Methods for Dummies Second level Analysis (for fMRI) Chris Hardy, Alex Fellows Expert: Guillaume Flandin

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 2

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 3

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1 st Level Analysis is within subject Y = X x β + fMRI scans Time (e.g. TR = 3s) Time Voxel time course 5

fMRI dataDesign MatrixContrast Images SPM{t} Subject 1 Subject 2 … Subject N 6

1 st Level Analysis Spatial preprocessing Movement in scanner Fitting to standard space (MNI) Smoothing etc for statistical power 7

Overview of today’s talk 1.Recap of first level analysis 2.Second level analysis 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Multiple conditions 6.Practical demonstration 7.Questions 8

RealignmentSmoothing Normalisation General linear model Statistical parametric map (SPM) Parameter estimates Design matrix Template Kernel Gaussian field theory p <0.05 Statisticalinference 9

2 nd level analysis – across subjects It isn’t enough to look just at individuals. So, we need to look at which voxels are showing a significant activation difference between levels of X consistently within a group. 1.Average contrast effect across sample 2.Variation of this contrast effect 3.T-tests 10

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 11

Fixed effects analysis Each subject in an experiment repeats trials of each type many times. The variation among the responses for each level of the design matrix (X) for a given subject gives us the within-subject variance, σ w 2. So, if we take the group effect size as the mean of responses across our subjects and analyse it with respect to σ w 2. 12

Fixed effects analysis (FFX) Subject 1 Subject 2 Subject 3 Subject N … Modelling all subjects at once 13

Fixed effects analysis (FFX) =+ Modelling all subjects at once  Simple model  Lots of degrees of freedom 14

Effect size, c ~ 4 σ w 2 : 0.9 For voxel v in the brain Subject 1 15

Effect size, c ~ 2 σ w 2 : 1.3 For voxel v in the brain Subject 2 16

Effect size, c ~ 4 σ w 2 : 0.7 For voxel v in the brain Subject 12 17

Whole Group – FFX calculation N subjects = 12 c = [4,3,2,5…] Within subject variability, σ w 2 = [0.9,1.2,1.5,0.5, …] Mean group effect = 2.67 Mean σ w 2 = 1.04 Standard Error Mean (SEM) = σ w 2 /(sqrt(N))=0.087 t=M/SEM = 30.69, p=

Fixed effects analysis (FFX) =+ Modelling all subjects at once  Large amount of data  Assumes common variance over subjects at each voxel 19

Random effects analysis (RFX) Synonymous with ‘mixed effects models’. Assumes our sample is a set of individuals taken at random from the population of interest. To do this we need to consider the between subject variance, σ b 2, as well as σ w 2 – and estimate the likely variance of the population from which our sample is derived. 20

Interim summary: Fixed vs Random (from Poldrack, Mumford and Nichol’s ‘Handbook of fMRI analyses’) 21

“Mixed effects models should be used whenever data are grouped within certain levels of a population and inferences are to be applied to the entire population.” - Mumford and Poldrack (2007) 22

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 23

Hierarchical Summary statistics approach Random effects analysis (RFX) - Methods 24

Methods for Random Effects Hierarchical Most accurate method – gold standard Set up a GLM containing parameters for the effects and variances at both the subject AND group levels, to all be estimated at the same time. Estimates subject and group statistics via “iterative looping” Computationally demanding 25

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Summary statistics 1 st level design for all subjects must be the SAME Uses the same sample means as in the first level analysis Gives exact same results as hierarchical model (when the within-subject variance is the same for all subjects) Validity is undermined by presence of extreme outliers. Methods for random effects 27

Whole Group – RFX Calculation N subjects = 12 c = [4,3,2,5…] Mean group effect = 2.67 Mean σ b 2 (SD) = 1.07 Standard Error Mean (SEM) = σ b 2 /(sqrt(N))=0.31 t=M/SEM = 8.61, p=

29 Contrast Images fMRI dataDesign Matrix Subject 1 … Subject N First level Generalisability, Random Effects & Population Inference. Holmes & Friston, NeuroImage,1998. Second level One-sample second level

Robustness Summary statistics Summary statistics Hierarchical Model Hierarchical Model Mixed-effects and fMRI studies. Friston et al., NeuroImage,

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 31

Button- pressing 32

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Directory To write to Design Scans: select con.*img from 1 st level Several design options: 1 sample t-test 2 sample t-test Paired t-test Multiple regression 1 way ANOVA Full or flexible factorial Set-up options 34

Set-up options Covariates Input covariates & nuisance variables here 1 value per con*.img Masking Specifies voxels within image which are to be assessed Implicit is default 35

(scroll down…) (For PET only) Specify 2 nd level Set-Up ↓ Save 2 nd level Set-Up ↓ Run analysis ↓ Look at the RESULTS 36

Results 37

e.g. 2 nd level 1-sample t-test Select t-contrast Define new contrast c = +1 (e.g. A>B) c = -1 (e.g. B>A) 38

Select options for displaying results: Correct for multiple comparisons – FWE/FDR. 39

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Summary For fMRI data, usually preferable to use RFX analysis, not FFX Hierarchical models provide gold standard for RFX, BUT computationally intensive Summary statistics = robust method for RFX group analysis 43

Overview of today’s talk 1.Recap of first level analysis 2.What is second level analysis? 3.Fixed vs. random effects 4.How do we analyse random effects? 5.Practical demonstration 6.Questions 44

Resources Previous MfD slides (Bex Bond and Samira Kazan) (Camilla Clark and Cat Slattery) Slides from Guillaume Flandin’s talk in Zurich, Feb 2014 Mumford, J. A., & Poldrack, R. A. (2007). Modeling group fMRI data. Social cognitive and affective neuroscience, 2(3), Friston, K. J., Stephan, K. E., Lund, T. E., Morcom, A., & Kiebel, S. (2005). Mixed-effects and fMRI studies. Neuroimage, 24(1),