The General Linear Model and Single-Subject & Group Analyses Flashcards
When are the data ready for statistical analysis?
After reconstruction, realignment, stereotactic normalisation and possibly smoothing
What are the two steps for statistical analysis?
- Statistics indicating evidence against a null hypothesis of no effect at each voxel are computed - this results in an image of statistics
- The statistic image must be assessed, reliably locating voxels where an effect is exhibited whilst limiting the possibility of false positives
What is used for analysis of functional mapping experiments?
SPM
What is auditory block-design experiment?
• One session
- One subject in the scanner
• Passive word listening versus rest
• 7 cycles of rest and listening
• Blocks of 6 scans with 7 second TR
• In the data template in the brain – what are the involved time series that correspond to the stimulus function
• There is greater response in the brain when you are listening to words vs resting
Why do we model the measured data?
Make inferences about effects of interest
How is the measured data modelled?
Decompose data into effects and error
Form statistic using estimates of effects and error
Partition the data into 2 points: noise and signal
Get data partition data using linear model
Why is statistic done?
To find the ratio between effects estimate divided by error estimate
When do you get a significant effect?
Very small effect size but a very consistent one
What happens after stimulus function?
- Data
- Linear model
- Effect estimate
- Error estimate - Statistic
(Mass univariate) voxel-wise time series analysis:
• As the data has been corrected for movement and normalised – look at the single voxel over time
• It corresponds to 1 time series – one time series per voxel
• Do modelling of the time series and describe how we expect it to evolve over time
- Model specification
- Parameter estimation
- Hypothesis
- Statistic
• Repeat the same procedure for each and every voxel in the brain
• We use the same voxel-wise analysis for all the voxel in the brain one at a time
What is a single voxel regression model?
• If I take a single voxel – volume density over time at a given voxel
• Express it as a linear combination of two things
- Stimulus function e.g. listening to words and resting [e.g. rest = 0, 1= listening]
- Regressor that is constant – it is one for every time point – model the average point of the time series
• Data that is observed can be expressed in a linear combination of 2 terms B1 + B2 [weights are given – unknown parameters]
• Error – noise that is found on top of our measurements
• Data y [time series] is expressed by linear combinations of B1 + B2 of 2 vectors – stimulus function and the average signal and plus error term linear model
What is the equation for the single voxel regression model?
Y = x1B1 + X2B2 + e
What is the General Linear Model (GLM) described by?
data Y – 1 x N data points expressed as a linear combination of X and B
• X contains all the columns – everything that you expect to see in the data
• B is the vectors of the unknown parameters – unknown quantities that we want to estimate – how much of the response is observed in a particular voxel
• Error term is the time series the same dimension as the data
What is the GLM model specified by?
- Design matrix X
2. Assumptions about e
How is the time series expressed as?
an image – dark regions mean small values and light regions means higher values FMRI time series in voxels
What is the time series expressed as?
Linear combination of stimulus function and the constant term + error
\why does the parameters need to be estimated?
Minimize the data - get the data to be as close as possible to 0
What are problems of this model with fMRI time series?
- The BOLD response has a delayed and dispersed shape
- The BOLD signal includes substantial amount of low-frequency noise (e.g. due to scanner drift)
- Due to breathing, heartbeat and unmodelled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the glm
The BOLD response has a delayed and dispersed shape
- The neuronal activity that occur in the brain will be the same thing as measured in FMRI data
- E.g. if you have a burst of activity at T=0
- The Haemodynamic response function (HRF) – the response to burst activity at time T=0
- Take neuronal activity and convolute with HRF to get an Expected BOLD
The BOLD signal includes substantial amounts of low-frequency noise (e.g. due to scanner drift
- At the start of the acquisition the mean data will be getting lower and lower because of heating and that has a direct effect on the intensity of the signal slow decay over time
- Implement a model by giving a number of regressors that correspond to a cosine function of different frequencies that allows us to model anything up to a certain frequency
- Get the matrix and fit it to the data
- Solution is high pass filtering
Due to breathing, heartbeat and unmodelled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM
- Noise is i.i.d = independent, identical, distributed
- The model has to take into account the correlation in time
What are contrasts?
Compare different conditions
• Contrasts are a way to quantify the parameter you are really interested in e.g. difference in response between 2 different conditions
• A contrast c is a vector of length p
• Compute the linear combination of this contrast in your B – unknown parameters
• A contrast selects a specific effect of interest
What is T-test?
• Passive word listening vs rest
• Compute the test statistics
• Q = activation during listening?
• Null hypothesis: B1=0
• How to specify B1=0
- Contrast of 1 followed by 0
• Compute test statistics equal to the effect size of interest B1 divided by the standard error
• Compute T for every voxel in the brain – get images of test statistics
• Get bilateral activity in the cortex corresponding to the task
What is inference at a single voxel?
- Null hypothesis H0 assumes that there is no activation
- If you compute a test statistic – you assume that you know the distribution of the statistic test – most of the time it will be 0
- It will be a larger or smaller value by chance
- Choose the significance value alpha – a typical value of 5
- Decision rule (threshold) u determines false positive rate alpha
- Choose u to give acceptable alpha under H0
- Compute test statistics and compare it with voxel threshold or below
- Voxel threshold – significant effect
- When you get an effect assign P – which is the area of the curve corresponding to statistical u
What is multiple tests?
• If we have 100,000 voxels, alpha=0.05 5,000 false positive voxels
- Scale the number of voxels
- No control over the number of false positives
• This is clearly undesirable; to correct for this we can define a null hypothesis for a collection of tests
What is family-wise null hyppthesis?
Activation is zero everywhere
What happens if a voxel null hypothesis at any voxel is rejected?
We reject the family-wise null hypothesis
What gives a family wise error (FWE)?
A FP anywhere in the image
What is the family-wise error rate (FWER)?
Corrected p-value
What is the Bonferroni correction?
an adjustment applied to p-values that is supposed to be applied, when two or more statistical analyses have been performed on the same sample of data
• Conducting multiple analysis on the same sample of data
What is Type 1 error?
Erroneously rejecting the null hypothesis with a statistical analysis, when the null hypothesis is in fact true in the population
How can you test the null hypothesis?
with an independent samples t-test
When do you reject the null hypothesis of equal means?
If the t-test p-values is less than .05.
Whatt is the familywise type 1 error rate known to be larger than?
per analysis error rate (i.e. alpha = 0.05)
What is the Familywise Error rate?
The probability of committing at least one type 1 error amongst two or more statistical analyses on the sample of data
What is the Random Field Theory?
Consider a statistic image as a discretisation of a continuous underlying random field
What is Euler characteristic?
- Topological invariant
- A number that describes a topological space’s shape or structure regardless of the way it is bent
- It is commonly denoted by X
- Compute at any threshold
- At high threshold, the Euler’s characteristic is counting the number of blobs
- If the threshold is really high, the Euler’s characteristic is either 1 or 0
What is Fixed effect analysis (FFX)?
- In order to test whether the obtained results are valid at the population level, the statistical procedure needs to consider that subjects constitute a randomly drawn sample from a large population
- Inferences drawn from obtained results are only valid for the included group of subjects since the group data is treated like a virtual single-case study
- Subjects are thus random quantities and the statistical analysis must assess the variability of observed
- Simple model
- lots of degrees of freedom
What is disadvantage of FFX?
- Large amount of data
- Assumes common variance over subject at each voxel
What can be compared with Fixed Effects Analysis (FFX)?
The group effect to the within-subject variability
It is not an inference about the population from the subjects were drawn
What is compared with Random Effects Analysis (RFX)?
Compare the group effect to the between-subject variability. It is an inference about the population from which subjects were drawn
Fixed-effects
- Only one source of random variation (over session)
- Measurement error
- Within-subject variance
Random Effects
- Two sources of random variation
- Measurement errors – within-subject variance
- Response magnitude (over subjects) – Between-subject variance
The summary statistics approach is exact if for each session/subject
- Within-subjects variances the same
2. First level design the same (e.g. number of trials )
What is summary statistics approach?
Robust against typical violations
