Structural Equation Modeling and Its Application to Network Analysis in Functional Brain Imaging

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Structural Equation Modelingand Its Application
to Network Analysis in Functional Brain Imaging

• A. R. McIntosh and F. Gonzalez-Lima
• Human Brain Mapping, 2, 2-22 (1994)
• ????!

2
Structural Equation Modeling

• Objective of SEM Plausibility of the
  hypothesized models in interpreting the
  covariances among variables
• Contribution from Charles E. Spearman and Sewall
  Wright

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Observed Covariance Matrix (Table I)
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Hypothesized Models (Figure 5)
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• Charles E. Spearman Factor Analysis
• Sewall Wright Path Analysis
• Comprehensive SEM
• Path Analysis Factor Analysis
• CFA, PA, are special cases of SEM

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Diagram Symbols

• Latent variables
• Observed variables
• Direct effects (Path)
• Correlations / Covariances

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An Example

• I-P Rel

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Steps in SEM

• Model Specification
• Model Identification
• Model Estimation
• Model Evaluation
• Model Modification / Respecification

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• Logic behind the use of SEM in functional brain
  imaging Brain function is the results of changes
  in the covariances of activity among neural
  elements
• Neuroanatomy is used to define a network and to
  express the interactions among brain regions

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Data Analysis in Neuroscience

• Mean differences approach (changes in regional
  activity) Brain functions are the
  responsibility of specialized areas
• Covariance-based approach Brain function is the
  results of changes in the covariances among
  neural elements

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Mathematical Extension
Equation (3)
Figure 1
Equation (2)
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Terminology

• Functional connections (connectivity)
  Correlations of activity between neural elements
• Effective connectivity the influence or effect
  one neural element has on another
• Functional network the pattern of covariances
  among evoked potential sites or obtained through
  other measures of neural activity

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Terminology continued

• Anatomical model the neuroanatomical
  connections between brain regions
• Functional model the influences of regions
  within the model on each other through the
  anatomical connections

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Software for SEM

• LISREL (Jöreskog Sörbom)
• EQS (Bentler)
• AMOS (Arbuckle)
• MPlus (Muthèn Muthèn)
• CALIS (SAS)

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An Example of Path Analysis
(Figure 2)
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Matrix Representation of Figure 2
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Comments on p. 6

• Path analysis vs. SEM PA can be viewed a
  special case of SEM with all the variables being
  observed variables
• Correlation matrix vs. Covariance matrix
• causal structure (established by neuroanatomy
  or other information)

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Selected Features of SEM

• (a) SEM could include loops
• (b) SEM could have stacked models multi-sample
  SEM
• (c) SEM could take measurement errors into
  account
• (d) SEM could have total effects decomposed into
  direct and indirect effects

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(a) SEM could include Loops
Equations (9) (10)
Figure 2
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(b) SEM could have stacked models

• Multiple group SEM (independent groups)
• Examples
• Clinical population vs. normal population
• Experimental groups vs. control group
• Chi-squares difference test

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(c) SEM could take measurement errors into account
Figure 4 (Multiple measures could be used.)
Equation (12)
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(d) SEM could have total effects decomposed
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Simulations for Incomplete Anatomical Models

• Omitted paths
• Omitted variables
• Cases of model misspecification

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Observed Covariance Matrix (Table I)
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Hypothesized Models (Figure 5)
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Estimates from Simulations (Table II)
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Effects of Model Misspecification

• Model A vs. Model B (Test model 1)
• Misspecification Omitted P54
• Effect Increased P51, MI for P54
• Model A vs. Model C (Test model 2)
• Misspecification Omitted V1
• Effect Increased P54
• Reason Omitted V1 mainly accounts for the
  covariation between V4 V5
• Model A vs. Model D (Test model 3)
• Misspecification Omitted P36
• Effect Magnitude change in P63, MI for P36

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Note that

• Modification indices and Lagrange Multiplier
  tests could be useful in identifying misspecified
  elements
• Suggestions from modification indices should take
  theoretical considerations into account
• The demonstrated simulation results are not
  universally applicable
• Effects of model misspecification depend on the
  nature of the misspecification

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Another Example SEM in Human PET Studies
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Data-Driven vs. Theory-Driven Modeling

• Functional systems involved obvious?
• Strategies for Model Specification (Jöreskog
  Sörbom)
• Strict model confirmation (rare in practice)
• Model generation (with caution)
• - Substantive justification
• - Cross validation
• Model comparison (could be useful)

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A few more words about SEM

• Model Specification
• Model Identification
• Model Estimation
• Distribution
• Large sample theory
• Model Evaluation
• Fit of the hypothesized model to data
• Model Modification / Respecification
• Cross validation

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What is the Benefit ofAnalytic Approaches to
the Study of Neural Interactions in
Neuroscience ?

• Brain function involves the cooperative
  interactions among many neural regions, though a
  particular area may be critical for a certain
  function

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The End

• Thank you.