34. Structural Equation and Path Models

Structural Equation and Path Models¶

Overview¶

Structural equation modeling is a family of techniques for modeling relationships — more formally, the variance–covariance structure — among a set of variables. Where the GLM predicts one outcome from a set of predictors, an SEM specifies a system of equations: multiple predictors, multiple outcomes, and possibly latent factors that are not observed directly but are expressed in the measured variables. The emphasis is on comparing models with different structures and finding one that gives a parsimonious, low-error account of the observed correlations. When all variables are observed (no latent factors), the SEM is called path analysis, and the two most widely used inferences in this framework are tests of mediation — whether the relationship between two variables is transmitted through a third — and moderation — whether one variable changes the strength of the relationship between two others.

Applied to fMRI, SEM is a classic tool for effective connectivity: a set of brain regions is chosen a priori, along with a hypothesized set of directed connections among them. The strength of each connection is a path coefficient — the expected change in activity in one region per unit change in a region that influences it. Writing $y$ for the vector of regional activities at one time point, the path model is

Rearranging so $y$ appears on only one side, $y = (I - B)^{-1}\zeta$, implies a model covariance matrix

Estimation minimizes the discrepancy between this model-implied covariance and the sample covariance, typically by maximum likelihood. Model fit is assessed with a $\chi^2$ test comparing the two covariance matrices — and, importantly, the logic is reversed from ordinary hypothesis testing: a nonsignificant result means the model is adequate (we cannot reject it), not that it is true or best. Specific edges are tested by comparing nested models with and without the connection using likelihood ratio tests.

Standard SEM has known limitations for fMRI time series: it assumes observations are independent across time (ignoring temporal autocorrelation, which distorts standard errors), it handles dynamics inflexibly, and it does not incorporate experimental inputs. Extensions such as the unified SEM, which couples a path model with a vector autoregressive model, and the extended unified SEM, which adds experimental inputs, address these gaps. Group analysis is another subtlety: naively concatenating subjects can yield results inconsistent with single-subject analyses, motivating stacked or multilevel SEMs — or simply t-tests across subject-level path estimates, the familiar summary-statistics approach.

Mediation analysis asks whether the effect of an exposure $X$ on an outcome $Y$ is at least partially transmitted through an intervening mediator $M$. The full model is a pair of regressions,

and the reduced model, without the mediator, is $Y_i = i_3 + c\,X_i + e_i$, where $i_3$ is its intercept and $c$ the total effect. Path $a$ is the effect of exposure on mediator, path $b$ the effect of mediator on outcome controlling for exposure, and $c'$ the direct effect — what remains of the $X \to Y$ relationship once $M$ is accounted for. The total effect decomposes exactly:

so the indirect (mediated) effect is the product $a \times b$, and testing mediation means testing $H_0\!: ab = 0$ (equivalently, $c - c' = 0$). The classical Sobel test uses a large-sample normal approximation for the standard error of $\widehat{ab}$, but it is overconservative; modern practice uses bootstrap tests — resampling cases, re-estimating $ab$ thousands of times, and forming confidence intervals and P values from the bootstrap distribution.

Mediation is a natural fit for fMRI because it links experiment, brain, and behavior in one model. A canonical brain-as-mediator design: participants are randomized to a stressful or innocuous challenge ($X$), task-evoked activity in an anterior cingulate cortex (ACC) region is the mediator ($M$), and stressor-evoked heart rate increase is the outcome ($Y$). Mediation holds when stress shifts ACC activity (path $a$) and ACC activity tracks heart rate within groups (path $b$) strongly enough that the direct effect $c'$ shrinks relative to $c$. Variables can be organized in other ways too — a brain region mediating the effect of another region on behavior, or one region mediating the link between two other regions. With trial-by-trial data, multilevel mediation estimates paths within person and lets path strengths vary (and be moderated) across people, and mediation effect parametric mapping searches the whole brain for voxels that act as mediators, analogous to a seed analysis.

These models are transparent and flexible, but their assumptions deserve respect. The causal reading of $a \times b$ requires, among other things, no unmodeled confounding of the mediator–outcome relationship: even when $X$ is randomized, $M$ is merely observed, so any third variable that drives both $M$ and $Y$ (arousal, attention, global signal) can manufacture a spurious “indirect effect.” These assumptions are easy to state but hard to verify and often violated, which is why mediation in neuroimaging is best framed as pathway discovery and description rather than definitive causal inference — you will create exactly this failure mode in the lab.

Finally, moderation asks a different question: does the $X$–$Y$ relationship change with the level of a third variable $M$? It is tested with an interaction term in a standard regression, $Y_i = b_0 + b_1 X_i + b_2 M_i + b_3 (X_i \times M_i) + e_i$, where $b_0$ is the intercept, $b_1$ and $b_2$ the main effects of exposure and moderator, and $b_3$ the interaction coefficient — rejecting $H_0\!: b_3 = 0$ establishes moderation. The time-series version is the widely used psychophysiological interaction (PPI) analysis: a seed region’s time course, a task variable, and their interaction enter a first-level GLM at every voxel, testing where connectivity with the seed depends on task context. Implementations differ on whether to deconvolve the HRF to form the interaction at the “neural” level (SPM) or interact the convolved signals directly (FSL) — deconvolution is only exact when the HRF is known and the system is linear — and generalized PPI (gPPI) models all task conditions with separate interaction terms for greater power in complex designs.

Hands-on tutorial¶

In this tutorial you will simulate a brain-as-mediator dataset ($X \to M \to Y$ with a direct path), estimate all four paths, and bootstrap the indirect effect — the core mediation workflow. The MATLAB code uses mediation.m from the CANlab Mediation Toolbox; the Python code builds the same analysis from regressions so you can see every moving part.

Step 1 — Simulate mediation data and estimate the paths. We generate $n = 200$ observations with true paths $a = 0.6$, $b = 0.5$, $c' = 0.2$, then recover them: path $a$ from the regression of $M$ on $X$, paths $b$ and $c'$ from the regression of $Y$ on $X$ and $M$, and the total effect $c$ from the reduced model.

% Requires CanlabCore + MediationToolbox on your MATLAB path
% Adapted from CANlab mediation tutorials (github.com/canlab/MediationToolbox)
rng(42);                               % fix the random seed for reproducibility
n = 200;                               % n = participants (one observation each)
X = randn(n, 1);                       % exposure (e.g., stressor intensity)
M = 0.6 * X + randn(n, 1);             % mediator (e.g., ACC activity), true a = 0.6
Y = 0.2 * X + 0.5 * M + randn(n, 1);   % outcome (e.g., heart rate), b = 0.5, c' = 0.2

% Estimate paths; 'boot' = bootstrap the indirect effect for inference
% (10,000 samples; use at least ~5,000 for stable tails)
[paths, stats] = mediation(X, Y, M, 'boot', 'verbose', 'bootsamples', 10000, ...
    'names', {'Stressor' 'Heart rate' 'ACC'});

% Columns of paths: [a  b  c'  c  a*b]
mediation_path_diagram(stats);         % path diagram with coefficients and stars

import numpy as np

rng = np.random.default_rng(42)                 # fix the random seed for reproducibility
n = 200                                         # n = participants (one observation each)
X = rng.standard_normal(n)                      # exposure (e.g., stressor intensity)
M = 0.6 * X + rng.standard_normal(n)            # mediator (e.g., ACC), true a = 0.6
Y = 0.2 * X + 0.5 * M + rng.standard_normal(n)  # outcome, true b = 0.5, c' = 0.2

def ols(y, *preds):                             # OLS betas, intercept first
    D = np.column_stack([np.ones(len(y)), *preds])
    return np.linalg.lstsq(D, y, rcond=None)[0]

a      = ols(M, X)[1]                           # Path a:   M ~ X
cp, b  = ols(Y, X, M)[1:3]                      # Paths c', b:  Y ~ X + M
c      = ols(Y, X)[1]                           # Path c:   total effect, Y ~ X

print(f"a={a:.3f}  b={b:.3f}  c'={cp:.3f}  c={c:.3f}")
print(f"indirect a*b={a * b:.3f}  check c - c'={c - cp:.3f}")   # identical

Example output:

a=0.519  b=0.531  c'=0.130  c=0.406
indirect a*b=0.275  check c - c'=0.275

The estimates land near the true values ($a = 0.6$, $b = 0.5$, $c' = 0.2$), and $\widehat{c} - \widehat{c'} = \widehat{a}\widehat{b}$ exactly — the decomposition is an algebraic identity for OLS.

Step 2 — Bootstrap inference for the indirect effect. The sampling distribution of $\widehat{ab}$ is skewed, so we resample cases with replacement, re-estimate $a \times b$ each time, and read the 95% confidence interval off the bootstrap distribution. mediation.m does this automatically with the 'boot' option; in Python we write the loop ourselves.

% mediation.m already bootstrapped a*b ('boot', 10,000 samples).
% stats fields hold [a  b  c'  c  a*b] estimates and bootstrap P values:
fprintf('a*b = %3.3f, bootstrap p = %3.4f\n', stats.mean(5), stats.p(5));

% The printed table also reports the Sobel-style parametric test --
% compare: the bootstrap test is typically less conservative.

n_boot = 5000                                   # bootstrap samples; ~5,000+ for stable CI tails
boot_ab = np.empty(n_boot)
for i in range(n_boot):
    idx = rng.integers(0, n, n)                 # resample cases with replacement
    a_i = ols(M[idx], X[idx])[1]
    b_i = ols(Y[idx], X[idx], M[idx])[2]
    boot_ab[i] = a_i * b_i

lo, hi = np.percentile(boot_ab, [2.5, 97.5])
p_boot = 2 * min((boot_ab <= 0).mean(), (boot_ab >= 0).mean())
print(f"a*b = {a * b:.3f}, 95% bootstrap CI [{lo:.3f}, {hi:.3f}], p = {p_boot:.4f}")

Example output:

a*b = 0.275, 95% bootstrap CI [0.176, 0.386], p = 0.0000

(None of the 5,000 bootstrap samples crossed zero, so the two-sided P value prints as 0 — report it as $p < 0.001$.)

The interval should exclude zero decisively — the data were built with real mediation. The full labs push further: they show how an unmodeled confounder of the $M$–$Y$ relationship produces a significant “indirect effect” when the true $b$ path is zero (and how adjusting for the confounder repairs it), and they fit the three-ROI path model of Figure 34.1 by regression equations, testing an individual edge by model comparison.

Thought questions¶

1. In a study where the exposure $X$ is randomized but the brain mediator $M$ is only measured, which of the paths $a$, $b$, and $c'$ inherit a causal interpretation from the randomization, and which do not? What kinds of variables could confound the $M$–$Y$ relationship in a typical fMRI experiment, and what would each do to $\widehat{ab}$?

2. Two SEMs with reversed arrows (ROI 1 → ROI 2 versus ROI 2 → ROI 1) can imply exactly the same covariance matrix and therefore fit the data equally well. What does this “equivalent models” problem mean for claims of effective connectivity based on SEM fit, and what could you add to the study design to break the symmetry?

3. Suppose you find significant $a$ and $b$ paths but the indirect effect $ab$ is small relative to the total effect $c$, as in the right panel of Figure 34.3. A colleague argues the brain region is “clearly involved” and should be called a mediator anyway. How would you adjudicate, and what additional evidence (statistical or experimental) would strengthen or weaken the mediation claim?

4. Multilevel mediation estimates paths from trial-by-trial data within each person, while single-level mediation uses one observation per person. What confounds does each approach protect against, and which is vulnerable to individual-difference confounds like age or head motion? When would the two give different answers?

5. SPM’s PPI deconvolves the seed time course to form the interaction at the “neural” level, while FSL interacts the HRF-convolved signals directly. Lay out the assumptions each approach makes, and explain why the choice matters more for event-related than for blocked designs.

Quiz yourself¶

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