... Averaging. Larry R. Price, PhD1, Angela R. Laird, PhD2

1
Neuroimaging Network Analysis using Bayesian
Model Averaging Larry R. Price, PhD1, Angela R.
Laird, PhD2, and Peter T. Fox, MD2 Texas State
University, San Marcos, Texas1 The University of
Texas Health Science Center at San Antonio
Research Imaging Center2

INTRODUCTION The aim of this investigation was to
present a method for graphical model selection of
the functional connectivity among regions of
interest in the human brain where little or no
previous theory exists. We propose and illustrate
a fully Bayesian approach for deriving network
models among regions of interest that allows
researchers to address the issue of uncertainty
that is inherent in a single-best model
strategy. We use the Bayesian Model Averaging and
Occams Window approach of Madigan and Raftery
(1994) and Bayesian Model Averaging (Madigan, et
al., 1996), to illustrate our method using data
acquired by Positron Emission Tomography (PET) on
a group of normal controls and a group exhibiting
neurogenic stuttering under varying conditions of
verbal activity. Our approach enables scientists
to examine different patterns of interregional
causal effects due to an intervening condition
within the network using Bayesian Structural
Equation Modeling (BSEM).
Examination of the autoregressive properties of
the three repeated measurements proceeded for
each ROI by fitting general Markov and Stationary
Markov (lag-2 and lag-3 time effects) models as a
function of each successive measurement within
each respective ROI, plus a random error
component (Arbuckle, 1996, Wothke, 2000). Based
on the results of these tests, aggregated
composite variables were created (Bagozzi
Edwards, 1998) using the three correlated
measurements obtained for each ROI using a
weighting scheme that incorporated the lag-1 time
effect. Eight regions of interest (ROIs) were
extracted from this analysis (Figure 1) using
Activation Likelihood Meta-Analysis corresponding
to areas of high brain activation during speech
production left and right primary motor cortex
(LM1 and RM1), left and right rolandic operculum
(LRO and RRO), left and right auditory cortex
(LAUD and RAUD), supplementary motor area (SMA),
and cingulate cortex (CING).
RESULTS Implementation of the Occams Window
algorithm proceeded using the Bayesian modeling
facility within the Analysis of Moment
Structures, version 7.0 (AMOS) computer program,
and the WinBUGS computer program (Spiegelhalter,
2003). Bayesian estimation proceeded using the
MCMC methodology (i.e., Metropolis-Hastings
algorithm) and Gibbs sampling (Geman Geman,
1984). The estimation process proceeded by using
centered parameterization allowing the composite
variables to have free intercepts and variances
and also, diffuse (non-informative) uniform
priors for the vector of parameters, including
the variance components. We followed guidelines
offered by Raftery and Lewis (1992) in
determining the selection of the number of MCMC
burn-in iterations with respect to establishing
the convergence criteria for the joint posterior
distribution of the model parameters. Table2
provides a summary of the models selected by
Occam's Window by group and experimental
condition. In all four models, the overall
chi-square fit statistic was not statistically
significant indicating an acceptable overall fit
of the model to the data. Application of the
Occams Window algorithm revealed that in all
four experimental conditions, model 1 was
observed to be optimum according to the Bayes
factor (BICL), Bayes Information Criterion
(BIC0), and probability values.
Data Acquisition and Experimental Design Data
were collected on 10 male subjects diagnosed with
chronic developmental stuttering (mean age 32.2
years age range 21-46 years) and 10 male
healthy control subjects who were normally fluent
(mean age 32.3 years age range 22-55 years).
All subjects gave informed consent according to
the approved procedures of the Institutional
Review Boards (IRB) of the University of Texas
Health Science Center at San Antonio and the
University of California, Santa Barbara. All
subjects were scanned while they performed three
tasks eyes-closed rest (Rest), unaccompanied
overt paragraph reading of a text passage (Solo),
and overt paragraph reading while accompanied by
a fluent audio recording of the same paragraph
(Chorus). Paragraphs for both the Solo and Chorus
conditions were presented on a video display
suspended above the subjects, approximately 14
inches from the eyes. In the Chorus condition, a
recording of a non-stutterer reading the same
passage was also presented via an earphone
inserted into the left ear. Each 40-second
condition was imaged three times in all subjects
each subject underwent nine PET scans in a
counterbalanced order.
Table 2
Figure 3 Posterior model 1 for control group
under chorus-rest condition
Figure 2 Posterior model 1 for stuttering group
under chorus-rest condition
METHOD A total of 90 observations across 10
subjects (3 trials/repeated measurements X 3
imaging conditions X 10 subjects) in 8 ROIs
constituted the raw data matrix for both the
clinical (N10) and control (N10) groups. Prior
to deriving the composite ROI variables, we
examined the data according to the following
statistical criteria (a) autoregressive
properties of the three trials/repeated
measurements for each ROI across each scan
condition, (b) the interaction effect between
scan condition and trial within each ROI, and (c)
the measurement and structural properties of each
ROI by specifying eight separate correlated-error
measurement models.
REFERENCES Arbuckle, J., Wothke, W. (1996).
Full information estimation in the presence of
incomplete data. In Advanced structural equation
modeling, G.A. Marcoulides and R.E. Schumacker,
eds. Mahwah, New Jersey Lawrence Erlbaum
Associates. Bagozzi, R., Edwards, J. (1998). A
general approach for representing constructs in
organization research. Organizational Research
Methods, 1 (1), 45-87. Fox P.T., Ingham R.J.,
Ingham J.C., Hirsch T.B., Downs J.H., Martin C.,
Jerabek P., Glass T., Lancaster J.L. (1996). A
PET study of the neural systems of stuttering.
Nature 382, 158 -161. et al. (1996). Geman, S.
and Geman, D. (1984). Stcohastic relaxation,
Gibbs distributions, and the Bayesian restoration
of images. IEEE Transactions on Pattern Analysis
and Machine Intelligence, 6, 721-741. Hoyle, R.,
Kenny, D. (1999). Sample size, reliability, and
tests of statistical mediation. In Statistical
strategies for small sample research, Rick Hoyle,
ed.,Thousand Oaks, California Sage. Madigan,
E.J. Raftery, A.E. (1994). Model selection and
accounting for model uncertainty in graphical
models using Occams window. Journal of the
American Statistical Association, 89 1535
1546. Raftery, A.E., Lewis, S.M. (1992a). How
many iterations in the Gibbs sampler? In
Bayesian statistics 4 (eds J.M. Bernardo, J.O.
Berger, A.P. Dawid, A.F.M. Smith), pp. 765 -
776. Oxford Oxford University Press. Spiegelhalte
r, D.J., Thomas, A., Best, N. (2003). WinBUGS
Windows-based Bayesian inference Using Gibbs
Sampling, version 1.4. Cambridge Medical
research Council Biostatistics Unit.
Figure 1
Figure 5 Posterior model 1 for control group
under solo-rest condition
Figure 4 Posterior model 1 for stuttering group
under solo-rest condition