First level analysis

1. First level analysis methods for dummies 2009, Tessa Dekker Model fitting at the within subject level

2. You have collected time series of bold-signal during your task in voxels in the brain: Time Time Time Time Run 1 Run 1 Run 2 Run 2 First level Subject 1 Subject n Second level group(s)

3. You have preprocessed your data (realigning, filtering, spatial normalization etc.) and are ready to go look for brain regions in which your experimental manipulation had an effect. First level analysis: Find out for each voxel in a subject to what extent it shows a response pattern you would expect based on your experimental manipulation or other things that you know may have impacted the signal. Now what?

4. Model specificationnn Parameter estimation Hypothesis Statistic First level, within subject analysis.From the observed response in a voxel to single subject SPM Time Time BOLD signal single voxel time series (Y in GLM) Second level analysis SPM

5. Outline: • Modeling the BOLD-response; the design matrix • The GLM and parameter estimation • T and F contrasts and parametric designs • Statistical inference • An SPM example • Some remarks on orthogonal contrasts

6. Predicting the Bold-response I presented a flashing visual stimulus I predict that a voxel that responds looks like this: And a voxel that doesn’t care looks like this:

7. This prediction is formulated in the design matrix X 0 0 1 1 0 0 1 1 0 0 1 1 0 0 11 0 0 . . .. . • The design matrix is simply a mathematical description of your experiment • E.g.: ‘visual stimulus on = 1’ (white) ‘visual stimulus off = 0’ (black) • Or ‘no =0, medium=1, much pain=2’ • Each column of the design matrix corresponds to an effect one has built into the experiment or that may confound the results (RT, motion parameters etc) • To minimise the error term of the model, you want to model as much of Y as possible using variables specified in X Time Original design variables Mean Other sources of variation, convolved HRF (next week)

8. b1 b2 The GLM for fMRI: Y= Xxβ+ ε Observed data: Y: time-series of BOLD signal in a single voxel Parameters: The contribution of each component of the design matrix to the signal Y (aim to minimise error) Error: Difference between the observed data, Y, and the time-course predicted by the model, Xβ. Design matrix: Variables that explain the observed data, i.e. the BOLD time series for the voxel (EV) N= nr of scans P= nr of regressors P ´ + = N N P N

9. Finding the best fitting model: SPM finds a for each EV so that our total model as defined in the design matrix optimally fits the observed data in the voxel.

10. Finding the best fitting model: These optimal fitting values are saved in beta image files for each EV. The residual signal variance in the voxel, unexplained by the model (within subject error) is saved in MSres image files.

11. Methods: • Variational Bayes: allows for spatial priors for regression and regularized voxel-wise models for noise processing, see SPM manual or book • Least squares method (classic): finds which linear combination of X variables in design matrix results in minimal difference between observed data and predicted data by Gives: Minimizing:

12. Contrasts • But usually we are not interested in the whole model • What variation in the signal is the result of a visual stimulus and not of movement? • A contrast selects a specific effect of interest: • a contrast c is a vector of the length of the number of EVs in the design matrix. • cTβis a linear combination of regression coefficients β. cTβ = 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb5 + . . . cT = [1 0 0 0 0 …] cTβ = 0xb1+-1xb2 + 1xb3 + 0xb4 + 0xb5 + . . . cT = [0 -1 1 0 0 …] • SPM saves these linear combinations of coefficients as contrast images and feeds them up to higher level analysis

13. Define contrasts - Factorial design a vector of the length of your (here 4) EVs MOTION NO MOTION • SIMPLE MAIN EFFECT • A – B • Simple main effect of motion (vs. no motion) in the context of low load • [ 1 -1 0 0] • MAIN EFFECT • (A + B) – (C + D) • The main effect of low load (vs. high load) irrelevant of motion •  Main effect of load • [ 1 1 -1 -1] • INTERACTION • (A - B) – (C - D) • The interaction effect of motion (vs. no motion) greater under low (vs. high) load • [ 1 -1 -1 1] A B C D LOW LOAD HIGH A B C D A B C D A B C D

14. V A C1 C2 C3 From design 2 design matrix 2 contrast • Imaging a 2x3 factorial design with factors Modality (Auditory, Visual) and Condition (Concrete, Abstract, Proper) You can model it like this…but is it the best way? C1: Concrete nouns Visual C2: Abstract nouns C3: Proper nouns C1: Concrete nouns C2: Abstract nouns Auditory C3: Proper nouns

15. What can we test with this design matrix? V A C1 C2 C3 We can test for main effects: - Visual > Auditory? C: [1 -1 0 0 0] - Concrete > Abstract? C: [0 0 1 -1 0] But we can’t test for interactions or simple main effects: Visual/concrete > Auditory/concrete???

16. An orthogonal design matrix C2 C3 C1 C1 C2 C2 C3 C3 V A V A V A V A Just like in SPSS, you need to cross your variables in order to model interactions SPM will do this for you automatically if you have a factorial design – just input the factors and the number of levels

17. Statistical inference • For each voxel, we estimated the value of the regression coefficient for each variable (column) in our model (design matrix) • We defined contrasts to pick out variables or combinations of variables of interest in our model • a large value of cX reflects a large effect size (% signal change) of our contrast of interest • But data is noisy- so can we trust these effects are not due to chance?? • To find this out SPM can calculate t and f-test statistical maps for each contrast using beta images, contrasts, residual variance maps and means.

18. contrast ofestimatedparameters T = varianceestimate T-statistic - one dimensional contrasts – SPM{t} box-car amplitude > 0 ? = b1 = cTb> 0 ? Question: cT = 1 0 0 0 0 0 0 0 b1b2b3b4b5 ... H0: cTb=0 Null hypothesis: Test statistic:

19. T-contrast in SPM • For a given contrast c: ResMS image beta_???? images spmT_???? image con_???? image SPM{t}

20. F-statistic • F-tests can be viewed as testing for the additional variance explained by a larger model compared to a simpler model • Tests a weighted sum of squares of one or several linear combinations of the regression coefficients: is bidirectional (squared t-test) • Contrasts and hypotheses:

21. F-contrast in SPM ResMS image beta_???? images ess_???? images spmF_???? images ( RSS0 - RSS) SPM{F}

22. Hypothesis Testing u • Type I Error α: Acceptable false positive rate α. Level  threshold uα Threshold uα controls the false positive rate  Null Distribution of T Observation of test statistic t, a realisation of T • P-value: A p-value summarises evidence against H0. This is the change of observing value more extreme than t under the null hypothesis. t P-val • The conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > uα Null Distribution of T

23. Statistics: 1 p-values adjusted for search volume set-level cluster-level voxel-level mm mm mm p c p k p p p T Z p ( ) corrected E uncorrected FWE-corr FDR-corr uncorrected º 0.000 10 0.000 520 0.000 0.000 0.000 13.94 Inf 0.000 -63 -27 15 SPMresults: 0.000 0.000 12.04 Inf 0.000 -48 -33 12 0.000 0.000 11.82 Inf 0.000 -66 -21 6 10 Height threshold T = 3.2057 {p<0.001} 0.000 426 0.000 0.000 0.000 13.72 Inf 0.000 57 -21 12 0.000 0.000 12.29 Inf 0.000 63 -12 -3 voxel-level 0.000 0.000 9.89 7.83 0.000 57 -39 6 20 mm mm mm 0.000 35 0.000 0.000 0.000 7.39 6.36 0.000 36 -30 -15 T Z p ( ) 0.000 9 0.000 0.000 0.000 6.84 5.99 0.000 51 0 48 uncorrected º 0.002 3 0.024 0.001 0.000 6.36 5.65 0.000 -63 -54 -3 0.000 8 0.001 0.001 0.000 6.19 5.53 0.000 -30 -33 -18 30 13.94 Inf 0.000 -63 -27 15 0.000 9 0.000 0.003 0.000 5.96 5.36 0.000 36 -27 9 0.005 2 0.058 0.004 0.000 5.84 5.27 0.000 -45 42 9 12.04 Inf 0.000 -48 -33 12 0.015 1 0.166 0.022 0.000 5.44 4.97 0.000 48 27 24 40 11.82 Inf 0.000 -66 -21 6 0.015 1 0.166 0.036 0.000 5.32 4.87 0.000 36 -27 42 13.72 Inf 0.000 57 -21 12 12.29 Inf 0.000 63 -12 -3 50 9.89 7.83 0.000 57 -39 6 7.39 6.36 0.000 36 -30 -15 60 6.84 5.99 0.000 51 0 48 6.36 5.65 0.000 -63 -54 -3 70 6.19 5.53 0.000 -30 -33 -18 5.96 5.36 0.000 36 -27 9 80 5.84 5.27 0.000 -45 42 9 5.44 4.97 0.000 48 27 24 0.5 1 1.5 2 2.5 5.32 4.87 0.000 36 -27 42 Design matrix T-test: a simple example • Passive viewing versus rest cT = [ 1 0 ] Q: activation during viewing ? Null hypothesis:

24. Final remarks: • Some designs are less efficient than others • If EVs are correlated it is hard to disentangle which one drives the effect in the voxel, resulting in less power to detect a unique effect of that EV (e.g.: button press and target detection) • SPM gives you an estimation of this problem in the orthogonality matrix • depicts the magnitude of the cosine of the angle between the EVs as indicator of correlation (Dark is close to 1, highly correlated. White means Evs are orthogonal or uncorrelated, good).

25. References • MFD slides 2007 • Short SPM course slides (by Guillaume) • FSL course slides • SPM manual • SPM book, chapter 1, 7 and 8