2 nd level analysis design matrix contrasts

2 nd level analysis – design matrix, contrasts and inference Irma Kurniawan MFD Jan 2009

Today’s menu • • • Fixed, random, mixed effects First to second level analysis Behind button-clicking: Images produced and calculated The buttons and what follows. . Contrast vectors, Levels of inference, Global effects, Small Volume Correction • Summary

Fixed vs. Random Effects • Subjects can be Fixed or Random variables • If subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject variance – But in f. MRI (unlike PET) the between-scan variance is normally much smaller than the between-subject variance Multi-subject Fixed Effect model Subject 1 Subject 2 Subject 3 • If one wishes to make an inference from a subject sample to the population, one needs to treat subjects as a Random variable, and needs a proper mixture of within- and between-subject variance • Mixed models: the experimental factors are fixed but the ‘subject’ factor is random. • In SPM, this is achieved by a two-stage procedure: 1) (Contrasts of) parameters are estimated from a (Fixed Effect) model for each subject 2) Images of these contrasts become the data for a second design matrix (usually simple t-test or ANOVA) Subject 4 Subject 5 Subject 6 error df ~ 300

Two-stage “Summary Statistic” approach 2 nd-level (between-subject) 1 st-level (within-subject) ^ b 2 ^ ( 2) ^ b 3 ^ ( 3) ^ b 4 ^ ( 4) ^ b 5 ^ ( 5) ^ b 6 ^ ( 6) contrast images of cbi ^ b 1 ^ ( 1) One-sample t-test N=6 subjects (error df =5) p < 0. 001 (uncorrected) ^ b SPM{t} pop ^ w = within-subject error WHEN special case of n independent observations per subject: var(bpop) = 2 b / N + 2 w / Nn

Relationship between 1 st & 2 nd levels • 1 st-level analysis: Fit the model for each subject. Typically, one design matrix per subject • Define the effect of interest for each subject with a contrast vector. • The contrast vector produces a contrast image containing the contrast of the parameter estimates at each voxel. Contrast 1 Subject 1 Con image for contrast 1 for subject 1 Subject 2 • 2 nd-level analysis: Feed the contrast images into a GLM that implements a statistical test. Contrast 2 Con image for contrast 1 for subject 2 Con image for contrast 2 for subject 1 Con image for contrast 2 for subject 2 You can use checkreg button to display con images of different subjects for 1 contrast and eye-ball if they show similar activations

Similarities between 1 st & 2 nd levels • Both use the GLM model/tests and a similar SPM machinery • Both produce design matrices. • The rows in the design matrices represent observations: – 1 st level: Time (condition onsets); within-subject variability – 2 nd level: subjects; between-subject variability • The columns represent explanatory variables (EV): – 1 st level: All conditions within the experimental design – 2 nd level: The specific effects of interest

Similarities between 1 st & 2 nd levels • The same tests can be used in both levels (but the questions are different) • . Con images: output at 1 st level, both input and output at 2 nd level • 1 st level: variance is within subject, 2 nd level: variance is between subject. • There is typically one 1 st-level design matrix per subject, but multiple 2 nd level design matrices for the group – one for each statistical test. B 1 B 2 B 3 A 1 1 2 3 A 2 4 5 6 For example: 2 X 3 design between variable A and B. We’d have three design matrices (entering 3 different sets of con images from 1 st level analyses) for 1) main effect of A 2) main effect of B 3) interaction Ax. B.

Difference from behavioral analysis • The ‘ 1 st level analysis’ typical to behavioural data is relatively simple: – A single number: categorical or frequency – A summary statistic, resulting from a simple model of the data, typically the mean. • SPM 1 st level is an extra step in the analysis, which models the response of one subject. The statistic generated (β) then taken forward to the GLM. – This is possible because βs are normally distributed. • A series of 3 -D matrices (β values, error terms)

Behind button-clicking… • Which images are produced and calculated when I press ‘run’?

1 st level design matrix: 6 sessions per subject

The following images are created each time an analysis is performed (1 st or 2 nd 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. • Res. MS. img An image of the variance of the error (NB: this image is used to produce spm. T images). • RPV. img The estimated resels per voxel (not currently used). • All images can be displayed using check-reg button

1 st-level (within-subject) ^ b 1 Beta images contain values related to size of effect. A ^ given voxel in each beta image will have a value related 1 to the size of effect for that explanatory variable. ^ b 2 ^ ^ b 3 ^ 3 The ‘goodness of fit’ or error term is contained in the Res. MS 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. ^ b 4 ^ b 5 ^ b 6 ^ = within-subject error w

t masks Mask. img Calculated using the intersection of 3 masks: 1) Implicit (if a zero in any image then masked for all images) default = yes 2) Thresholding which can be i) none, ii) absolute, iii) relative to global (80%). 3) Explicit mask (user specified) Single subject mask Segmentati on of structural images Group mask Note: You can include explicit mask at 1 st- or 2 nd-level. If include at 1 st-level, the resulting group mask at 2 ndlevel is the overlapping regions of masks at 1 stlevel so, will probably much smaller than single subject masks.

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.

Res. MS. img = residual sum of squares or variance image and is a measure of withinsubject error at the 1 st level or betweensubject error at the 2 nd. Con. value is combined with Res. MS value at that voxel to produce a T statistic or spm. T. img.

spm. T. img Thresholded using the results button.

spm. T. img and corresponding spm. F. img

So, which images? • beta images contain information about the size of the effect of interest. • Information about the error variance is held in the Res. MS. img. • beta images are linearly combined to produce relevant con. images. • The design matrix, contrast, constant and Res. MS. img are subjected to matrix multiplication to produce an estimate of the st. dev. associated with each voxel in the con. img. • The spm. T. img are derived from this and are thresholded in the results step.

The buttons and what follows. . • Specify 2 nd-level • Enter the output dir • Enter con images from each subject as ‘scans’ • PS: Using matlabbatch, you can run several design matrices for different contrasts all at once • Hit ‘run’ • Click ‘estimate’ (may take a little while) • Click ‘results’ (can ‘review’ first before this)

A few additional notes…

Effort How to enter contrasts… E 1 E 2 Reward R 1 R 2 E 1 E 2 Main effect 1 1 -1 -1 of Reward Main effect 1 of Effort x 1 Reward -1 1 -1 -1 -1 1 Interaction: RE 1 x RE 2 = (R 1 E 1 – R 1 E 2) – (R 2 E 1– R 2 E 2) = R 1 E 1 – R 1 E 2 – R 2 E 1 + R 2 E 2 = 1 - 1 + 1 = [ 1 -1 -1 1]

Levels of Inference • Three levels of inference: – extreme voxel values ® voxel-level (height) inference – big suprathreshold clusters voxel-level: P(t 4. 37) =. 048 ® cluster-level (extent) inference – many suprathreshold clusters n=1 2 ® set-level inference n=82 Set level: At least 3 clusters above threshold Cluster level: At least 2 cluster with at least 82 voxels above threshold Voxel level: at least cluster with unspecified number of voxels above threshold Which is more powerful? Set > cluster > voxel level Can use voxel level threshold for a priori hypotheses about specific voxels. n=32 cluster-level: P(n 82, t u) = 0. 029 set-level: P(c 3, n k, t u) = 0. 019

Example SPM window

Global Effects • May be global variation from scan to scan • Such “global” changes in image intensity confound local / regional changes of experiment • global Adjust for global effects (for f. MRI) by: Proportional Scaling • • Can improve statistics when orthogonal to effects of interest (as here)… …but can also worsen when effects of interest correlated with global (as next) Scaling global

Global Effects • Two types of scaling: Grand Mean scaling and Global scaling • Grand Mean scaling is automatic, global scaling is optional • Grand Mean scales by 100/mean over all voxels and ALL scans (i. e, single number per session) • Global scaling scales by 100/mean over all voxels for EACH scan (i. e, a different scaling factor every scan) • Problem with global scaling is that TRUE global is not (normally) known… • …we only estimate it by the mean over voxels • So if there is a large signal change over many voxels, the global estimate will be confounded by local changes • This can produce artifactual deactivations in other regions after global scaling • Since most sources of global variability in f. MRI are low frequency (drift), high-pass filtering may be sufficient, and many people to not use global scaling

Small-volume correction • If have an a priori region of interest, no need to correct for wholebrain! • But can correct for a Small Volume (SVC) • Volume can be based on: – An anatomically-defined region – A geometric approximation to the above (eg rhomboid/sphere) – A functionally-defined mask (based on an ORTHOGONAL contrast!) • Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…(cf. Random Field Theory slides) • . . but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough)

Example SPM window

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.

Statistical inference: imaging vs. behavioural data • Inference of imaging data uses some of the same statistical tests as used for analysis of behavioral data: – t-tests, – ANOVA – The effect of covariates for the study of individualdifferences • Some tests are more typical in imaging: – Conjunction analysis • Multiple comparisons poses a greater problem in imaging (RFT; small volume correction)

With help from … • Rik Henson’s slides. • Debbie Talmi & Sarah White’s slides • Alex Leff’s slides • SPM manual (D: spm 5man). • Human Brain Function book • Guillaume Flandin & Geoffrey Tan