lecture 5 - GLM

Question 1

the signal intensity of time series

Answer 1

1. changes with specific experimental conditions, stimuli, or tasks the subject was performing during the scan to identify brain regions that are involved in particular cognitive processes or responses.
2. However, the signal can also be influenced by noise, movement artifacts, and physiological changes unrelated to neural activity (breathing), which requires careful preprocessing and analysis to ensure accurate interpretation.

• so, how much of the signal is explained variance

Question 2

averaging

Answer 2

1. take 2 periods
2. overlay and average
3. test difference between two values (t-test)

Question 3

problem with averaging fMRI data (+ solution)

Answer 3

• all types of events are being averaged together
  -> cue-related and stimulus-related responses
• solution: EPOCH AVERAGING
  Instead of averaging the whole time series, take the data 20 seconds after an event (cue or stimulus) and average those epochs.
  –> Segmenting the time series data into shorter intervals (“epochs”) around the time of the stimulus or cue and then averaging the data within those epochs.
  –> This method allows for the examination of the time-locked response to specific events
  (i.e., epoched average of cue-locked response and stimulus-locked response separately)

Question 4

problem with epoch averaging

Answer 4

it is impossible to unambiguously assign data to overlapping events within epochs

–> if stimuli and cues are presented in quick succession, their related BOLD responses can overlap in time.
–> if data responses fall into multiple epochs, they are ‘counted’ multiple times

Question 5

GLM/Linear Time-Invariant (LTI) System

Answer 5

• neuronal activity acts as input/impulse
• HRF acts as the impulse response function
• the expected fMRI signal at time t is modeled as the convolution of a canonical response and neural activity

Question 6

GLM/Linear Time-Invariant (LTI) System - principle of linearity

Answer 6

GLM assumes that the BOLD responses elicited by different events or stimuli within the experiment can simply be added together in a linear way to produce the overall BOLD response
–> BOLD response is directly proportional to the magnitude of the neural activity

Question 7

convolution

Answer 7

• neural activity * ‘canonical’ response
• shows how the BOLD signal changes over time in response to neural activity induced by the stimulus.
• because we assume that the link between neural firing and BOLD response is an LTI system, we can use convolution with an HRF to model the predicted BOLD response

Question 8

meat of the GLM analysis

Answer 8

1. multiple events of which you’ll be interested in differential responses to these events
2. form explicit expectations for different conditions/events/predictors/regressors/explanatory variables
   –> here, you assume the HRF shape
3. see whether the voxel’s response pattern is more similar to predictor (regressor) 1 or 2.
   -> This comparison is done by convolving the stimulus timing for each condition with the canonical HRF to create predicted BOLD time-courses, which are then compared to the actual observed time-course of BOLD signals from the voxel.

Question 9

GLM analysis goal

Answer 9

1. to try to explain the measured BOLD signal as a combination of our explanatory variables
2. Modeling voxel time courses with a linear combination of hypothetical time-series (regressors).
3. Same model for each voxel –> One beta estimate per regressor per voxel

Question 10

y_i = a_i ⋅ x_1 + b_i ⋅ x_2 + c_i + N(0,σ)

Answer 10

• y_i = observed BOLD response for the ith voxel at a specific time point
  –> single voxel
• x_1, x_2 = regressors
• a, b, c = scaling factors (beta weights) that specify whether a voxel is e.g., ‘house’ selective or ‘face’ selective
  –> Indicate how much each regressor contributes to the observed signal
• c = intercept: describes the average signal and has nothing to do with the BOLD signal
• N(0,σ) = normally distributed noise

Question 11

beta weights (scaling factors)
y-hat_i = β_i,1 ⋅ x_1 + β_i,2 ⋅ x_2 + β_i,3

Answer 11

1. allow you to say whether a voxel is a ‘face’ or a ‘house’ voxel
2. these are the values you need to multiply with your explanatory variables to best explain the data
3. optimal combination of betas gives us the explained signal
4. indicates the responsiveness of a voxel to each condition

• y-hat_i = the predicted BOLD response for a single voxel
  –> explains a cewrtain portion of the signal’s variance

Question 12

explained & unexplained variance for
y-hat_i = β_i,1 ⋅ x_1 + β_i,2 ⋅ x_2 + β_i,3

Answer 12

• the optimal combination of betas gives us the modeled/explained signal
• this y-hat_i explains a certain portion of the original signal’s variance.
• the better our model, the higher the ratio of explained variance

Question 13

GLM analysis
Y = X ⋅ β + ε

Answer 13

• We can perform this analysis across all voxels at the same time ( = fast!)
  –> simultaneous solution for multiple voxels.

Question 14

GLM methods

Answer 14

minimizing squared error

||y = Xβ||^2

y = Xβ + e
β-hat = (X’X)^-1⋅X’y (unbiased estimator)

Question 15

fMRI Donders’ subtraction method

Answer 15

1. to zoom into a specific cognitive process, we often want to vary only a single thing in our experiments
2. therefore, we need a control condition to subtract
   –> task - control task

Question 16

contrast vectors

Answer 16

• GLM subtraction method
• used to encode the comparisons between different conditions by assigning weights to the regressors (conditions).
• The weights in the contrast vector sum to zero
• we want to test whether there is a stronger response to X than Y
  –> β_x > β_y
  –> β_x - β_y > 0

Question 17

contrast outcomes

Answer 17

1. depend on the correlation between regressors
2. negative t-statistics are also possible
3. ‘stronger activation for β_x than for β_y’

Question 18

family-wise error rate

Answer 18

• the rate at which false positives occur
• increases with more comparisons, which means you are more likely to claim that there is activation in voxels when there is not.

Question 19

bonferroni correction

Answer 19

changing (lowering) the alpha level
–> very conservative

α_new = α/v
–> v = number of voxels

Question 20

bonferroni correction: pro’s and cons

Answer 20

pro: no false positives - you can be confident whenever you find something

con: false negatives - you can be confident you’re throwing away actual findings

the trade-off is that by being so strict to avoid false positives, t

Question 21

why is the Bonferroni correction too conservative?

Answer 21

it assumes independent tests across voxels

Question 22

false discovery rate (FDR) correction

Answer 22

approach used in multiple hypothesis testing to control the expected proportion false positives.

adjusts the p-values to ensure that among all positive results (i.e., rejected null hypotheses), the proportion that are false positives is at most q = FDR

1. FDR = FP / (FP + TP)
2. The acceptability of false discoveries depends on the total number of discoveries.
   –> 2 false discoveries out of 4 = bad
   –> 2 false discoveries out of 50 = okay (FDR = 2/50 = 0.04)

Question 23

FDR pro’s and cons

Answer 23

pro’s: sensitive if large amounts of voxels are significant, conservative if all voxels show noise
–> (i.e.,: good when there’s a large number of significant results (it’s sensitive) and even when there’s a lot of noise among the results (it’s not too strict))

cons: more complicated algorithms, outcome depends on distribution of p-values

Question 24

FDR vs Bonferroni correction

Answer 24

bonferroni is the most conservative and thus will show the least activation compared to FDR correction

Question 25

deconvolution

Answer 25

• fitting the HRF shape, instead of assuming a fixed HRF shape
• is called deconvolution, since we are interested in finding the impulse response function of the LTI system
  –> essentially working backward from the BOLD signal to the neural events that caused it.
• we have to describe the shape, which takes more regressors (BASIS FUNCTIONS)

Question 26

different basis functions for deconvolution

Answer 26

Basis functions are mathematical concepts used to represent any function as a combination of basic functions – used to model the Hemodynamic Response Function (HRF)

1. by ones:
   finite impulse response (FIR) fitting:
   uses separate regressors for each time point after an event to individually capture the response at that specific time, with ‘0’ indicating no response and ‘1’ indicating a response at that particular time point along with the intensity of that response (0.0-1.0).
2. by frequencies:
   Fourier Basis Set & Discrete Cosine Set:
   regressors are sine and cosine functions
3. Gamma
   use gamma shape as backbone and add a regressor of neural activity convolved with the HRFs time derivative

Question 27

disadvantage of Finite Impulse Response (FIR) fitting

Answer 27

“hard to fit” because it involves many different regressors, which makes the model complex and can lead to overfitting if not managed properly.

Question 28

Gamma basis functions

Answer 28

• since FIR takes a lot of regressors, we can use the gamma shape as a backbone and add a regressor of neural activity convolved with the HRF’s time derivative
• this allows us to change the timing of the HRF, so that we better capture the subject’s HRF shape
  –> flexibility but not too many regressors

Question 29

Gamma basis function: dispersion derivative

Answer 29

an extra regressor that changes the shape of the HRF more than its timing

Question 30

we perform a GLM analysis instead of epoched averaging because

Answer 30

1. we can leverage our knowledge of the HRF shape to disambiguate responses from consecutive events
2. the HRF lasts so long that responses to consecutive events in your analysis will overlap, these overlapping responses need to be teased apart

Question 31

SSE

Answer 31

quantifies the unexplained signal

Question 32

GLM is typically two-level hierarchical analysis

Answer 32

1. within subject
   - single subject, single voxel
2. across subjects

can be done in stages. the hierarchical model ocmbines stages into one model

Question 33

GLM

Answer 33

• treats the data as a LINEAR COMBINATION of predictors (model functions) plus noise
• the model functions are assumed to have KNOWN SHAPES but their AMPLITUDES ARE UNKNOWN and need to be estimated

Question 34

mass univariate approach in GLM

Answer 34

construct a separate model for every voxel

Question 35

block design

Answer 35

similar groups are grouped together

–> two condition block design is optimal for power

Question 36

event-related design

Answer 36

events are mixed

Question 37

rapid presentation in event-related design

Answer 37

standard event-related design has many extraneous processes, leading to activation for the wrong reasons

solution = rapid presentation

1. prevents subjects from engaging in spontaneous processes (getting distracted)
2. maximizes time on the task’s process of interest

Question 38

psychological considerations

Answer 38

1. stimulus predictability
2. time on task
3. participant strategy
4. temporal precision of psychological manipulations
5. unintended psychological activity

Question 39

basic tradeoffs

Answer 39

1. fewer conditions and contrasts: more power, but less generalizability and specificity
2. many comparisons, high potential for specificity of inference, but low power