Variance components Wellcome Dept. of Imaging Neuroscience Institute of Neurology, UCL, London Stefan Kiebel

Modelling in SPM pre-processing general linear model general linear model SPMs functional data templates smoothed normalised data smoothed normalised data design matrix variance components hypotheses adjusted P-values adjusted P-values parameter estimation parameter estimation

general linear model = + model specified by 1.design matrix X 2.assumptions about  model specified by 1.design matrix X 2.assumptions about  N: number of observations p: number of regressors N: number of observations p: number of regressors error  normally distributed error  normally distributed

Summary Sphericity/non-sphericity Restricted Maximum Likelihood (ReML) Estimation in SPM2

Summary Sphericity/non-sphericity Restricted Maximum Likelihood (ReML) Estimation in SPM2 Sphericity/non-sphericity

‚sphericity‘ ‚sphericity‘ means: Scans i.e.

‚non-sphericity‘ non-sphericity means that the error covariance doesn‘t look like this * : non-sphericity means that the error covariance doesn‘t look like this * : *: or can be brought through a linear transform to this form

Example: serial correlations with autoregressive process of order 1 (AR(1)) autocovariance- function

Restricted Maximum Likelihood (ReML) Summary Sphericity/non-sphericity Estimation in SPM2

Restricted Maximum Likelihood observed ReML estimated

t-statistic (OLS estimator) c = approximate degrees of freedom following Satterthwaite approximate degrees of freedom following Satterthwaite ReML- estimate

Variance components Variance components Q model the error Variance components Q model the error model for sphericity model for inhomogeneous variances (2 groups) model for inhomogeneous variances (2 groups) The variance parameters are estimated by ReML.

Example I Stimuli: Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards Subjects: e.g. “Book” and “Koob” e.g. “Book” and “Koob” fMRI, 250 scans per subject, block design Scanning: U. Noppeney et al. (i) 12 control subjects (ii) 11 blind subjects (i) 12 control subjects (ii) 11 blind subjects

Population differences 1 st level: 2 nd level: Controls Blinds

Estimation in SPM2 Summary Sphericity/non-sphericity Restricted Maximum Likelihood (ReML)

Estimating variances EM-algorithm E-step M-step K. Friston et al. 2002, Neuroimage Assume, at voxel j: Assume, at voxel j:

Time Intensity Time Time series in one voxel Time series in one voxel voxelwise model specification parameter estimation parameter estimation hypothesis statistic SPM

Spatial ‚Pooling‘ Assumptions in SPM2: global correlation matrix V local variance Assumptions in SPM2: global correlation matrix V local variance observed ReML estimated global local in voxel j:

Estimation in SPM2 ‚quasi‘-Maximum Likelihood Ordinary least-squares ReML (pooled estimate) optional in SPM2 one pass through data statistic using (approximated) effective degrees of freedom optional in SPM2 one pass through data statistic using (approximated) effective degrees of freedom 2 passes (first pass for selection of voxels) more precise estimate of V 2 passes (first pass for selection of voxels) more precise estimate of V

t-statistic (ML-estimate) c = ReML- estimate

Example II Stimuli: Auditory Presentation (SOA = 4 secs) of words Subjects: fMRI, 250 scans per subject, block design fMRI, 250 scans per subject, block design Scanning: U. Noppeney et al. (i) 12 control subjects MotionSoundVisualAction “jump”“click”“pink”“turn” Question: What regions are affected by the semantic content of the words? What regions are affected by the semantic content of the words?

Repeated measures Anova 1 st level: 2 nd level: Visual Action ?=?= ?=?= ?=?= Motion Sound