Hypothesis testing in variance components with constraints

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Hypothesis testing in variance components with constraints

M. Przystalski¹ M. Fonseca²

1The Research Center for Cultivar Testing, S lupia Wielka, Poland

2Centro de Mathemática e Aplica¸cões, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboba, Portugal

LinStat 2014

M. Przystalski, M. Fonseca Hypothesis testing in variance components with constraints LinStat 2014 1 / 32

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Introduction

For testing hypotheses related to variance components one uses e.g.

permutation test (e.g. Fitzmaurice at al., 2007), or generalized p-values (see e.g. Volaufova, 2009).

For inference on variance component the usual chi-square distribution of likelihood ratio statistic under the null is used.

E.g. Self and Liang (1987) had shown that this not necessarily hold. For LMM with one variance component, Crainiceanu and Ruppert (2004) had shown that the distribution of the LR statistic is mixture of two chi-squares.

This is due to the fact that the values of the variance component under the null lie on the boundry of the parameter space.

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Introduction

E.g. Self and Liang (1987) had shown that this not necessarily hold. For LMM with one variance component, Crainiceanu and Ruppert (2004) had shown that the distribution of the LR statistic is mixture of two chi-squares.

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Introduction

E.g. Self and Liang (1987) had shown that this not necessarily hold.

For LMM with one variance component, Crainiceanu and Ruppert (2004) had shown that the distribution of the LR statistic is mixture of two chi-squares.

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Introduction

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Introduction

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Introduction

Aims

1 Construct statiscal tests for variance components with posistivity constraints.

2 Investigate the perfomance of the proposed procedures.

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Introduction

Aims

1 Construct statiscal tests for variance components with posistivity constraints.

2 Investigate the perfomance of the proposed procedures.

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One-way model

Statisitcal model

y =µ1+Zα+e, where

y - vector of observed values, µ- general mean

1_N - vector of one’s

α - vector of random effects, Z=Ia⊗1n is design matrix, e - vector of errors

We assume that α ande are indendent and that α∼N 0, σ_a²I_a and e ∼N 0, σ²_eI_N

,respectively.

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One-way model

Statistical model

The problem of interest is:

H₀:σ²_a = 0 vs. H₁:σ²_a >0.

Alternatively, the problem of interest can be written as: H₀⁰ :γ1 =γ2 vs. H₁⁰ :γ1 > γ2,

whereγ₁ =nσ²_a+σ_e² andγ₂=σ_e² (see Khuri et al., 1998).

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One-way model

Statistical model

H₀:σ²_a = 0 vs. H₁:σ²_a >0.

Alternatively, the problem of interest can be written as:

H₀⁰ :γ1=γ2 vs. H₁⁰ :γ1 > γ2,

whereγ₁=nσ²_a+σ²_e andγ₂=σ_e² (see Khuri et al., 1998).

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One-way model

Statistical model

H₀:σ²_a = 0 vs. H₁:σ²_a >0.

H₀⁰ :γ1=γ2 vs. H₁⁰ :γ1 > γ2, whereγ1=nσ²_a+σ²_e andγ2=σ_e² (see Khuri et al., 1998).

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One-way model

Statistical model

The maximum likelihood estimators (MLE)

bµ= 1

N1⁰_Ny, γb1 = 1

a−1y⁰P1y bγ2= 1

N−ay⁰P2y, where

P₁ = I_a⊗J_n n − J_N

N P₂=I_a⊗I_n−I_a⊗J_n n .

E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;

E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²²

. By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have

Vγb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Statistical model

bµ= 1

N1⁰_Ny, γb1 = 1

N−ay⁰P2y, where

N P₂=I_a⊗I_n−I_a⊗J_n n . E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;

E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²²

.

By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have

Vγb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Statistical model

bµ= 1

N1⁰_Ny, γb1 = 1

N−ay⁰P2y, where

N P₂=I_a⊗I_n−I_a⊗J_n n . E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²² .

By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have

Vγb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Statistical model

bµ= 1

N1⁰_Ny, γb1 = 1

N−ay⁰P2y, where

N P₂=I_a⊗I_n−I_a⊗J_n n . E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²² . By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have

Vγb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Statistical model

bµ= 1

N1⁰_Ny, γb1 = 1

N−ay⁰P2y, where

N P₂=I_a⊗I_n−I_a⊗J_n n . E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²² . By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have V

γb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Statistical model

bµ= 1

N1⁰_Ny, γb1 = 1

N−ay⁰P2y, where

N P₂=I_a⊗I_n−I_a⊗J_n n . E(bγ1) =γ1 andV(bγ1)≈ _a−1^2γ²¹;E(bγ2) =γ2 and; V(γb2)≈ _N−a^2γ²² . By the independence of bγ1 andbγ2 (see Theorem 2.4.1, Khuri et al., 1998) we have V

γb =diag_2γ2 1

a−1,_N−a^2γ²²

V^−1/2

γb (γb−γ)→^d N(0,I₂). (1)

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One-way model

Likelihood ratio

Using the Self and Liang approach (Self and Liang, 1987) and (1)

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

In case γ₁ > γ₂

⇒ eγ₁ =bγ₁ andeγ₂=bγ₂ ⇒`₁₁= 0.

In case γ₁ ≤γ₂

⇒ γ₁ =γ₂ =γ

`12=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

⇒ eγ₁ =bγ₁ andeγ₂=bγ₂ ⇒`₁₁= 0.

⇒ γ₁ =γ₂ =γ

`12=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

⇒ γe₁ =bγ₁ andeγ₂=bγ₂ ⇒`₁₁= 0. In case γ₁ ≤γ₂

⇒ γ₁ =γ₂ =γ

`12=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

In case γ₁ > γ₂ ⇒ γe₁ =bγ₁ andeγ₂=γb₂

⇒`₁₁= 0. In case γ₁ ≤γ₂

⇒ γ₁ =γ₂ =γ

`12=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

In case γ₁ > γ₂ ⇒ γe₁ =bγ₁ andeγ₂=γb₂ ⇒`₁₁= 0.

⇒ γ₁ =γ₂ =γ

`12=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

⇒ γ₁ =γ₂ =γ

`₁₂=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

In case γ₁ ≤γ₂ ⇒ γ₁ =γ₂ =γ

`₁₂=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

`₁₂=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ1(a−1)

bγ1

γ −1

−bγ2(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

`1=−a−1 2

bγ1

γ1

−1 2

−N−a 2

γb2

γ2

−1 2

`₁₂=−a−1 2

bγ1

γ −1 2

−N−a 2

bγ2

γ −1 2

d`12

dγ⁻¹ =−bγ₁(a−1)

bγ1

γ −1

−bγ₂(N−a)

bγ2

γ −1

= 0.

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One-way model

Likelihood ratio

γ₁≤γ₂ ⇒

eγ = (a−1)bγ₁²+ (N−a)bγ₂² (a−1)bγ1+ (N−a)bγ2

, `12=−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

Under the null hypothesis

`₀=−a−1 2

bγ₁

γ −1 2

−N−a 2

bγ₂

γ −1 2

eγ = (a−1)γb₁²+ (N−a)bγ₂²

(a−1)γb₁+ (N−a)bγ₂ `0 =− (a−1) (N−a) (bγ1−bγ2)² 2 (a−1)bγ₁²+ (N−a)bγ₂²

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One-way model

Likelihood ratio

γ₁≤γ₂ ⇒

, `12=−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

`₀=−a−1 2

bγ₁

γ −1 2

−N−a 2

bγ₂

γ −1 2

eγ = (a−1)γb₁²+ (N−a)bγ₂²

(a−1)γb₁+ (N−a)bγ₂ `0 =− (a−1) (N−a) (bγ1−bγ2)² 2 (a−1)bγ₁²+ (N−a)bγ₂²

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One-way model

Likelihood ratio

γ₁≤γ₂ ⇒

, `12=−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

`₀=−a−1 2

bγ₁

γ −1 2

−N−a 2

γb₂

γ −1 2

eγ = (a−1)γb₁²+ (N−a)bγ₂²

(a−1)bγ₁+ (N−a)bγ₂ `0 =− (a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)bγ₂²

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One-way model

Likelihood ratio

γ₁≤γ₂ ⇒

, `12=−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

`₀=−a−1 2

bγ₁

γ −1 2

−N−a 2

γb₂

γ −1 2

eγ = (a−1)γb₁²+ (N−a)bγ₂²

(a−1)bγ₁+ (N−a)bγ₂ `0 =− (a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)bγ₂²

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One-way model

Likelihood ratio

γ₁≤γ₂ ⇒

, `12=−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

`₀=−a−1 2

bγ₁

γ −1 2

−N−a 2

γb₂

γ −1 2

eγ = (a−1)γb₁²+ (N−a)bγ₂²

(a−1)bγ₁+ (N−a)bγ₂ `₀ =−(a−1) (N−a) (bγ1−γb2)² 2 (a−1)bγ₁²+ (N−a)γb₂²

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One-way model

Likelihood ratio

Proposition

The LRT rejects the null hypothesis for large values of λgiven by:

λ=







0 bγ1≤bγ2

(a−1)(N−a)(bγ1−bγ2)²

(^(a−1)bγ₁²+(N−a)bγ₂²) bγ₁>bγ₂.

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One-way model

Random permutations

y = [y_ij] – lexicographical order

i – first factor j – replicates Permutation

y_l =Ply, where Pl is a random permutation matrix.

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One-way model

Permutation procedure

1 Calculate ˆγ₁ and ˆγ₂

2 Calculateλ

3 Generate a pseudo-sample of size M of

`_l =`(y_l) =`(P_ly)

4 Retrieve cα, the pseudo-sample’s empirical (1−α)-th quantile

5 Compare with the observed value λwith c_α

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One-way model

Application of delta method

Combining the delta method (see van der Vaart, 1998) with the Self and Liang approach (Self and Liang, 1987) we obtain the following proposition.

Proposition

The LRT rejects the null hypothesis H0 for a large values of κgiven by

κ=

(0 bγ1 ≤bγ2,

(a−1)(N−a)(logbγ1−logbγ2)²

2(N−1) bγ₁ >bγ₂.

Theorem

The distribution of the LRT κ is a mixture of χ²₀ andχ²₁ with coefficients p and 1-p, with

p =P(bγ1 ≤bγ2).

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One-way model

Proposition

The LRT rejects the null hypothesis H0 for a large values ofκ given by

κ=

(0 γb1 ≤bγ2,

2(N−1) γb₁ >bγ₂.

Theorem

p =P(bγ1 ≤bγ2).

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One-way model

Proposition

The LRT rejects the null hypothesis H0 for a large values ofκ given by

κ=

(0 γb1 ≤bγ2,

2(N−1) γb₁ >bγ₂.

Theorem

p =P(bγ1 ≤bγ2).

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Two-fold nested model

Statistical model

y =µ1_N+Z₁α+Z₂β+e, where

y - vector of observed values, µ- general mean,

1_N - vector of one’s,

α andβ- vectors of random effects,

Z1 =Ia⊗1b⊗1r andZ2 =Ia⊗Ib⊗1r - are design matrices, e - vector of errors

We assume that α,β ande are indendent and thatα∼N 0, σ_a²I_a , β∼N 0, σ_b²I_b

ande ∼N 0, σ_e²I_N

,respectively.

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Two-fold neste model

Statistical model

H₀:σ²_b= 0 vs. H₁:σ²_b>0.

Alternatively, the problem of interest can be written as: H₀⁰ :γ2 =γ3 vs. H₁⁰ :γ2 > γ3,

whereγ₁ =brσ_a²+rσ²_b+σ_e²,γ₂ =rσ²_b+σ²_e andγ₃=σ²_e, respectively (see Khuri et al, 1998).

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Two-fold neste model

Statistical model

H₀:σ²_b= 0 vs. H₁:σ²_b>0.

H₀⁰ :γ2=γ3 vs. H₁⁰ :γ2 > γ3,

whereγ₁=brσ_a²+rσ²_b+σ²_e,γ₂ =rσ²_b+σ²_e andγ₃=σ²_e, respectively (see Khuri et al, 1998).

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Two-fold neste model

Statistical model

H₀:σ²_b= 0 vs. H₁:σ²_b>0.

H₀⁰ :γ2=γ3 vs. H₁⁰ :γ2 > γ3, whereγ₁=brσ_a²+rσ²_b+σ²_e,γ₂ =rσ²_b+σ²_e andγ₃=σ²_e, respectively (see Khuri et al, 1998).

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Two-fold nested model

Statistical model

µb= 1

N1⁰_Ny, bγ₁ = y⁰P₁y

a−1 , bγ₂= yP₂y

a(b−1), bγ₃ = y⁰P₃y ab(r−1), whereP1= Ia−¹_aJa

⊗_br¹Jbr,P2=Ia⊗ Ib−¹_bJb

⊗¹_rJr, P₃=I_ab⊗ I_r −¹_rJ_r

.

E(bγ1) =γ1 andV(bγ1)≈ ^2γ_a¹²;

E(bγ2) =γ2 and;V (bγ2)≈ ^2γ_ab²²; E(bγ3) =γ3 and; V(bγ3)≈ ^2γ_abr²³.

Vγb =diag_2γ2 1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

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Two-fold nested model

Statistical model

µb= 1

.

E(bγ1) =γ1 andV(bγ1)≈ ^2γ_a¹²;

E(bγ2) =γ2 and;V (bγ2)≈ ^2γ_ab²²; E(bγ3) =γ3 and; V(bγ3)≈ ^2γ_abr²³.

Vγb =diag_2γ2 1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

(45)

Two-fold nested model

Statistical model

µb= 1

.

E(bγ1) =γ1 andV(bγ1)≈ ^2γ_a¹²;E(bγ2) =γ2 and;V (bγ2)≈ ^2γ_ab²²;

E(bγ3) =γ3 and; V(bγ3)≈ ^2γ_abr²³. Vγb =diag_2γ2

1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

(46)

Two-fold nested model

Statistical model

µb= 1

.

E(bγ1) =γ1 andV(bγ1)≈ ^2γ_a¹²;E(bγ2) =γ2 and;V (bγ2)≈ ^2γ_ab²²; E(bγ3) =γ3 and;V (bγ3)≈ ^2γ_abr²³.

Vγb =diag_2γ2 1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

(47)

Two-fold nested model

Statistical model

µb= 1

.

Vγb =diag_2γ2 1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

(48)

Two-fold nested model

Statistical model

µb= 1

.

Vγb =diag_2γ2 1

a ,^2γ_ab²²,^2γ_abr²³

V^−1/2

γb (γb−γ)→^d N(0,I3). (2)

(49)

Two-fold nested model

Likelihood ratio. H₁

bγ₁ ≥γb₂ ≥bγ₃

eγ₁ =bγ₁ eγ₂ =bγ₂ eγ₃ =bγ₃

−2`₁ = 0

(50)

Likelihood Ratio

H1

bγ₁ <bγ₂,bγ₂ ≥bγ₃

eγ12= bγ²₁+bγb₂² bγ1+bγb2

eγ₃ =bγ₃

−2`₁ = ab²(bγ₁γb₂−bγ²₂)²+ab(bγ₁bγ₂−bγ₁²)² (bγ₁²+bγb₂²)²

(51)

Two-fold nested model

bγ₁ ≥ ^b^γ²²^+r^b^γ³²

bγ2+rbγ3,bγ₂ <bγ₃

eγ1 =bγ1

eγ₂₃= bγ₂²+rbγ₃² bγ₂+rbγ₃

−2`₁= abr²(bγ₂γb₃−bγ²₃)²+abr(bγ₂bγ₃−bγ₂²)² (bγ₂²+rbγ₃²)²

(52)

Two-fold nested model

bγ1 < ^b^γ²²^+r^b^γ³²

bγ2+rbγ3,bγ2 <bγ3

γe₁₂₃= bγ₁²+bbγ₂²+brbγ₃² bγ1+bbγ2+brbγ3

−2`₁ = a b(bγ₁bγ₂−bγ₂²) +br(γb₁bγ₃−bγ₃²)2

(bγ₁²+bγb₂²+brbγ₃²)² +ab (bγ₁bγ₂−bγ₁²) +br(γb₂bγ₃−bγ₃²)2

(bγ₁²+bbγ₂²+brγb₃²)²

+abr (bγ1bγ2−bγ₁²) +b(γb1bγ2−bγ₂²)2

(53)

Two-fold nested model

Likelihood ratio. H₀ :γ₂ =γ₃

bγ₁ ≥ ^b^γ²²^+r^b^γ³²

bγ2+rbγ3

eγ1 =bγ1

eγ₂₃= bγ₂²+rbγ₃² bγ₂+rbγ₃

−2`₁= abr²(bγ₂γb₃−bγ²₃)²+abr(bγ₂bγ₃−bγ₂²)² (bγ₂²+rbγ₃²)²

(54)

Two-fold nested model

Likelihood ratio. H₀ :γ₂ =γ₃

bγ1 < ^b^γ²²^+r^b^γ³²

bγ2+rbγ3

γe₁₂₃= bγ₁²+bbγ₂²+brbγ₃² bγ1+bbγ2+brbγ3

−2`₀ = a b(bγ₁bγ₂−bγ₂²) +br(γb₁bγ₃−bγ₃²)2

(bγ₁²+bγb₂²+brbγ₃²)² +ab (bγ₁bγ₂−bγ₁²) +br(γb₂bγ₃−bγ₃²)2

+abr (bγ1bγ2−bγ₁²) +b(γb1bγ2−bγ₂²)2

(55)

Two-fold nested model

Random Permutations

y = [yijk] – lexicographical order

i – first factor j – second factor k – replicates Selective permutation

y_l = (I_a⊗P_l)y, where P_l is a random permutation matrix.