A log-normal distribution and twosample tests for the full diffusion tensor 227 T-AM Armin Schwartzman, Robert F. Dougherty, Jonathan E. Taylor Departments of Statistics and Psychology, Stanford University, California, USA PURPOSE Normal distribution for symmetric matrices: To develop formal testing tools for group comparisons of DTI maps using eigenvalues and eigenvectors, or both. BACKGROUND • DTI data: 3 × 3 positive definite matrix at each voxel. • Previous DTI group studies use only scalar measures (e.g. FA – Deutsch et al.) and principal diffusion direction (Schwartzman et al.). • Simulated example: ⎛ 4 0⎞ ⎟⎟ , σ = 0.5 , n = 100 exp(M ) = ⎜⎜ ⎝ 0 1⎠ (2π)q / 2 σ q q=6 (# independent entries) ⎛ 1 2⎞ exp⎜ − 2 tr (Y − M ) ⎟ ⎝ 2σ ⎠ f (Y ) = Y1 = V1 Λ1V1 ' , Y2 = V2 Λ2V2 ' , Y = VΛV ' • Test H0: M1=M2: The space of positive definite matrices: T= ⎧a > 0, b > 0 ⎛a c⎞ ⎜⎜ ⎟⎟ > 0 ⇔ ⎨ 2 ⎝ c b⎠ ⎩ab − c > 0 ⎛ 1 .1 0 ⎞ ⎟⎟ X 1 = ⎜⎜ ⎝ 0 0.4 ⎠ 1 ⎛ N (0,1) N (0, 12 ) N (0, 12 ) ⎞ ⎜ ⎟ Z =⎜ ∗ N (0,1) N (0, 12 ) ⎟ ⎜ ∗ ∗ N (0,1) ⎟⎠ ⎝ • Mean M, variance σ2: • Estimates (assume common variance σ2): METHODS c ( ) 1 ⎛ 1 ⎞ f (Z ) = exp⎜ − tr Z 2 ⎟ (2π)q / 2 ⎝ 2 ⎠ Two-sample testing: • New statistical methods needed for testing full set of eigenvalues and eigenvectors in tensor imaging data. • E.g.: 2 × 2 case: • Standard normal: n1n2 2 tr (Y1 − Y2 ) ~ F (q, q(n − 2)) H0 qns 2 ⎛ 0 .4 0 ⎞ ⎟⎟ X 2 = ⎜⎜ ⎝ 0 1.1⎠ n ⎞ 1 ⎛ n1 ⎜ ∑ tr (Yi − Y1 )2 + ∑ tr (Yi − Y2 )2 ⎟ ⎜ ⎟ q(n − 2) ⎝ i =1 i = n1 +1 ⎠ • Test H0: M1, M2 have same eigenvectors (assume same eigenvalues): T= • Test H0: M1, M2 have same eigenvalues (regardless of eigenvectors): nn 2 T = 1 22 tr ( Λ1 − Λ2 ) pns s2 = n →∞ ~ F ( p, q(n − 2)) H0 n (q − p )s 2 ⎛⎛ n Λ + n Λ ⎞ ⎞ tr⎜ ⎜ 1 1 2 2 ⎟ − Λ2 ⎟ ⎜⎝ ⎟ n ⎠ ⎝ ⎠ 2 n →∞ ~ F (q − p, q(n − 2)) H0 3 Y1 60 Y Y2 DATA EXAMPLE a b Arithmetic mean Geometric mean 0 ⎞ ⎛ 0.75 ⎟ ⎜⎜ 0 0.75 ⎟⎠ ⎝ 0 ⎞ ⎛ 0.65 ⎟ ⎜⎜ 0 0.65 ⎟⎠ ⎝ • Straight line violates boundaries. • Log-straight line stays within space. Log transform: • Scalars: • Matrices: x∈ℝ+ X∈Sym+(p) log exp log exp y∈ℝ Y∈Sym(p) • Positive definite matrices are transformed into symmetric matrices with no restrictions on their coefficients. X i = Vi ΛiVi ' ⇒ Yi = log( X i ) = Vi log( Λi )Vi ' References • • • • • Source Deutsch et al.: 2 groups, 6 subjects in each group. • Voxelwise test for equality of eigenvectors between group means. • E.g.: voxel with direction differences in corpus callosum / corona radiata area (Schwartzman et al.) Group 1 Group 2 Test statistic map Matrix geometric mean across subjects (principal diffusion direction shown) p-values 1 10-1 10-2 10-3 10-4 10-5 CONCLUSIONS We have developed an integrated representation of diffusion tensors in log-space including: • Removal of positive-definiteness constraints. • A normal distribution for symmetric matrices. Website: • Formal two-sample tests of full tensor, eigenvalues or eigenvectors. www.stanford.edu/~armins Chikuse (2003). Statistics on special manifolds. Springer-Verlag, New York. Deutsch et al. (2005). Correlations between white matter microstructure and reading performance in children. Cortex, 41(3): 354-363. Schwartzman et al. (2005). Analysis tools for DTI maps using the full diffusion tensor. Poster, HBM 11th meeting, Toronto, Canada. Schwartzman et al. (2005). Cross-subject comparison of principal diffusion direction maps. Mag Reson Med, 53: 1423-1431. Acknowledgments William R. and Sara Hart Kimball Stanford Graduate Fellowship, NIH EY-015000 and the Schwab Foundation for Learning.