Statistical Analysis of fMRI Data: The General Linear Model

Question 1

What is the design matrix?

Answer 1

It is a matrix that contains variation over time of all independent variables

Question 2

What are regressors of interest?

Answer 2

They are the values in the design matrix associated with independent variables that the researcher is interested in. Each regressor of interest is usually associated with one experimental condition

Question 3

What are covariates (nuisance regressors)?

Answer 3

They are other variables that the researcher is not interested in, but that still are expected to predict variations of the signal

Question 4

Are regressors paired exactly with the occurrence of the independent variable?

Answer 4

No, they are convoluted to account for the distortion in the signal caused by the Haemodynamic Response Function

Question 5

When are nuisance regressors actually useful? Can you make some examples?

Answer 5

Nuisance regressors are useful when they help predict some variance in the Y (BOLD signal) and reduce epsilon (error). Movement of a participant when blinking eyes or moving a hand to perform a task, heart rate and respiration are examples of useful nuisance regressors

Question 6

What are the Beta values telling us? Make an example, what happens if column 3 of the design Matrix has a big beta?

Answer 6

Beta values are the coefficient of a regressor. They are telling us how much the variation in the independent variable predicts change in the system. For instance a big beta in column three means that a small change in the independent variable 3 is associated with a big change in the signal Y

Question 7

What is the difference in terms of relationship between variables and signal in a simple linear regression and a multiple linear regression?

Answer 7

In a simple linear regression values of the independent variables are each linearly associated directly with changes of the signal, in a multiple linear regression the interactions are also among the variables

Question 8

What is a constant regressor?

Answer 8

It is a regressor variable with an associated beta that is used to account for a background noise in the signal

Question 9

How is epsilon assumed to be?

Answer 9

Independent and identically distributed according to zero mean normal distribution with unknown variance.

Question 10

In a geometric interpretation of the GLM, how would you interpret the vector of residuals?

Answer 10

The shortest line in the hyperplane connecting the modeled signal with the actual signal

Question 11

What is the mathematical and non mathematical objective of the process called Ordinary Least Squares (OLS)?
Explain its mathematical expression 𝑋T𝜀Ƹ= 0
𝑋T( 𝑦 − 𝑋𝛽)= 0

Answer 11

The mathematical objective is to reduce the squared (so that it accommodates for positive and negative) error in the GLM.
The non mathematical is to reduce to the minimum the discrepancy between modeled and actual signal, which is what the second equation is saying

Question 12

What does efficiency measure? What does it mean to have high efficiency?

Answer 12

Efficiency measures the ability of a regressor to predict the signal independently of other regressors. A low efficiency is bad because it gets impossible to tell which condition elicited a certain BOLD response (imagine for instance nuisance regressors covarying strongly with a regressor of interest.

Question 13

What is a t-contrast measuring in GLM?

Answer 13

If there is a difference between two regressors, it calculates the probability of differences between two conditons.

Question 14

What is an F-contrast measuring in GLM?

Answer 14

If there is an effect of first or second regressor, as the t-contrast test it it tested against a null hypothesis.
It’s a statistical test used to measure the overall significance of a set of predictors or conditions in explaining the observed variance in the data.
It tests the significance of a linear combination of multiple predictors or contrasts. It evaluates whether a combination of predictors, taken together, has a significant impact on the response variable. F-contrasts can be used to test hypotheses about the overall effect of a group of conditions or to compare models with different sets of predictors. (Last part is ChatGPT)

Question 15

Slow drifts, what are they and how to take care of them

Answer 15

Low frequency signals, pure noise for our purposes, we can set a low pass filter and also apply a Discrete Cosine Transform (a nuisance regressor) and regress the slow drifts out of the signal “temporal filtering”

Question 16

How to deal with the fact that stimulus response functions are boxcar shaped and not reflecting the haemodynamic response function?

Answer 16

They are convoluted with te HRF, so also temporally shifted, to give a more realistic prediction of the BOLD signal.

Question 17

How to deal with the fact that despite spatial realignemnt there are still some movement artifacts in the signal?

Answer 17

Include the movement artifacts as some nuisance regressor

Question 18

Serial auto correlation, what is it due to and how is it approached?

Answer 18

Consecutive fMRI scans are not statistically
independent, i.e. fMRI time series have serial correlation.
An auto-regressive model of order one [AR(1)] is fit
to account for serial correlations.

Question 19

What is the sphericity assumption?

Answer 19

Key assumption in the analysis of repeated measures data, (in this case time series data). It assumes that the variances of the differences between all pairs of conditions are equal. In other words, it assumes that the covariance structure of the data is spherical or circular.

Question 20

How do we get from a i.i.d noise model to an enhanced noise model?

Question 21

What is the problem of multiple comparisons due to?

Answer 21

When running t tests on multiple voxels, despite the p number being really low, the probability of type 1 errors is multiplied by the number of voxels serially analyzed. atype of correction, e.g Bonferroni one, is needed

Question 22

What is Gaussian Random Field Theory?

Answer 22

It is a theory that tries to estimate the maximum amount of correlation between voxels (due to both proximity and smoothing in preprocessing) and then taking it into account to temper statistical corrections that would be too conservative. e.g. untempered Bonferroni

Question 23

Parametric Modulators

Answer 23

parametric modulator is a regressor that captures the relationship between a specific experimental condition or stimulus and a CONTINUOUS variable of interest. It allows for modeling the effect of the variable on brain activity in a more nuanced and flexible manner.

Question 24

Second Level GLM, what are they?

Answer 24

Once the first-level analysis is completed for each participant, the second-level GLM is used to combine and analyze the results across participants. It allows for making inferences about the group or population as a whole. The primary goal of the second-level analysis is to identify consistent patterns of activation or effects across participants and assess their statistical significance.

The second-level GLM involves modeling the individual-level results or parameters obtained from the first-level analysis as the dependent variable. This typically includes the estimated coefficients, contrast estimates, or statistical maps representing the activation or effect size at each voxel or region of interest.

The second-level GLM can include various statistical methods, such as t-tests, ANOVA (analysis of variance), mixed-effects models, or nonparametric tests, depending on the research question and study design. It allows for testing group-level hypotheses, such as identifying brain regions showing significant activation across the entire group or comparing the effects between different conditions, groups, or populations.

In summary, the second-level GLM in fMRI data analysis is performed after the first-level analysis and aims to combine and analyze the results across participants to make group-level inferences. It enables researchers to identify consistent patterns of activation or effects and assess their statistical significance in the context of the entire study population