GLM Modelling and Statistics Flashcards
general linear model
y = X * beta + error
error ~ N(0, varianz * In) (independent and identically distributed)
- y and error: 1 col, n rows = number of scans/time points
- X: n col = number of regressors, n rows = scans / time points
- beta: 1 col, n rows = number of regressors
OLS
beta^ = (X^T * X)^-1 * X^T * y
- Estimated optimal regression coefficients are equal to the covariance of the design matrix with the measured signal y divided by the covariance of the design matrix with itself
t-contrast
- [number of regressors] x 1 contrast vector c
- possible research question: Is there a difference between the 1st and 2nd regressor (e.g. condition)?
- directional
F-contrast
- [number of regressors] x [number of ORs + 1]
- possible research question: Is there an effect of the 1st or 2nd regressor
What are the challenges for GLM when it comes to fMRI data?
- low-frequency drifts
- hemodynamic response
- residual movement effects
- serial auto-correlations
- multiple comparisons
GLM problem - low-frequency drifts
- there are slow trends in the measured fMRI signal that can make model estimation less accurate
solution:
- build a discrete cosine transform (DCT) set and regress it out of the signal (“temporal filtering”)
GLM problem - hemodynamic response
- fMRI responses do not have rectangular shape
solution:
- stimulus functions are convolved with the hemodynamic response function (HRF) to give rise to a more realistic prediction of measured fMRI signals
GLM problem - residual movement effects
- despite spatial realignment, there are still movement artifacts in the signal, esp. at tissue boundaries
solution:
- include realignment parameters as regressors in the design matrix of the GLM.
GLM problem - serial auto-correlations
- consecutive fMRI scans are not statistically independent, i.e. fMRI time series have serial correlation
solution
- auto regressive model of order one [AR(1)] is fit to account for serial correlations
GLM problem - multiple comparisons
- when running many parallel statistical tests, there will be a large number of spurious findings
solution:
- instead of controlling the false positive rate per test, control the family wise errorrate (FWE; probability of at least 1 false positive in n tests)
- for independent tests: FWE = 1 - (1 - alpha)^n
- common corrections are too conservative
- Gaussian field theory: do not correct alpha for the number of voxels but the number of resels
GLM - parametric modulators
- sometimes, there is a trial-wise variable possibly influencing the BOLD signal (e.g. stimulus intensity)
solution:
- add a parametric modulator regressor describing possible trial-wise effects.
second-level GLM
- usually, we want to provide evidence for activation differences at the group level (i.e. multi subject analyses)
solution:
- fit a second level model over first level contrast maps to obtain group level statistics.
