Analysis of fMRI data with linear models

Typical fMRI processing steps• Image reconstruction• Slice time correction• Motion correction• Temporal filtering• 1st level (individual subject) linear modeling• Conversion to % signal change• Image normalisation (Talairach/MNI)• Spatial blurring• 2nd level (group) linear modeling• Extraction of “activated” clusters

Typical fMRI processing steps• Image reconstruction• Slice time correction• Motion correction• Temporal filtering• 1st level (individual subject) linear modeling• Conversion to % signal change• Image normalisation (Talairach/MNI)• Spatial blurring• 2nd level (group) linear modeling• Extraction of “activated” clusters

1st level linear modeling

For each brain voxel we have a time series of BOLD signal that we want to explain in terms of the experimental manipulations (events)

1st level linear modelingFitting a general linear model to individual voxel time series

We postulate a number of predictor variables : ideal time series that represent what we think the response should look like to each type of experimental stimulus.

Then compute the weighted sum of these predictor variables that produces the closest match to the actual data time series.

Signal = ß1F1 + ß 2F2 + ß3F3 + constant + error

Where F1, F2 and F3 are the postulated predictor variables or functions.

1st level linear modeling• Although only one set of predictor variables is posited for every

voxel in the analysis, the model is individually fit for every voxel. This gives a unique set of weights (beta coefficients) for each voxel.

• Each beta weight for a voxel represents the BOLD signal change due to the corresponding experimental condition

• We can also form linear contrasts of beta weights at each voxel (e.g. the subtraction of one beta from another, representing the difference in signal change between two conditions)

• Thus we have one or more statistical brain maps of beta weights and/or contrasts

• Associated with each beta and contrast is a t-statistic, that tells us to what extent such a value could be expected by chance alone – these are only interesting for single subject analyses.

2nd level linear modeling

• The goal of 2nd level analysis is to determine the extent to which beta-weights or contrasts are consistent across subjects in a group, and to what extent they differ across different groups

• The method is simply a matter of running t-tests, ANOVA or regression on the beta weights or contrasts for each voxel

• The result is one or more group statistical maps

2nd level linear modeling: t-tests

• Used to test whether the means of two groups are equal.

• 3 types of t-test:

– Unpaired – tests equality of means of two independent groups

– Paired – tests equality of means of two dependent groups (e.g. two time points on same group, two groups that are pairwise matched)

– Single group – tests if mean of one group is equal to zero

– Can be used directionally – 1-tailed vs. 2-tailed test

2nd level linear modeling: ANOVA• Used to test whether the means of two or more groups are

equal.

• 3 types of ANOVA:

– Fixed effects

– Random effects

– Mixed effects (i.e. both random and fixed)

• Most usual for fMRI data is mixed effects

– Experimental conditions are fixed

– Subjects are random

• Does not provide directional test – must look at mean differences

2nd level linear modeling: regression• Used to test association between BOLD signal change and some

external measure (e.g. trait/state measure, behavioral measure, physiological measure)

• Provides a directional measure of association

• In simplest form is simple correlation

• Can be used to account for undesirable variance

• Very sensitive to outliers – you MUST extract the voxel values and examine the scatterplots

• Can be used to create more complicated ANOVA models, including ANCOVA