Random Effects Analysis Will Penny Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 2004

Summary Statistic Approach 1st Level 2nd Level Data Design Matrix Contrast Images 1 ^ SPM(t) 1 ^ 2 ^ 2 ^ 11 ^ 11 ^  ^ One-sample t-test @2nd level 12 ^ 12 ^

Validity of approach ^ ^ Effect size Gold Standard approach is EM – see later – estimates population mean effect as MEANEM the variance of this estimate as VAREM For N subjects, n scans per subject and equal within-subject variance we have VAREM = Var-between/N + Var-within/Nn In this case, the SS approach gives the same results, on average: Avg[a] = MEANEM Avg[Var(a)] =VAREM ^ ^ Effect size

Example: Multi-session study of auditory processing SS results EM results Friston et al. (2004) Mixed effects and fMRI studies, Submitted.

Two populations Estimated population means Contrast images Two-sample t-test @2nd level Patients Controls One or two variance components ?

The General Linear Model y = X  + e N  1 N  L L  1 N  1 Error covariance N 2 Basic Assumptions Identity Independence N

Multiple variance components K y = X  + e N  1 N  L L  1 N  1 =1 Error covariance N Errors can now have different variances and there can be correlations N K=2

Estimating variances y = X  + e EM algorithm E-Step ( ) y C X T 1 - = e q h M-Step r for i and j { } { Q tr J g i j ij k å + l y = X  + e N  1 N  L L  1 N  1 Friston, K. et al. (2002), Neuroimage

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

} } Population Differences Covariance Matrix Design matrix Controls Blinds 1st Level 2nd Level } Contrast vector for t-test Covariance Matrix } Design matrix Difference of the 2 group effects