Biometrics 74, 342–353 DOI: 10.1111/biom.12713
Spatial Bayesian Latent Factor Regression Modeling of
Coordinate-Based Meta-Analysis Data
Silvia Montagna ,
Lisa Feldman Barrett,
Timothy D. Johnson,
and Thomas E. Nichols
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, U.K.
Department of Psychology and Neuroscience, University of Colorado at Boulder, Boulder, Colorado 80309, U.S.A.
Department of Psychology, Northeastern University, Boston, Massachusetts 02115, U.S.A.
Biostatistics Department, University of Michigan, Ann Arbor, Michigan 48109, U.S.A.
Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K.
Summary. Now over 20 years old, functional MRI (fMRI) has a large and growing literature that is best synthesised with
meta-analytic tools. As most authors do not share image data, only the peak activation coordinates (foci) reported in the
article are available for Coordinate-Based Meta-Analysis (CBMA). Neuroimaging meta-analysis is used to (i) identify areas of
consistent activation; and (ii) build a predictive model of task type or cognitive process for new studies (reverse inference). To
simultaneously address these aims, we propose a Bayesian point process hierarchical model for CBMA. We model the foci from
each study as a doubly stochastic Poisson process, where the study-speciﬁc log intensity function is characterized as a linear
combination of a high-dimensional basis set. A sparse representation of the intensities is guaranteed through latent factor
modeling of the basis coeﬃcients. Within our framework, it is also possible to account for the eﬀect of study-level covariates
(meta-regression), signiﬁcantly expanding the capabilities of the current neuroimaging meta-analysis methods available. We
apply our methodology to synthetic data and neuroimaging meta-analysis datasets.
Key words: Bayesian modeling; Factor analysis; Functional principal component analysis; Meta-analysis; Reverse
inference; Spatial point pattern data.
Functional magnetic resonance imaging (fMRI) has become
an essential, non-invasive, tool for learning patterns of acti-
vation in the working human brain (e.g., Pekka, 2006; Wager
et al., 2015). Whenever a brain region is engaged in a par-
ticular task, there is an increased demand for oxygen in that
region which is met by a localised increase in blood ﬂow. The
MRI scanner captures such changes in local oxygenation via
a mechanism called the Blood Oxygenation Level-Dependent
(BOLD) eﬀect; see, for example, Brown et al. (2007) for a
brief introduction on fMRI. The great popularity that fMRI
has achieved in recent years is supported by various software
packages that implement computationally eﬃcient analysis
through a mass univariate approach (MUA). Speciﬁcally,
MUA consists of ﬁtting a general linear regression model at
each voxel independently of every other voxel, thus produc-
ing images of parameter estimates and test statistics. These
images are then thresholded to identify signiﬁcant voxels or
clusters of voxels, and signiﬁcance is typically determined via
random ﬁeld theory (Worsley et al., 1996) or permutation
methods (Nichols and Holmes, 2001). Despite its simplicity,
the MUA lacks an explicit spatial model. Even though the
activation of nearby voxels is correlated, estimation with the
MUA ignores the spatial correlation; crucially inference later
accounts for it when random ﬁeld theory or permutation pro-
cedures deﬁne a threshold for signiﬁcant activation.
The relatively high cost of MRI scanner time, however, pose
some limitations to single fMRI studies. The main limita-
tion is the small number of subjects that can be recruited
for the study, often fewer than 20 (Carp, 2012). As a result,
most fMRI studies suﬀer from inﬂated type II errors (i.e.,
low power) and poor reproducibility (Thirion et al., 2007).
To overcome these limitations there has been an increasing
interest in the meta-analysis of neuroimaging studies. By com-
bining the results of independently conducted studies, meta-
analysis increases power and can be used to identify areas of
consistent activation while discounting chance ﬁndings.
In addition to the identiﬁcation of areas of consistent
activation (a.k.a. forward inference), there has been intense
interest in the development of meta-analytic methods to
implement proper reverse inference (Yarkoni et al., 2011).
Reverse inference refers to inferring which cognitive process
or task generated an observed activation in a certain brain
region. Suppose that researchers develop a task to probe cog-
nitive process A and ﬁnd that brain area X is activated.
A common but misguided practice in neuroimaging is to
conclude that activation of brain region X is evidence that
cognitive process A is engaged. However, this logic is wrong
and the resulting inference is faulty. In fact, a single region
may be activated by a range of diﬀerent tasks (Yeo et al.,
2017, The International Biometric Society