2nd 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 fMRI (unlike PET) the between-scan variance is normally much smaller than the between-subject variance Multi-subject Fixed Effect model Subject 1 error df ~ 300 Subject 2 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 3 Subject 4 Fixed effect: A variable with fixed values E.g. levels of an experimental variable. Box on the right: Fixed-effect analysis; The factor ‘subject’ treated like other experimental variable in the design matrix. Within-subject variability across condition onsets represented across rows. Between-subject variability ignored Case-studies approach: Fixed-effects analysis can only describe the specific sample but does not allow generalization. Random effect: A variable with values that can vary. E.g. the effect ‘list order’ with lists that are randomized per subject The effect ‘Subject’ can be described as either fixed or random Subjects in the sample are fixed Subjects are drawn randomly from the population Typically treated as a random effect in behavioural analysis Random effects analysis: Generalization to the population requires taking between-subject variability into account. The question: Would a new subject drawn from this population show any significant activity? Mixed models: the experimental factors are fixed but the ‘subject’ factor is random. Mixed models take into account both within- and between- subject variability  better! Subject 5 Subject 6

Two-stage “Summary Statistic” approach 1st-level (within-subject) 2nd-level (between-subject) b1 ^ b2 b3 b4 b5 b6 contrast images of cbi  N=6 subjects (error df =5) One-sample t-test ^ (1) w = within-subject error (2) (3) (4) (5) (6) p < 0.001 (uncorrected) SPM{t} ^ bpop  WHEN special case of n independent observations per subject: var(bpop) = 2b / N + 2w / Nn

Relationship between 1st & 2nd levels 1st-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. 2nd-level analysis: Feed the contrast images into a GLM that implements a statistical test. Con image for contrast 1 for subject 1 Con image for contrast 2 for subject 2 Con image for contrast 1 for subject 2 Con image for contrast 2 for subject 1 Contrast 1 Contrast 2 Subject 2 Subject 1 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 1st & 2nd levels Both use the GLM model/tests and a similar SPM machinery Both produce design matrices. The rows in the design matrices represent observations: 1st level: Time (condition onsets); within-subject variability 2nd level: subjects; between-subject variability The columns represent explanatory variables (EV): 1st level: All conditions within the experimental design 2nd level: The specific effects of interest

Similarities between 1st & 2nd levels The same tests can be used in both levels (but the questions are different) .Con images: output at 1st level, both input and output at 2nd level 1st level: variance is within subject, 2nd level: variance is between subject. There is typically only one 1st-level design matrix per subject, but multiple 2nd level design matrices for the group – one for each statistical test. 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 1st level analyses) for main effect of A main effect of B interaction AxB. A1 A2 1 2 4 5 3 6 B2 B3 B1

Difference from behavioral analysis The ‘1st 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 1st 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) A single number: categorical or frequency Whether the subject reports pain or not; the number of times the subject reported feeling pain A summary statistic, resulting from a simple model of the data, typically the mean: The mean reaction time in a classification task.

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

1st level design matrix: 6 sessions per subject

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). All images can be displayed using check-reg button

1st-level (within-subject) ^ b2 b3 b4 b5 b6 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. 1 ^ 2 3 4 5 6 w = within-subject error 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.

Explicit masks Single subject mask Group mask Mask.img Calculated using the intersection of 3 masks: Implicit (if a zero in any image then masked for all images) default = yes Thresholding which can be i) none, ii) absolute, iii) relative to global (80%). Explicit mask (user specified) Single subject mask Group mask Note: You can include explicit mask at 1st- or 2nd-level. If include at 1st-level, the resulting group mask at 2nd-level is the overlapping regions of masks at 1st-levelso, will probably much smaller than single subject masks. A sensible option here is to use a segmentation of structural images to specify a within-brain mask. Explicit masks are other images containing (implicit) masks that are to be applied to the current analysis. Segmentation of structural images

Con. value = summation of all relevant betas. 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.

spmT.img Thresholded using the results button.

spmT.img and corresponding spmF.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 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.

The buttons and what follows.. Specify 2nd-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…

How to enter contrasts… Effort How to enter contrasts… E1 E2 R1 Reward R2 R1 R2 E1 E2 Main effect of Reward 1 -1 Main effect of Effort Effort x Reward Interaction: RE1 x RE2 = (R1E1 – R1E2) – (R2E1– R2E2) = R1E1 – R1E2 – R2E1 + R2E2 = 1 - 1 - 1 + 1 = [ 1 -1 -1 1]

set-level: P(c  3, n  k, t  u) = 0.019 Levels of Inference Three levels of inference: extreme voxel values voxel-level (height) inference big suprathreshold clusters cluster-level (extent) inference many suprathreshold clusters set-level inference voxel-level: P(t  4.37) = .048 n=82 n=32 n=12 set-level: P(c  3, n  k, t  u) = 0.019 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. cluster-level: P(n  82, t  u) = 0.029

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 Adjust for global effects (for fMRI) 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) global global Scaling

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 fMRI 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 whole- brain! 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 Degrees of freedom. p value associated with t and Z scores is dependent on 2 parameters: Degrees of freedom. 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 individual-differences 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:\spm5\man). Human Brain Function book Guillaume Flandin & Geoffrey Tan