General Linear Model I: principles and fMRI

Introduces the General Linear Model as the workhorse of fMRI analysis: the model, its assumptions, how parameters are estimated, and what leverage and influence mean for design and outliers. Then shows how fMRI predictors are built by convolving stimulus functions with a hemodynamic response function, how a two-level (within-subject, then group) analysis is organized, and which files SPM writes at each stage.

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Take-aways

The GLM finds betas that minimize squared error; independence of errors is the assumption that matters most.
fMRI predictors are stimulus functions convolved with an HRF that peaks at 4 to 6 s and returns to baseline by 20 to 30 s.
Analyses are hierarchical: single-subject betas and contrasts (con images) feed a random-effects group test.

Key terms

General Linear Model
design matrix
beta (parameter estimate)
pseudoinverse
leverage and influence
hemodynamic response function (HRF)
convolution
linear time-invariant system
mass univariate
contrast image

**Micrograph of cortical vasculature.** Lecture 1.8 slides (Wager)

Outline

What the session covers

01Framework: two-level strategy

Within-person questions: is the task effect greater than zero, or does brain activity track performance across trials?
Between-person questions: do task effects differ between groups or correlate with individual differences?
Level 1: fit a GLM in each voxel for each person and save summary statistics (betas, regression slopes).
Level 2: analyze those summary statistics across people (random effects) to determine what is significant at the population level.

02The GLM family and its assumptions

The GLM treats data as a linear combination of predictors plus noise; t-tests, ANOVA, ANCOVA, and OLS regression are all special cases.
Key assumption: errors are independent; violations invalidate the statistics. Normality and equal variance matter less.
The correlated-errors variant assumes a known dependence structure (repeated measures or time series), which is the fMRI situation.
Logistic regression is a GLM with a link function; a generalized inverse handles design matrices that are not full rank.
A decision flowchart maps predictor and outcome types to the classic named tests.

**Canonical hemodynamic response function.** Lecture 1.8 slides (Wager)

**GLM schematic: data equals design matrix times betas.** Lecture 1.8 slides (Wager)

03The model and how it is fit

y = X beta + epsilon: data equal design matrix times parameters plus residuals; predictors are known, parameters are estimated.
Predictors can be categorical (sex) or continuous (performance); the intercept is a column of ones.
Fitting minimizes the sum of squared residuals; beta = pinv(X) times y, geometrically a projection of y onto the space spanned by X.
Crime-rate versus police-expenditure example: a strong association, but beware causal inference when causes like urbanicity are unmodeled.
In MATLAB the whole fit is X = [i x]; b = pinv(X) * y; e = y - X*b.

04Leverage, influence, and power

Leverage is how far an observation's predictor values are from the predictor mean; high-leverage points can strongly affect the fit.
Influence is how much an observation actually changes the betas; it requires both high leverage and a large residual (Cook's distance).
An outlier at the mean of x has zero leverage and does not change the slope.
Design take-home: extreme x values create power; observations at the mean add little, which is why experiments beat observational correlations per observation.
Analysis take-home: check for predictors with uneven leverage and for influential outliers.

**Voxel-wise effects across an axial brain slice.** Lecture 1.8 slides (Wager)

05Building fMRI predictors: the HRF and convolution

Brief neural events produce a delayed, dispersed BOLD response; the HRF reflects blood oxygenation, flow, and volume (Buxton et al. 1998).
Empirical HRF: peak at 4 to 6 s, baseline by 20 to 30 s, sometimes an initial undershoot; similar in V1, A1, S1 but not necessarily elsewhere.
Vasodilation involves PGE2, COX-2, EETs, nitric oxide, and potassium; both glutamatergic and GABAergic transmission can drive it, and neuromodulators (ACh, NA, 5-HT, DA) shape it.
Linear time-invariant assumption: the predictor is the stimulus function (blocks or events) convolved with the HRF.
Each event type gets its own regressor (e.g., famous versus non-famous faces) and betas are compared with contrasts; factorial designs work the same way.
Assumptions to keep in mind: the neural activity function is right, the HRF is right, and the system is LTI.

06The GLM pipeline in SPM

SPM is mass univariate: the model is fit one voxel at a time to produce a statistical image.
Design specification is saved in SPM.mat; estimation writes beta_????.img and ResMS.img.
T-contrasts (c'b) write con_????.img and spmT_????.img; F-contrasts write ess.img and spmF_????.img.
Each participant's score is usually a contrast between conditions (e.g., task minus control) that goes to the group analysis.
Variance in the two-level model comes from unexplained error (E), design-related inefficiency (D), and individual differences (I).

Three axial slices with simulated effect patterns — Lecture 1.8 slides (Wager)

From the instructors' research