GLM modelling / Statistics Flashcards
What is the objective of ordinary least squares?
Minimize the sume of squared errors
How is OLS achieved?
OLS = ordinary least sqaures
Because the residual vector and the vector describing the modelled signal must be orthogonal to each other geometrically, we can state, that the transpostion of the residual and each column (=regressor; p) of X must be 0.
-> as the residual is the same as the differense between the actual and the modelled signal, we can insert this term and then perform some calculations on this term.
-> the result is, that the ß (regressor coefficient for each) regressor (p) can be calculated by multiplying the inverse transposition of X and why with the transposition of X and X
Mathematically the last part is equivalent to dividing the covariance of X and Y with the covaraince of X and X!
What are two typical contrast typed in GLM modelling?
t-contrast and F-contrast
Name 5 problems that can occur in statistical analysis with fMRI data.
- Low frequency drifts
- Model is not convolved with HRF
- Residual movement effects
- Serial autocorrelation
- Multiple comparisions
What are the steps of statistical analysis?
- model specification
- model estimation
- statistical inference
What is a stimulus function?
It is a sign function that returns 1 when a the subject was in a certain experimental conditiom, 0 in other conditions
What would be a naive (heuristic) model?
measured signal = variation at condition 1 * weight + variation at condition 2 * weight + baseline signal + noise
How does a naive parametre estimation work?
It will be comparison between measured and modelled signal => search for best-fit parameters
How to assess statistical inference
in a naive way, of course
subtract condition 2 from condition 1
What are the ideas behind multiple linear regression models?
- generalize two-condition model to multiple linear regression with arbitrary number of predictors/regressors
- make some assumptions about errors/noise
What does GLM do?
GLM decomposes BOLD signal measured from a voxel to
- variation coming from experimental conditions (controlled factors) and potential confounds, weighted by a vector, and
- noise (cannot be decomposed)
What is the vector of residuals?
Explain in a geometrical sense
Vector of residuals is the shortest distance between hyperplane spanned by design matrix and actually measured signal (modelled signal and residual errors are orthogonal)
What is the low-frequency drift problem and how to tackle it?
Problem: Slow trends in the measured fMRI signal undermining the accuracy of model estimation
Solution: A discrete cosine transform set regressed out of the signal (temporal filtering)
What is the hemodynamic response problem and how to tackle it?
Problem: fMRI responses do not have a rectangular shape
Solution: Stimulus functions are convolved with the hemodynamic response function to give rise to a more realistic prediction of measure fMRI signal (stimulus function * HRF = actual regressor)
What is the residual movement effects problem and how to tackle it?
Problem: Despite spatial realignment, there are still movement artefacts in the signal, esp. at tissue boundaries
Solution: We include realignment parameters as regressors in the design matrix of the GLM
What is the serial auto-correlations problem and how to tackle it?
Problem: Consecutive fMRI scans are not statistically independent (fMRI has serial correlation due to slow hemodynamic response and biological factors)
Solution: An auto-regressive model of order one is fit to account for serial correlations (an independent and identically distributed noise model is replaced by an enhanced noise model)
What is the multiple comparisons problem and how to tackle it?
Problem: When running many parallel statistical tests, there will be a large number of spurious findings (with more tests, the probability of obtaining of positive results decreases)
Solution: Instead of controlling the false-positive rate per test, control the family-wise error (FWE) rate
Gaussian random field theory (RFT): model of the joint distribution of noise across voxels to quantify voxel dependencies -> do not correct for the number of voxels, but for the number of resolution elements (resels < voxels due to smoothing)
(conservative solutions: Bonferroni correction, Sidak correction)
What are parametric modulators?
Problem: Sometimes, a trial-wise variable possibly influences the BOLD signal (e.g., stimulus intensity).
Solution: Add a parametric modulator regressor describing possible trial-wise effects.
What is the second-level GLM? How is it different from the first-level GLM?
Problem: Usually, we want to provide evidence for activation differences at the group level (i.e. multi-subject analyses).
Solution: Fit a second-level (across subjects) model over first-level (single subject) contrast maps to obtain group-level statistics.
First-level statistics: single-subject inference, we test whether the mean (difference) across images is different from zero
Second level statistics: population inference, we test whether the mean across subjects is different from zero
What GLM is usually used to model and analyze data? What alternatives are there?
Usually, we use condition-wise GLM. But we can also use trial-wise GLM or multi-voxel pattern analysis
Trial-wise GLM:
- extends the analysis to the level of individual trials within each condition
- provides a more fine-grained analysis, allowing for the investigation of trial-specific effects or variability within a condition
MVPA:
- focuses on the analysis of patterns of activity across multiple voxels rather than relying solely on average activation levels
