1. Functional Brain Signal Processing: EEG & fMRI Lesson 15 Kaushik Majumdar Indian Statistical Institute Bangalore Center kmajumdar@isibang.ac.in M.Tech. (CS), Semester III, Course B50

2. Group Statistics: An Example Poldrack et al., 2011, Section 6.1

3. Fixed Effect vs. Mixed Effect Model is fixed effect variance in average hair length in a gender group. is within group variance, which is assumed to be 1 here. is the average mixed effect variance, where is the between the groups variance.

4. Mixed Effect Model in fMRI Poldrack et al., 2011 Here β = a.

5. Linear Regression Analysis http://en.wikipedia.org/wiki/File:Linear_regression.svg

6. Multilevel GLM for Group Analysis Y k is a vector of T time points. k є {1,…..,N}, where N is the number of subjects in the group. M k is the design matrix of model functions and e k is the error vector (for the kth subject with one element in the vector for each time point.

7. Two Level Model Level 1: individual analysis Level 2: group analysis M G is the group level design matrix, a G is a vector of group level parameters d is the residual vector of group level parameters. Beckmann et al., 2003 (1)

8. Two Level Model as A Single Level Model This is equivalent to the two level model described in the previous slide.

9. Parameter Estimation at Two Levels Linear spaces generated by the columns of M and M T M are the same (Rao, 1974, p. 222). Proof: Let λ be an eigenvalue of M. Then there is an eigenvector v such that Mv = λv or M = λI M T = λI. So, M T Mv = M T λv = λ 2 v. In other words M and M T M have same eigenvectors and therefore generate the same eigen space.

10. Parameter Estimation (cont) In general Y = Ma may be inconsistent (may not have unique solution), but M T Ma = M T Y always has a unique solution in a, because M T Y is in the space generated by columns of M T M. Let â be a solution of M T Ma = M T Y, then (Y – Ma) T (Y – Ma) = [Y – Mâ + M(â – a)] T [Y – Mâ + M(â – a)] = (Y – Mâ) T (Y – Mâ) + (â – a) T M T M(â – a) ≥ (Y – Mâ) T (Y – Mâ). This shows that the minimum of (Y – Ma) T (Y – Ma) is (Y – Mâ) T (Y – Mâ) and is attained for

11. Parameter Estimation (cont) a = â, which is unique for all solutions â of M T Ma = M T Y.

12. Solution for Two Level GL Model For individual. For group. This together with (1) gives the second level estimation of the parameters.

13. Inference of BOLD Activation Poldrack et al., 2011

14. Nature of BOLD Signal Buxton, 2009 CBF = Cerebral blood flow. CMRO 2 = Cerebral metabolic rate of O 2. CBV = Cerebral blood volume.

15. BOLD Components BOLD response is primarily driven by CBF, but also strongly modulated by two other factors:, and M, which reflects level of deoxyhemoglobin at the baseline.

16. BOLD is Best Captured in Gradient Recall Echo (GRE) Imaging Buxton, 2009

17. References R. A. Poldrack, J. A. Mumford and T. E. Nichols, Handbook of Functional MRI Data Analysis, Cambridge University Press, Cambridge, New York, 2011. Chapter 6. C. F. Beckmann, M. Jenkinson and S. M. Smith, General multilevel linear modeling for group analysis in fMRI, NeuroImage, 20: 1052 – 1063, 2003.

18. References (cont) C. R. Rao, Linear Statistical Inference and Its Applications, 2e, Wiley Eastern Ltd., New Delhi, 1974, Chapter 4 (Theory of least squares and analysis of variance).

19. THANK YOU This lecture is available at http://www.isibang.ac.in/~kaushikhttp://www.isibang.ac.in/~kaushik