Methods for Dummies Second level Analysis (for fMRI)

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

Overview of today’s talk • Recap of first level analysis • What is second level analysis? • Fixed vs. random effects • How do we analyse random effects? • Practical demonstration • Questions

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

SPM{t} fMRI data Design Matrix Contrast Images Subject 1 Subject 2 … Subject N

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

Overview of today’s talk • Recap of first level analysis • Second level analysis • Fixed vs. random effects • How do we analyse random effects? • Multiple conditions • Practical demonstration • Questions

Statistical parametric map (SPM) Design matrix Kernel Realignment Smoothing General linear model Gaussian field theory Statistical inference Normalisation p <0.05 Template Parameter estimates

2nd 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. • Average contrast effect across sample • Variation of this contrast effect • T-tests

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, σw2. • So, if we take the group effect size as the mean of responses across our subjects and analyse it with respect to σw2.

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

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

Subject 1 For voxel v in the brain Effect size, c ~ 4 σw2 :0.9

Whole Group – FFX calculation • N subjects = 12 c = [4,3,2,5…] Within subject variability, σw2 = [0.9,1.2,1.5,0.5, …] • Mean group effect = 2.67 • Mean σw2= 1.04 • Standard Error Mean (SEM) =σw2 /(sqrt(N))=0.087 t=M/SEM = 30.69, p=10-16

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

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, σb2, as well as σw2 – and estimate the likely variance of the population from which our sample is derived.

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

“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)

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

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

Methods for random effects Summary statistics • 1st 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.

Whole Group – RFX Calculation • N subjects = 12 c = [4,3,2,5…] • Mean group effect = 2.67 • Mean σb2(SD) = 1.07 • Standard Error Mean (SEM) =σb2 /(sqrt(N))=0.31 t=M/SEM = 8.61, p=10-6

First level Second level Contrast Images fMRI data Design Matrix One-sample t-test @ second level Subject 1 … Subject N Generalisability, Random Effects & Population Inference. Holmes & Friston, NeuroImage,1998.

Robustness Summary statistics Hierarchical Model Mixed-effects and fMRI studies. Friston et al., NeuroImage, 2005.

Button- pressing

Set-up options • Directory • To write to • Design • Scans: select con.*img from 1st 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 • Covariates • Input covariates & nuisance variables here • 1 value per con*.img • Masking • Specifies voxels within image which are to be assessed • Implicit is default

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

Results

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

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

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

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), 251-257. Friston, K. J., Stephan, K. E., Lund, T. E., Morcom, A., & Kiebel, S. (2005). Mixed-effects and fMRI studies. Neuroimage, 24(1), 244-252.

Methods for Dummies Second level Analysis (for fMRI)