Transcript Document 7493957

The M3 Toolbox: the Multi-level Mediation/Moderation Framework for
Connectivity Analyses in fMRI Data
324 Schermerhorn Hall
Department of Psychology
1190 Amsterdam Ave.
New York, NY 10027
Matthew Davidson, Lauren Atlas, Martin Lindquist, Niall Bolger & Tor Wager
Departments of Psychology and Statistics, Columbia University, New York
Columbia Psychology SCAN group
http://www.scan.psych.columbia.edu/
Download this poster: http://www.columbia.edu/cu/psychology/tor/
Introduction
Multilevel Mediation/Moderation, cont.
BACKGROUND
• Early imaging analyses focused on identifying regional neuronal correlates of psychological
processes. However, this is an incomplete picture, providing little detail in terms of the
interrelationship of various brain regions. As a result, interest has shifted towards identifying and
describing related regions in terms of their pathways and circuits.
PRIOR WORK
• Existing techniques/software focus on univariate methods (SPM, AFNI, FSL VoxBo and
BrainVoyager)
• Even the tools that are multivariate (ICA variants, PPI, DCM1, SEM, Granger causality models)
typically lack certain key properties:
1. The ability to search for pathways rather than confirm a priori pathways - useful when the
paths are not known
2. Identifying mediating brain regions
3. Adjust for differences in hemodynamic response between brain regions - differing HRFs can
be problematic for between-region correlations2
4. Multilevel modeling - to properly account for intersubject variance3
MODERATION
• Test whether relationship between two variables depends on a third
• In a multilevel analysis, moderators can be:
1. 1st level (within-subjects)
2. 2nd level (between-subjects)
•
•
•
Variable latency model
• Conduct a time-shifted search between data sources
• Assumes HRF shape the same, up to a delay d
• x and m are replaced by f(x, d1) and f(m, d2), where f() is a time-shifting
function implemented by linear interpolation
• Equations become:
1. y = c * f(x, d1) + ey
2. f(m, d2) = a * f(x, d1) + em
3. y = b * f(m, d2) + c' * f(x, d1) + e'y
• d1 and d2, are estimated with a genetic algorithm that maximizes -log(SSET)
Since the path coefficients are computed on l:
• lx = g-1(x, h(B, px))
Then the equations become:
1. g-1(y, h(B, py)) = c * g-1(x, h(B, px)) + ey
2. g-1(m, h(B, pm)) = a * g-1(x, h(B, px)) + em
3. g-1(y, h(B, py)) = b * g-1(m, h(B, pm)) + c' * g-1(x, h(B, px)) + e'y
B consists of two gamma functions such that
• HRFx = h(B, px) = Bp/∑Bp
BOLD Activity
A

B
Group ICA,
tensor ICA
Advantages
Distributed patterns
Y
N
N
N
N
Seed
Analysis
Bivariate interactions
w/ 1 area
Y
N
N
N
N
PPI
Single moderator of
biv connectivity
Y
N
N*
N
N
Granger
causality
DCM
SEM
M3
Bivariate interaction
w/ time lag/diff HRFs
Powerful modeling of
multi-region activity
Exploratory and
confirmatory
Y
N
Y
N
This experiment looked at the relationship between 4 different levels of applied heat and
reported pain. In the mediation diagram, X is the level of heat applied to the subject, Y is the
level of pain reported, and M is any brain voxel mediating the relationship between heat and
pain.
Below are the results of a search across the brain for any voxels mediating the heat-pain
relationship. E.g., you can see that the ACC is a strong mediator. It correlates strongly with
applied heat, and with reported pain (controlling for heat). The product of the two paths is
significant at the group level, and hence, we can infer that the ACC is a mediator of the heatpain relationship.
A
Brain regions mediating a heat stimulus - pain report relationship
B

C
D
D

Time (s)
Fig. 2. BOLD activity deconvolved into an HRF and
neural activity. Mediation in the latent model then
uses the neural activity instead of the BOLD.

Single-trial analysis
• As an alternative to the complex and computationally intensive full
deconvolution or latency models, a single-trial analysis can be used.
• In the single-trial analysis, the response to each trial is fitted with a set of
basis functions, and certain HRF parameters, such as height, delay, width,
and area under the curve (AUC) are estimated.
• Then, instead of using a BOLD signal, the mediation will use the trial-level
parameters. This is illustrated below, in Fig. 3:
Heat-brain path
N = 18
N
Basis set
N
Y
Y
N
N
N
Y
N
N
N
Y
Y
Y
Y
Y
HRF Parameters
Brain-report path
N = 18
Subjs
1,2,3…N
Mediating
brain regions
Controlling for
stim. intensity
Trial-level amplitude estimates
Amplitude
Technique
Search for
Identify
Handle
Non-param
brain regions mediators HRF diffs Multilevel options
Neural activity
C
Time (s)
Latent model
• Deconvolves the hemodynamic response function using a small 2-param
basis set (see Fig. 2)
• Run the mediation/moderation analyses directly on computed neural
activity
• Similar in principle to DCM
• If g() represents the convolution operation and h(B,[1,2…p]T) the HRFgenerating function given p basis parameters and an n (time points) x p
matrix of basis functions B, x is the observed timeseries, lx is the latent
‘metabolic’ signal, and px is the vector of basis parameters, then:
• x = g(lx, h(B, px))
Assumed HRF
(Basis functions)
Real-world Experiment
Noxious
stimulation
(4 levels)
Reported
pain
Multi-level path diagram
Multilevel Mediation/Moderation
MEDIATION
• Simple, three-variable form of SEM extended to the multilevel setting, making it feasible to
treat linkages (i.e., connectivity between regions) as random effects.
• Uses two key concepts:
1. Mediation/moderation in path analysis
2. Mixed-effects (or hierarchical) models
• The M3 analysis merges the two approaches, building on recent developments in multilevel mediation analyses in psychology4
• Mediation provides tests of whether relationship between two variables is explained
(mediated) by a third, thus establishing either a direct or indirect linkage5
• A test for mediation should satisfy the following criteria:
1. X should be related to M (the a pathway in Fig. 1 below)
2. b should be significant after controlling for X
3. The indirect relationship (a*b)should be significant
• This is generally assessed with the Sobel test, or more efficiently, with a bootstrap
test6
Fig. 3. Single-trial analysis. Each trial's response is fitted to a basis set (left), then key HRF parameters are computed (middle), and then the
resulting timeseries of trial parameter estimates (e.g., amplitude - right) are used in the mediation instead of the BOLD signal.
Software
Simulations
SPM TOOLBOX
The M3 toolbox is currently available as a toolbox for SPM5, downloadable from
http://www.columbia.edu/cu/psychology/tor/software.htm. It piggybacks off of the SPM job
manager, and thus, presents no new learning curve to those familiar with SPM5. The M3
toolbox supports both single- and multi-level analyses, shifting and latent correlations, and
contains built-in checks to prevent insertion of bad data.
Summary
Bootstrapping
•
Three linear equations:
1. y = cx + ey
2. m = ax + em
3. y = bm + c'x + e'y
• If the relationship between x and y can be
accounted for by an indirect relationship
through m as described by slope coefficients a
and b, then c - c’(the product ab) will be
statistically different from zero.
MULTILEVEL
• Equations:
1. ci = c + u0i
2. ai = a + u1i
3. bi = b + u2i
4. c'i = c' + u3i
• The u's are between-subjects error terms
• Subject-level path coefficients are random
effects, enabling population inference
Power and False Positive Rate as a function of the number of subjects. N =
10, 20, or 40.
Sign perm
Bootstrapping
Sign perm
Power and FPR by permutation option (Bootstrapping or Sign Permutation)
and number of iterations.1k = 1000 iterations, 5k = 5000 iterations, 10k =
10000 iterations.
M
X
References
b
a
Total: c
Direct: c'
•M3 provides tests of population inference on within-subject pathways and their moderators.
•Tests are efficient and valid for unbalanced designs because the method is explicitly
designed for multilevel connectivity.
•The M3 framework provides the capability for hybrid exploratory and confirmatory
approaches to identifying functional pathways when the exact voxels that comprise such a
pathway are unknown.
•Provisions are made for specific characteristics of fMRI data, such as HRF and inter-regional
latency differences.
Y
Mo (Moderator)
Fig.1. The basic unit of analysis in
the mediation / moderation
framework is a 3-variable system.
Bootstrapping
Power and FPR by shift option. N = 20. Norm = normal, +/-1,+/- 2, and +/-3
are max shift amounts in the latency model.
Sign perm
Bootstrapping
Sign perm
Power and FPR by permutation option (Bootstrapping or Sign Permutation)
and number of subjects. N = 10, 20, 40, 60.
1. Friston, K.J., L. Harrison, and W. Penny, Dynamic causal modelling. Neuroimage, 2003. 19(4):
p. 1273-302.
2. Gitelman, D.R., et al., Modeling regional and psychophysiologic interactions in fMRI: the
importance of hemodynamic deconvolution. Neuroimage, 2003. 19(1): p. 200-7.
3. Raudenbush, S.W. and A.S. Bryk, Hierarchical Linear Models: Applications and Data Analysis
Second ed. Methods. 2002, Newbury Park, CA: Sage.
4. Kenny, D.A., J.D. Korchmaros, and N. Bolger, Lower level mediation in multilevel models.
Psychol Methods, 2003. 8(2): p. 115-28.
5. Baron, R.M. and D.A. Kenny, The moderator-mediator variable distinction in social
psychological research: conceptual, strategic, and statistical considerations. J Pers Soc
Psychol, 1986. 51(6): p. 1173-82.
6. Shrout, P.E. and N. Bolger, Mediation in experimental and nonexperimental studies: new
procedures and recommendations. Psychol Methods, 2002. 7(4): p. 422-45.