Hypothesis testing in multilevel models with block circular covariance structures

Transcription

1 1/ 25 Hypothesis testing in multilevel models with block circular covariance structures Yuli Liang 1, Dietrich von Rosen 2,3 and Tatjana von Rosen 1 1 Department of Statistics, Stockholm University 2 Department of Energy and Technology, Swedish University of Agricultural Sciences 3 Department of Mathematics, Linköping University Presentation at LinStat2014, Linköping (24-28 August, 2014)

2 2/ 25 Circular dependence: Circular Toeplitz (CT) matrix τ 0 τ 1 τ 2 τ 1 τ 1 τ 0 τ 1 τ 2 τ 2 τ 1 τ 0 τ 1 τ 1 τ 2 τ 1 τ 0

3 3/ 25 CT matrix, cont. An n n matrix T of the form t 0 t 1 t 2 t 1 t 1 t 0 t 1 t 2. T = t 2 t 1 t t 1 t 2 t 1 t 0 = T oep(t 0, t 1, t 2..., t 1 ) is called a symmetric circular Toeplitz matrix. The matrix T = (t ij ) depends on [n/2] + 1 parameters, where [.] stands for the integer part, and for i, j = 1,..., n, t ij = { t j i j i [n/2], t n j i otherwise.

4 4/ 25 A specific structure Flower 1 Flower Flower n 2 P 2 P 2 P 2 P 3 P 1 P 3 P 1 P 3 P 1 P 4 P 4 P 4

5 5/ 25 Outline Model and hypotheses Model setup Hypotheses

6 6/ 25 Outline Model and hypotheses Model setup Hypotheses Previous work of symmetry model

7 7/ 25 Outline Model and hypotheses Model setup Hypotheses Previous work of symmetry model External test

8 8/ 25 Outline Model and hypotheses Model setup Hypotheses Previous work of symmetry model External test Simulation study

9 9/ 25 Outline Model and hypotheses Model setup Hypotheses Previous work of symmetry model External test Simulation study Internal test

10 10/ 25 Balanced three-level model y k = µ1 p + Z 3 α + Z 2 β + ɛ k, k = 1,..., n, where p = n 2 n 1, Z 3 = I n2 1 n1 and Z 2 = I n2 I n1, Cov(α) = V 3 0, Cov(β) = V 2 0 and V ar(ɛ k ) = σ 2 I p > 0, α and β are independent. y k N p (µ1 p, Σ) and Σ = Z 3 V 3 Z 3 + V 2 + σ 2 I p. Y N p,n (µ1 p 1 n, Σ, I n ), where Y = (y 1 : y 2 :... : y n ) are n independent samples.

11 11/ 25 External test Hypotheses at macro-level : test the global structures of Σ H 1 : Σ I = I n2 Σ 1 + (J n2 I n2 ) Σ 2, where Σ h, h = 1, 2, is a n 1 n 1 unstructured matrix. H 2 : Σ II = I n2 Σ 1 + (J n2 I n2 ) Σ 2, where Σ h, h = 1, 2, is a CT matrix and depends on r parameters, r = [n 1 /2] + 1. H 3 : Σ III = I n2 Σ 1 + (J n2 I n2 ) Σ 2, where Σ h, h = 1, 2, is a CS matrix and can be written as Σ h = σ h1 I n1 + σ h2 (J n1 I n1 ). The number of parameters are n 1 (n 1 + 1), 2r and 4, respectively.

12 12/ 25 If n 2 = 4, n 1 = 4, then τ 0 τ 1 τ 2 τ 1 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 1 τ 0 τ 1 τ 2 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 2 τ 1 τ 0 τ 1 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 1 τ 2 τ 1 τ 0 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 3 τ 4 τ 5 τ 4 τ 0 τ 1 τ 2 τ 1 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 4 τ 3 τ 4 τ 5 τ 1 τ 0 τ 1 τ 2 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 5 τ 4 τ 3 τ 4 τ 2 τ 1 τ 0 τ 1 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 4 τ 5 τ 4 τ 3 τ 1 τ 2 τ 1 τ 0 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 0 τ 1 τ 2 τ 1 τ 3 τ 4 τ 5 τ. 4 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 1 τ 0 τ 1 τ 2 τ 4 τ 3 τ 4 τ 5 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 2 τ 1 τ 0 τ 1 τ 5 τ 4 τ 3 τ 4 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 1 τ 2 τ 1 τ 0 τ 4 τ 5 τ 4 τ 3 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 0 τ 1 τ 2 τ 1 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 1 τ 0 τ 1 τ 2 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 2 τ 1 τ 0 τ 1 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 4 τ 5 τ 4 τ 3 τ 1 τ 2 τ 1 τ 0

13 13/ 25 Selected previous work of symmetry model Olkin and Press (1969), Olkin (1973) Andersson (1975), Perlman (1987) Nahtman (2006) and Nahtman and von Rosen (2008) studied properties of some patterned covariance matrices arising under different symmetry restrictions in balanced mixed linear models. Roy and Fonseca (2012): double exchangeability

14 14/ 25 Canonical reduction and equivalent hypotheses Lemma (Arnold, 1973) Suppose Y N p,n (µ1 p 1 n, Σ I, I n ), where p = n 2 n 1 and µ is an unknown scalar parameter. Let Γ 1 be an n 2 n 2 orthogonal matrix whose first column is proportional to 1 n1 and put (Y 1.Y 2) = (Γ 1 I n1 )Y, where Y 1 : n 1 n and Y 2 : n 1 (n 2 1) n. Then, Y 1 and Y 2 are independently distributed, and Y 1 ( N n1,n n2 µ1 n1 1 ) n, 1, I n, (1) Y 2 N n1 (n 2 1),n (0, I n2 1 2, I n ), (2) where 1 = Σ 1 + (n 2 1)Σ 2 and 2 = Σ 1 Σ 2. Moreover, Σ I is positive definite if and only if both 1 and 2 are positive definite.

15 15/ 25 Theorem Let Γ 2 be the orthogonal matrix whose columns v 1,..., v n1 are the known orthonormal eigenvectors of any n 1 n 1 CT matrices. To test H i, i = 1, 2, 3, is respectively equivalent to test H 1 : (Γ 1 I n1 )Σ(Γ 1 I n1 ) = block diag( 1, 2 = = 2 ), H 2 : Γ 2 1 Γ 2 = diag(λ 1,, λ n1 ), λ i = λ n1 i+2, Γ 2Λ 2 Γ 2 = = Γ 2 2 Γ 2 = diag(λ n1+1,, λ 2n1 ) = = diag(λ (n2 1)n 1+1,, λ n2n 1 ) and λ (j 1)n1+i = λ jn1 i+2, i = 2,..., [n 1 /2] + 1, j = 2,..., n 2, assuming H 1, H 3 : λ 2 = = λ n1, λ n1+1 = λ 2n1+1 = = λ (n2 1)n 1+1, λ n1+2 = = λ 2n1 = = λ (n2 1)n 1+2 = = λ n2n 1 assuming H 2.

16 16/ 25 Likelihood ratio test H 0 : Σ = Σ II and H A : Σ = Σ I, i.e. test one exchangeable hierarchy in data and the other level is circularly correlated versus one exchangeable hierarchy and the other unstructured level in data. λ 21 = T 2 n(n 2 1) 2r i=1 n 2 1 n/2 T 1 n ( ), mi t2i nm i where t 21 = tr(t 1 P n1 ), t 2i = tr(t 1 (v i v i + v n 1 i+2v n 1 i+2 )), t 2(r+1) = tr(t 2 P n1 ) and t 2(r+i) = tr(t 2 (v i v i + v n 1 i+2v n 1 i+2 )), i = 2,..., r.

17 17/ 25 LRT, cont. H 0 : Σ = Σ III and H A : Σ = Σ I, i.e. test the doubly exchangeable hierarchical data versus one exchangeable hierarchy and the other unstructured level in data. λ 31 = T 2 n(n 2 1) 4 j=1 n 2 1 n/2 T 1 n ( ), t3j mj nm j where t 31 = tr(t 1 P n1 ), t 32 = tr(t 1 Q n1 ), t 33 = tr(t 2 P n1 ) and t 34 = tr(t 2 Q n1 ).

18 18/ 25 LRT, cont. H 0 : Σ = Σ III and H A : Σ = Σ II, i.e. test the doubly exchangeable hierarchical data versus one exchangeable hierarchy in data and the other level is circularly correlated. λ 32 = ( t32 n(n 1 1) r i=2 ( t2i nm i ) mi 2r i=r+2 ) n1 1 ( t 34 n(n 2 1)(n 1 1) ( t2i nm i ) mi ) (n2 1)(n 1 1) n/2.

19 19/ 25 n 2 = 3, n 1 = 4 Tabela : Type I error probabilities of LRT for α = 5% n λ 21 λ 32 λ

20 20/ 25 Internal test Hypotheses at micro-level : testing the specific variance components given Σ has a block circular structure. Assuming V 3 = σ 1 I n2 + σ 2 (J n2 I n2 ) and V 2 has the same structure of Σ II. Σ = In2 Σ 1 + (J n2 I n2 ) Σ 2, where Σ h, h = 1, 2, is also a CT matrix. Σ1 = T oep(σ 2 + σ 1 + τ 1, σ 1 + τ 2,..., σ 1 + τ 2 ) and Σ 2 = T oep(σ 2 + τ r+1, σ 2 + τ r+2,..., σ 2 + τ r+2 ). θ = (σ 2, σ 1, σ 2, τ 1,..., τ 2r ) and η = (η 1,..., η 2r )

21 1/ 25 Remarks The model does not have explicit expression since the number of unknown parameters in Σ (2r + 3) is more than the number of distinct eigenvalues of Σ (2r). (Szatrowski, 1980) Different restricted models are considered. (Liang et al., 2014) Testing hypotheses means to impose restrictions on restricted models. possible to derive equivalent hypotheses through the distinct eigenvalues η

22 22/ 25 Equivalent hypotheses Theorem For the model under the restriction K 1 θ = 0, the following hypotheses are equivalent: (i) σ 2 = 0, (ii) η 1 = η r+1. λ 2/n 1 d [ X(1 X) n2 1, with probability P F (n 1, n(n 2 1)) [ 1, with probability 1 P F (n 1, n(n 2 1)) where X Beta( n 1 2, n(n2 1) 2 ). ] n n 1, n n 1 ]

23 23/ 25 Theorem For the model under the restriction K 1 θ = 0, the following hypotheses are equivalent: (i) τ 2 =... = τ r and τ r+2 =... = τ 2r, (ii) η 2 =... = η r and η r+2 =... = η 2r. λ 2/n 2 d i B m i i, i {2,..., r} {r + 2,..., 2r}. where B i Beta( nm i 2, n( r i=2 m i m i ) 2 ).

24 References 24/ 25 Andersson, S. (1975). Invariant normal models. Annals of Statistics, 3, Arnold, S. F. (1973). Application of the theory of products of problems to certain patterned covariance matrices. Annals of Statistics, 1, Nahtman, T. (2006). Marginal permutation invariant covariance matrices with applications to linear models. Linear Algebra and its Applications, 417, Nahtman, T. and von Rosen, D. (2008). Shift permutation invariance in linear random factor models. Mathematical Methods of Statistics, 17, Olkin, I. (1973). Testing and estimation for structures which are circularly symmetric in blocks. In D. G. Kabe and R. P. Gupta, eds., Multivariate statistical inference , North Holland, Amsterdam. Olkin, I. and Press, S. (1969). Testing and estimation for a circular stationary model. The Annals of Mathematical Statistics, 40, Perlman, M. D. (1987). Group symmetry covariance models. Statistical Science, 2, Roy, A. and Fonseca, M. (2012). Linear models with doubly

25 25/ 25 Thank you for your attention!