set of K features for study i
Model
• Each xi is the realisation of a Poisson point process Xi defined on the brain B:
Xi ∼ Pois (B, λi )
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• We model study i ’s intensity at location ξ ∈ B as:


∗
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K
X
X
βk (ξ) zik +
βk zik ,
λi (ξ) = exp 
k=K ∗+1
k=0
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where βk (·) are spatially varying coefficients and βk are spatially-constant coefficients.
We model each βk (·) as a Gaussian process
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Inference
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• Inference is done under the Bayesian paradigm
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K∗
{βk (·)}k=0
• Let θ include all model parameters
and {βk }K
k=K ∗+1. Posterior is given by:


Z
Y
ni
I
Y
exp − λi (ξ)dξ
π (θ | data) ∝
λi (xij )π (θ)
B
i =1
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Emotion
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Executive control
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Difference
ROI analysis
• Based
• Analytically intrectable, we use MCMC to draw samples from this posterior
on the properties of the spatial Poisson process, several quantities of interest can
be obtained
• For the whole brain the expected number of foci for emotion studies is 7.14 while for
executive control 11.9
Algorithm details
• For any ROI of interest we can find P (n ≥ 1), the probability of at least one focus:
• Discretise βk (·) on a grid B with V voxels of volume A mm3 and approximate as:
β k = µk +
ROI
1/2
σk Rk γ k ,
where γ k ∼ NV (0, I) and (Rk )i j = exp −ρk d(vi , vj )δk
• Flat priors on µk , σk , ρk ∼ Uni [0.01, 100] and fixed δk = 2
• Parameters jointly updated with Hamiltonian Monte Carlo [Neal, 2010]
• Matrix-vector products handled with Circulant Embedding [Wood & Chan, 1994]
Left amygdala
OFC lateral left
Left thalamus
Left putamen
Cingulate anterior
OFC medial
Emotion Executive
0.228
0.235
0.144
0.104
0.210
0.094
0.022
0.167
0.216
0.143
0.291
0.011
ROI
Right amygdala
OFC lateral right
Right thalamus
Right putamen
Cingulate Posterior
Precuneus
Emotion Executive
0.203
0.158
0.125
0.086
0.105
0.181
0.013
0.184
0.177
0.116
0.092
0.374
Application: fMRI meta-analysis
Discussion
Data
Conclusions
• 1199 fMRI/PET studies of 2 types:
• New approach with good interpretability, that can quantify the effect of covariates locally
or globally throughout the brain
ã 860 emotion studies
ã 339 executive control studies
• Differences between different types of studies directly obtained from posterior. No need
to run 2 different models
• 10775 foci
ã 6459 from emotion
ã 4316 from executive control
• For the real data, very distinct localisation between emotion and executive control
Future work includes:
• Average 16 participants per experiment
Objectives
Emotion
• T scores can be attached to the foci as marks
• Find regions of consistent activations
• Model extension for detection of co-activation patterns
• Localise difference between emotions/executive
control
Implementation
References
• 2 spatially varying coefficients for covariates
zi 1, zi 2 where:
(
1 study i is emotion
zi 1 =
,
0 otherwise
• Carp, J. (2012). NeuroImage, 63(1), 289-300
• Neal, R. (2011). Handbook of Markov Chain Mote Carlo, Chapter 5, 113-162
• Raemaekers, M. et al (2007). NeuroImage, 36(3), 532-542
Executive control
and zi 2 = 1 − zi 1
• HMC: 15,000 iterations, 10,000 burn-in
• Predictive distribution for type classification
• Wager, T. et al (2009). NeuroImage, 45(1, Supplement 1), S210-S221
• Wood, A. and Chan, G. (1994). Journal of computational and Graphical Statistics, 3(4),
409-432