2nd level analysis – design matrix, contrasts and inference Alexander Leff

2nd level analysis – design matrix, contrasts and inference

Alexander Leff

Overview

• Why do 2nd-level analyses?

• Image based view of spm stats.

• Practical examples of 2nd level design matrices.

• Correction for multiple comparisons.

2nd Level inference

Study Group

Sample Group

InferenceSampling

In order for inferences to be valid about the study group (the population you want you findings to pertain to), the sample population (those who you actually study) should be representatively drawn from the study population.

Theoretical basis for 2nd level analyses

• Depends on whether you want to generalize your findings beyond the subjects you have studied (sample group).

• Usually this is the case, however:– Karl’s ‘talking dog’.– In a human, this happens: In humans, this happens.– In this group of patients… : In patients…

1st level design matrix:6 sessions per subject

15 auditory contrasts 2 target contrasts 6 movement parameters

A B

A = convolved with HRFB = not convolved with HRF

Betas are calculated for each column of the design matrix 23 betas x 6 sessions = 138 + 6 constants = 144 beta images.

Data, Y

A regressor, X, = timeseries of expected activation.

Y = X + e

The following images are created each time an analysis is performed (1st or 2nd level)

• beta images (with associated header), images of estimated regression coefficients (parameter estimate). Combined to

produce con. images.• mask.img This defines the search space for the

statistical analysis.• ResMS.img An image of the variance of the

error (NB: this image is used to produce spmT images).

• RPV.img The estimated resels per voxel (not currently used).

1st-level (within-subject)

1

2

3

4

5

6

1

^

^

^

^

^

^

wwithin-subject error^

Beta images contain values related to size of effect. A given voxel in each beta image will have a value related to the size of effect for that explanatory variable.

The ‘goodness of fit’ or error term is contained in the ResMS file and is the same for a given voxel within the design matrix regardless of which beta(s) is/are being used to create a con.img.

Design efficiency

Mask.imgCalculated using the intersection of 3 masks: 1) user specified, 2) Implicit (if a zero in any image then masked for all images) default = yes, 3) Thresholding which can be i) none, ii) absolute, iii) relative to global (80%).

Specify this contrastfor each session

Contrast 1:Vowel - baseline

Beta value = % change above global mean.In this design matrix there are 6 repetitions of the condition so these need to be summed.

Con. value = summation of all relevant betas.

ResMS.img =residual sum of squares or variance image and is a measure of within-subject error at the 1st level or between-subject error at the 2nd.

Con. value is combined with ResMS value at that voxel to produce a T statistic or spm.T.img.

2ˆi

spmT.imgThresholded using theresults button.

Eg random noiseEg random noise

Gaussian10mm FWHM(2mm pixels)

pu = 0.05

Gaussian10mm FWHM(2mm pixels)

pu = 0.05pu = 0.05

spmT.img and corresponding spmF.img

]c[

]c[

]c[ ]c[

ConX)(XCon ˆ]c[

voxel@ sMSReˆ1-TT2

2

i

i

ii

ii

ii

stdt

Varstd

Var

ci = voxel value in contrast image at voxel iCon = contrast applied to design matrix

]c[

]c[

]c[ ]c[

ConX)(XCon ˆ]c[

voxel@ sMSReˆ1-TT2

2

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i

ii

ii

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stdt

Varstd

Var

ci = voxel value in contrast image at voxel iConT = contrast applied to design matrixC = Constant term

]c[

]c[

]c[ ]c[

ConX)(XCon ˆ]c[

voxel@ sMSReˆ1-TT2

2

i

i

ii

ii

ii

stdt

Varstd

Var

ci = voxel value in contrast image at voxel iConT = contrast applied to design matrixC = Constant term

Efficiency termContrast specific(See Tom Jenkins’/Paul Bentley’s slides).

]c[

]c[

]c[ ]c[

ConX)(XCon ˆ]c[

voxel@ sMSReˆ1-TT2

2

i

i

ii

ii

ii

stdt

Varstd

Var

ci = voxel value in contrast image at voxel iConT = contrast applied to design matrixC = Constant term

]c[

]c[

]c[ ]c[

ConX)(XCon ˆ]c[

voxel@ sMSReˆ1-TT2

2

i

i

ii

ii

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stdt

Varstd

Var

ci = voxel value in contrast image at voxel iConT = contrast applied to design matrixC = Constant term

Summary

• beta images contain information about the size of the effect of

interest.

• Information about the error variance is held in the ResMS.img.

• beta images are linearly combined to produce relevant con.

images.

• The design matrix, contrast, constant and ResMS.img are subjected

to matrix multiplication to produce an estimate of the st.dev.

associated with each voxel in the con.img.

• The spmT.img are derived from this and are thresholded in the

results step.

Contrast 2:Vowel - Tones

Vowel - baseline Tone - baseline Vowel - ToneVowel – baseline

Contrast images for the two classes of stimuli versus baseline and versus each other(linear summation of all relevant betas)

Vowel - baseline Tone - baseline Vowel - Tone

spmT images for the two classes of stimuli versus baseline and versus each other(these are not linearly related as the st.dev. of the voxel value in each con.img varies with each contrast).

2nd level analysis – what’s different?• Maths is identical.• Con.img at the first level are output files, at the second level

they are both input and output files.• 1st level: variance is within subject, 2nd level: variance is

between subject.• Different types of design matrix (3 examples).

Specify 2nd level: One-sample t-testSimplest example, most parsimonious.

NB: Mask file will only include voxels common to all subjects.

Group mask Single subject mask

Estimate:

Results:Vowel – Tone contrast.

NB: beta and con images are identical.

beta.img con.img

Plot from voxel shows error variance.spmT.img

Specify 2nd level: Full factorialMore than one contrast per subject, can cause a problem with sphericity assumptions.Can analyse systematically: simple main effects then interactions.Mask one contrast with another etc.

Vowels Formants Tones

Specify 2nd level: One-sample t-testSimplest example, most parsimonious

Test with a conjunction term:Which voxels are activated in both contrasts

Vowels Formants TonesPlot:

Specify 2nd level: One sample t-testwith a covariate added.Test correlations between task specific activations and some other measure(age, performance, etc.).

Vectors added here.Needs to be mean corrected by hand.(in this case age squaredthen mean corrected).

Main effect of grip (1st level analysis event related Correlation between grip and age of subjectdesign: grip vs. rest)

This plot correlates betas related to grip (y-axis) with a measure of age (x-axis)

2nd level analysis summary

• Many different ways of entering the contrast images of

interest generated by the first level design matrix.

• Choice depends primarily on:

1. Initial study design.

2. Parsimonious models vs. more complex ones.

Voxels = volume/8: FDR-corr

Resels = calculated from the estimated smoothness (FWHM): FEW-corr

volume = defined by mask.img

Small volume correction: box, sphere, image.

Canonical 152 T1.img STG mask (created in MRIcro. NB: must reorient origin in spm).

SVC summary

• p value associated with t and Z scores is dependent on 2

parameters:

1. Degrees of freedom.

2. How you choose to correct for multiple comparisons.

Sources

• Rik Henson’s slides.

• MfD past and present.

• SPM manual (D:\spm5\man).

• Will Penny.