Latent model

1
The M3 Toolbox the Multi-level
Mediation/Moderation Framework for Connectivity
Analyses in fMRI Data Matthew Davidson, Lauren
Atlas, Martin Lindquist, Niall Bolger Tor
WagerDepartments of Psychology and Statistics,
Columbia University, New York
324 Schermerhorn Hall Department of
Psychology 1190 Amsterdam Ave. New York, NY 10027
Download this poster http//www.columbia.edu/cu/p
sychology/tor/
Introduction
Multilevel Mediation/Moderation, cont.
Real-world Experiment

• Since the path coefficients are computed on l
• lx g-1(x, h(B, px))
• Then the equations become
• g-1(y, h(B, py)) c g-1(x, h(B, px)) ey
• g-1(m, h(B, pm)) a g-1(x, h(B, px)) em
• 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

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 heat-pain
relationship.

• MODERATION
• Test whether relationship between two variables
  depends on a third
• In a multilevel analysis, moderators can be
• 1st level (within-subjects)
• 2nd level (between-subjects)

• 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
• The ability to search for pathways rather than
  confirm a priori pathways - useful when the paths
  are not known
• Identifying mediating brain regions
• Adjust for differences in hemodynamic response
  between brain regions - differing HRFs can be
  problematic for between-region correlations2
• Multilevel modeling - to properly account for
  intersubject variance3

• 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
• y c f(x, d1) ey
• f(m, d2) a f(x, d1) em
• y b f(m, d2) c' f(x, d1) e'y
• d1 and d2, are estimated with a genetic algorithm
  that maximizes -log(SSET)

• 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,1,2pT) the HRF-generating 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))

Technique Advantages Search for brain regions Identify mediators Handle HRF diffs Multilevel Non-param options
Group ICA, tensor ICA 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 Bivariate interaction w/ time lag/diff HRFs Y N Y N N
DCM Powerful modeling of multi-region activity N Y Y N N
SEM Powerful modeling of multi-region activity N Y N N N
M3 Exploratory and confirmatory Y Y Y Y Y

• 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

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
• Mediation/moderation in path analysis
• Mixed-effects (or hierarchical) models
• The M3 analysis merges the two approaches,
  building on recent developments in multi-level
  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
• X should be related to M (the a pathway in Fig. 1
  below)
• b should be significant after controlling for X
• The indirect relationship (ab)should be
  significant
• This is generally assessed with the Sobel test,
  or more efficiently, with a bootstrap test6

Software
Simulations
SPM TOOLBOX The M3 toolbox is currently
available as a toolbox for SPM5, downloadable
from http//www.columbia.edu/cu/psychology/tor/sof
tware.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

• 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.

• Three linear equations
• y cx ey
• m ax em
• 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
• ci c u0i
• ai a u1i
• bi b u2i
• c'i c' u3i
• The u's are between-subjects error terms
• Subject-level path coefficients are random
  effects, enabling population inference

References
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.