The Geometry of the Wilks’s Λ Random Field

F. CarbonellInstitute for Cybernetics, Mathematics and Physics

Havana

L. GalanCuban Neuroscience Center

Havana

K. J. WorsleyMcGill University

Montreal

August 19, 2005

Abstract

The statistical problem addressed in this paper to approximate the P-value of the maximumof a smooth random field of Wilks’s Λ statistics. So far results are only available for the usualunivariate statistics (Z, t, χ2, F ) and a few multivariate statistics (Hotelling’s T 2, maximumcanonical correlation, Roy’s maximum root). We derive results for any differentiable scalarfunction of two independent Wishart random fields, such as Wilks’s Λ and Pillai’s trace. Weapply our results to a problem in brain shape analysis.

Key words and phrases: Multivariate Random Fields, Excursion Sets, Euler Characteristic,Derivatives of Matrix Functions.

MSC 2000: 60G60, 60D05,62M40,

1 Introduction

The motivation for this work came from practical applications in brain imaging. Changes in brainshape can be inferred from the 3D non-linear deformations required to warp a patient’s MRI scanto an atlas standard. Using these vector deformations as a dependent variable, we can use amultivariate multiple regression model to detect regions of the brain whose shape is correlatedwith external regressors, such as disease state. The statistic of choice is Wilks’s Λ, the likelihoodratio statistic (Anderson 1984), evaluated at each voxel in the brain. The challenging statisticalproblem is how to assess the P-value of local maxima of a smooth 3D random field (RF) of Wilks’sΛ statistics.

This problem of assessing the P-value of local maxima was studied in the seminal works ofAdler & Hasofer (1976) and Hasofer (1978) for the case of Gaussian RFs. The key idea consistson approximating the P-value that the local maximum exceeds a high threshold value via thegeometry of the excursion set, the set of points where the RF exceeds the threshold value. For highthresholds, the excursion sets consists of isolated regions, each containing a local maxima. TheEuler characteristic (EC) of the excursion sets counts the number of such connected components.For high thresholds near the maximum of the RF, the EC takes the value one if the maximum is

1

above the threshold, and zero otherwise. Thus, for high thresholds, the expected EC approximatesthe P-value of the maximum of the RF.

During the last decade this approach has became an important tool in different areas of appliedprobability. This is due to the many applications that have emerged principally in medical imagingand astrophysics (Gott, Park, Juskiewicz, Bies, Bennett, Bouchet & Stebbins 1990, Beaky, Scherrer& Villumsen 1992, Vogeley, Park, Geller, Huchira & Gott 1994, Worsley 1995a, Worsley 1995b,Worsley, Cao, Paus, Petrides & Evans 1998, Cao & Worsley 2001). So far, results are availablefor some fields of standard univariate statistics such as Z, t, χ2 and F RFs (Worsley 1994). Someother useful extensions were obtained in Cao & Worsley (1999a) and Cao & Worsley (1999b) forHotelling’s T 2 and the cross-correlation RFs, respectively. However, the extension of these resultsto other RFs of classical multivariate statistics has not so far been fully studied (Worsley, Taylor,Tomaiuolo & Lerch 2004). RFs of multivariate statistics appear when dealing with multivariate RFdata and multiple contrasts, where the classical example is the Wilks’s Λ statistic mentioned above.

The purpose of this work is to extend Adler’s results to a wide class of RFs of multivariatestatistics to which the Wilks’s Λ RF belongs. We build these random fields from Gaussian randomfields in the same way as we build the statistics themselves. First let Z = Z(t), t ∈ RN , be asmooth isotropic Gaussian random field with zero mean, unit variance, and with

Var(Z) = IN , (1)

where dot represents derivative with respect to t, and IN is the N ×N identity matrix. Note thatif we have another isotropic Gaussian ransom field Z∗ with Var(Z∗) = v2IN and v 6= 1, then we canalways re-scale the parameter t by dividing by v to define Z(t) = Z∗(t/v) so that (1) is satisfied.

Now suppose that Z(t) is an ν ×m matrix independent Gaussian random fields each with thesame distribution as Z(t). Then the Wishart random field with ν degrees of freedom is defined as

W (t) = Z(t)′Z(t) ∼ Wishartm(Im, ν) (2)

at each fixed t (Cao & Worsley 1999b). Let U(t) and V(t) be independent Wishart random fieldswith ν and η degrees of freedom, respectively, whose component Gaussian random fields all havethe same distribution as Z(t) above. We shall be concerned with the class of RFs Λ(t) defined as

Λ (t) = Λ (U (t) ,V (t)) , (3)

where Λ is any differentiable function of two matrix arguments. Our aim is to provide approximatevalues for the expected EC of the excursion sets of Λ(t).

The plan of the paper is the following. In Section 2 we shall state the notation as well as somepreliminaries facts about RF theory and matrix differential theory. In the Section 3 we obtainexpressions for the derivatives up to second order of the random field Λ (t), while, in the Sections 4and 5 we get the EC densities of Λ (t). Finally, we applied these results to detecting brain damage ofa group of 17 non-missile trauma patients compared to a group of 19 age and sex matched controlsin the Section 6.

2 Preliminaries

In this section we shall settle the notation to be used throughout the paper, as well as some basicfacts about RFs and matrix differentiation.

2

2.1 Basic Notations

We use Normalm×n(µ,Σ) to denote the multivariate normal distribution of a matrix X ∈ Rm×n

with mean µ and covariance matrix Σ, and write Σ = M(Imn) when cov(Xij,Xkl) = ε(i, j, k, l) −δijδkl with ε(i, j, k, l) symmetric in its arguments and with δij = 1 if i = j and 0 otherwise. LetWishartm (ν,Σ) be represent the Wishart distribution of a m × m matrix with expectation νΣand degrees of freedom ν, and use Betam

(12ν, 1

2η)

for the multivariate beta distribution of a m×mmatrix with degrees of freedom ν and η. For any vector we shall use subscripts j and |j to denotethe jth component and the first j components, respectively. In a similar way, for any matrix, thesubscript |j shall denote the matrix of the first j rows and columns. For any scalar b, let b+ = b ifb > 0 and zero otherwise. For any symmetric matrix B , let B− = B if B is negative definite andzero otherwise and let detrj (B) be the sum of the determinants of all j × j principal minors of Band define detr0 (B) = 1. We shall denote by B1/2 the square root of the matrix B, which is definedby B1/2(B1/2)′ = B. For any m×m symmetric matrix B and any multi-indices κ = (k1, k2, . . . , km),k1 ≥ k2 ≥ · · · ≥ km, we denote with Cκ(B) the zonal polynomials corresponding to κ (see detailsin Muirhead (1982)), which satisfy

tr(B)k =∑

|κ|=k

Cκ(B). (4)

For any real a and natural k, (a)k shall be defined by (a)k = a(a + 1) . . . (a + k − 1), (a)0 = 1. Forthe multi-index κ = (k1, k2, . . . , km) define

(a)κ =m∏

i=1

(a− 12(i− 1))ki

.

Finally, for any real a > 12(m− 1), Γm(a) will denote the multivariate Gamma function, defined by

Γm(a) = πm(m−1)/4

m∏i=1

Γ(a− 12(i− 1)),

where Γ(.) denotes the usual Gamma function.

2.2 Random Field Theory

Let Y = Y (t) , t ∈ S ⊂ RN be a real valued isotropic random field and let S be a compact subset of

RN with a twice differentiable boundary ∂S. Denote the first derivative of Y by.

Y and the N ×Nmatrix of second-order partial derivatives of Y by Y . For j > 0 define the j-dimensional EulerCharacteristic (EC) density as

ρj(y) = E(Y +j det(−Y | j−1) | Y|j−1 = 0, Y = y) θ|j−1(0, y) (5)

where E (·) denotes mathematical expectation and θ|j−1 (·, ·) is the joint probability density function

of Y|j−1 and Y. For j = 0 define ρ0(y) = P(Y ≥ y). Let aj = 2πj/2/Γ(j/2) be the surface area of a unit(j− 1)−sphere in RN . Define µN (S) to be the Lebesgue measure of S and µj (S) , j = 0, . . . , N − 1to be proportional to the j−dimensional Minkowski functional or intrinsic volume

µj (S) =1

aN−j

∫

∂S

detrN−1−j (C) dt

3

with C denoting the inside curvature matrix of ∂S at the point t. The expected EC χ of theexcursion set of Y above the threshold y inside S (Ay(Y, S)) is given by

E(χ(Ay(Y, S))) =N∑

j=0

µj (S) ρj(y).

Then, the P-value of the maximum of Z inside S is well approximated by

P(

maxt∈S

Y (t) ≥ y

)≈ E(χ(Ay(Y, S))) (6)

(Adler 2000, Taylor, Takemura & Adler 2005).In this paper Y is a function of independent RFs each with the same distribution as the stan-

dard Gaussian RF Z, and the EC densities are ρj(y) are calculated assuming unit variance of thederivative (1). If instead

Var(Z) = v2Im (7)

then t can be re-scaled by dividing t by v to satisfy (1). Thus is equivalent to multiplying µj(S)by vj, to give

P(

maxt∈S

Y (t) ≥ y

)≈ E(χ(Ay(Y, S))) =

N∑j=0

µj(S)vjρj(y). (8)

For this reason all the subsequent theory for the EC densities will assume v = 1.

2.3 Matrix Functions

Throughout the next section we shall need the following result, which follows easily from the prop-erties of the operators vec (·) (vectorization) and ⊗ (Kronecker tensor product). If the p× q matrixX is such that X ∼ Normalp×q(0, Ipq) then for any s× p matrix A and q × r matrix B,

AXB ∼ Normals×r (0, (B′B)⊗ (AA′)) . (9)

We shall also make intensive use the differential theory of matrix argument functions proposed inMagnus & Neudecker (1988).). It is based on expressing the derivative of a matrix-valued functionof a matrix argument F (X) : Rp×q −→ Rm×n by the mn× pq matrix

DXF (X) =d (vec (F (X)))

d ( vec (X)).

3 Representations of Derivatives

In this section we obtain expressions for the first and second derivatives of the random field Λ (t)with respect to t, denoted by Λ and Λ respectively. These expressions generalize, for instance, theformulas for the derivatives of the F field obtained in Worsley (1994).

Theorem 1 The two first derivatives of the random field Λ can be expressed as

Λ = A1vec(M0(W,B)),

4

Λ = tr(M1(W,B))IN + tr(M2(W,B))1/2H + tr(M3(W,B))P + tr(M4( W,B))Q

+A1M5(W,B)A′1 + A1M6(W,B)A′

2 + A2M7(W,B)A′1 + A2M8(W,B)A′

2,

where equality means equality in law,

A1, A2 ∼ NormalN×m2(0, INm2), H ∼ NormalN×N (0,M (IN2)) ,

P ∼ WishartN (ν −m, IN) , Q ∼ WishartN(η −m, IN),

W ∼ Wishartm(ν + η, Im), B ∼ Betam(12ν, 1

2η),

independently, and Mi(W,B), i = 0, . . . , 8 are matrices that depends on B and W.

Proof. For each j = 1, . . . , N we have

Λj = DUΛ (U,V) DtjU + DVΛ (U,V) Dtj V. (10)

According to Neudecker (1969) one is to able to find two matrix functions G1(U,V), G2(U,V),

G1,G2 : Rm×m × Rm×m −→ Rm×m

such that

DUΛ (U,V) = vec(G1)′,

DVΛ (U,V) = vec(G2)′.

From Lemma 3.2 in Cao & Worsley (1999a) we make the substitutions

DtjU = vec(U1/2AUj + (U1/2AU

j )′), (11)

DtjV = vec(V1/2AVj + (V1/2AV

j )′),

where(AU

1 , . . . ,AUN), (AV

1 , . . . ,AVN) ∼ Normalm×mN (0, Im2N) ,

independently. Thus,

Λj = vec(G1)′vec(U1/2AUj + (U1/2AU

j )′) + vec(G2)′vec(V1/2AVj + (V1/2A)′)

= tr(((G1)′ + G1)U1/2AUj ) + tr(((G2)′ + G2)V1/2AV

j )

= tr(GU1/2AUj + FV1/2AV

j )

= tr(C−11 A1

j),

where

G = (G1)′ + G1,

F = (G2)′ + G2,

C1 = (GUG + FVF)−1/2,

A1j = C1(GU1/2AU

j + FV1/2AVj ),

j = 1, . . . , N . It is easily verified from (9) that, conditional on U,V, A1j ∼ Normalm×m (0, Im2) ,

which yieldsΛ = A1vec(C−1

1 ),

5

whereA1 = (vec(A1

1), . . . , vec( A1N))′ ∼ NormalN×m2(0, Im2N).

Using the fact that

W = U + V ∼ Wishartm(ν + η, Im),

B = (U + V)−1/2U(U + V)−1/2 ∼ Betam(12ν, 1

2η),

independently (Olkin & Rubin 1962), we obtain the first identity by rewriting the matrix C−11 in

terms of W and B by means of the substitutions

U = W1/2BW1/2, V = W1/2(Im −B)W1/2. (12)

For the second identity we obtain from (10)

Λkj = (DtjU)′D2UUΛ (U,V) DtkU + (D2

VUΛ (U,V))DtkV (13)

+(DtjV)′D2UVΛ ( U,V))DtkU + (D2

VVΛ ( U,V))DtkV

+ (DUΛ (U,V)) D2tktj

U + (DVΛ (U,V)) D2tktj

V,

From Lemma 3.2 in Cao & Worsley (1999a) we make the substitutions

D2tktj

U = vec(Pjk + Pkj + U1/2HUkj + (U1/2HU

kj)′ + (AU

j )′AUk + (AU

k )′AUj − 2δkjU), (14)

D2tktj

V = vec(Qjk + Qkj + V1/2HVkj + (V1/2HV

kj)′ + (AV

j )′AVk + (AV

k )′AVj − 2δkjV ),

where

P = (Pjk) ∼ WishartmN(ν −m, ImN),

Q = (Qjk) ∼ WishartmN(η −m, ImN),

HUkj,H

Vkj ∼ Normalm×m (0,M (Im2)) .

The expressions (11) and (14) yields

(DtjU)′(D2UUΛ)Dtk U = vec(U1/2AU

j )′(Im2 + K′m)(D2

UUΛ) (Im2 + Km) vec(U1/2AUk ), (15)

(DtjU)′(D2VUΛ)Dtk V = vec(U1/2AU

j )′(Im2 + K′m)(D2

VUΛ) (Im2 + Km) vec(V1/2AVk ),

(DtjV)′(D2UVΛ)Dtk U = vec(V1/2AV

j )′(Im2 + K′m)(D2

UVΛ) (Im2 + Km) vec(U1/2AUk ),

(DtjV)′(D2VVΛ)Dtk V = vec(V1/2AV

j )′(Im2 + K′m)(D2

VVΛ) (Im2 + Km) vec(V1/2AVk ),

and

DUΛ(D2tktj

U) = −2δkjvec(G1)′vec (U) + vec(G)′vec(Pjk) + vec(G)′vec(U1/2HUjk)

+vec(G)′vec((AUj )′AU

k ),

DVΛ(D2tktj

V) = −2δkjvec(G2)′vec (V) + vec(F)′vec(Qjk) + vec(F)′vec(V1/2HVjk)

+vec(F)′vec((AVj )′AV

k ). (16)

We can now find a linear combination of AUj and AV

j , j = 1, . . . , N that is independent of A1j ,

namely,

A2j = C2(G

−1U−1/2AUj − F−1V−1/2AV

j ),C2 = ((GUG)−1 + (FVF)−1)−1/2.

6

This implies that the matrices AUj , AV

j can be written in terms of A1j and A2

j by suitable expressions(unequivocally determined), which when substituted in each of the right members of (15) and (16)give

Λkj = −2δkjtr(G1U + G2V) + tr(C−1

1 Hkj) + tr(GPjk) + tr(FQjk) (17)

+(vec(A1j))

′S1vec(A1k) + (vec(A1

j))′S2vec(A2

k)

+(vec(A2j))

′S3vec(A1k) + (vec(A2

j))′S4vec(A2

k),

for certain matrices S1,S2,S3,S4 that only depends on U,V, and Hkj given by

Hkj = C1(GU1/2HUjk + FV1/2HV

kj).

It can be easily shown that the matrix (tr(C−11 Hkj))kj can be rewritten as

(tr(C−11 Hkj))kj = tr(C−2

1 )1/2H,

withH = tr(C−2

1 )−1/2(IN ⊗ (vec(C−11 ))′)(vec(Hkj))kj, (18)

which, conditional on U,V, is distributed as NormalN×N (0,M (IN2)) . On the other hand, notingthat

(tr(GPkj))kj = (IN ⊗ vec(G1/2)′)(Im ⊗ Pkj)kj(IN ⊗ vec( G1/2)),

(tr(FQkj))kj = (IN ⊗ vec(F1/2)′)(Im ⊗ Qkj)kj(IN ⊗ vec( F1/2)),

(Im ⊗ Pkj)kj ∼ Wishartm2N (ν −m, Im2N) ,

(Im ⊗ Qkj)kj ∼ Wishartm2N(η −m, Im2N),

we define the matrices

P = (tr(G−1))−1(IN ⊗ vec(G1/2)′)(Im ⊗ Pkj)kj(IN ⊗ vec(G1/2)), (19)

Q = (tr(F−1))−1(IN ⊗ vec(F1/2)′)(Im ⊗ Qkj)kj(IN ⊗ vec(F1/2)),

which distribute independently as WishartN (ν −m, IN) and WishartN(η − m, IN), respectively.It is also easily verified that

(vec(A1j)′S1vec(A1

k))kj = A1S1A′1,

(vec(A2j)′S2vec(A2

k))kj = A2S2A′2,

(vec(A1j)′S3vec(A2

k))kj = A1S3A′2,

(vec(A2j)′S4vec(A1

k))kj = A2S4A′1,

with A2 = (vec(A21), . . . , vec(A2

N))′ ∼ NormalN×m2 (0, Im2N) independently of A1. Combiningthese last expressions with (18) and (19) and substituting in (17) we obtain

Λ = −2tr(GUU + GVV)IN + tr(C−21 )1/2H + tr

(G−1

)P + tr

(F−1

)Q

+A1S1A′1 + A1S2A

′2 + A2S3A

′1 + A2S4A

′2.

The substitutions (12) conclude the proof.

7

4 Euler Characteristic Densities

In this section, we obtain expressions for the EC densities of the random field Λ. We begin with thefollowing proposition.

Proposition 2 The following equalities hold

Λ|j = tr (M2)1/2 z

Λ|j | (Λ|j = 0) = tr(M1)Ij + tr (M2)1/2 H + tr (M3)P + tr (M4)Q + X1M5X

′1

+X1M6X′2 + X2M7X

′1 + X2M8X

′2,

where

X1 ∼ Normalj×m2(0,Σ⊗ Ij), X2∼Normalj×m2(0, Ijm2),

z ∼ Normalj(0, Ij), H ∼ Normalj×j(0,M(Ij2)),

P ∼ Wishartj(ν −m, Ij), Q ∼ Wishartj(η −m, Ij)

W ∼ Wishartm(ν + η, Im), B ∼ Betam

(12ν, 1

2η)

independently, with Σ given by

Σ = Im2×m2 − tr(M2)−1vec(M0)vec(M0)

′.

Proof. Define the vector b and the matrix C by

b = A|j1 vec (M0) and C = (A|j

1 ,b),

respectively. It is easily seen from the definition of K0 and K2 in the theorem above that

vec (M0)′ vec (M0) = tr(M2),

which impliesb∼Normalj(0, tr(M2) Ij).

The first equality follows by noting that Λ|j = b. Moreover it is very easy to check that

C∼Normalj×m2+1(0, Σ),

where

Σ =

(Ijm2×jm2 vec(K0)⊗ Ij

vec(K0)′ ⊗ Ij tr(K2)Ij

).

From Theorem 3.2.3 and Theorem 3.2.4 in Mardia & Bibby (1979) we obtain

A|j1 | (b = 0)∼Normalj×m2(0,Σ1), (20)

independently of b, where

Σ1 = Ijm2×jm2 − (vec(M0)⊗ Ij)(tr(M2)Ij)−1(vec(M0)

′ ⊗ Ij)

= Ijm2×jm2 − tr(M2)−1vec(M0)vec(M0)

′ ⊗ Ij

= (Im2×m2 − tr(M2)−1vec(M0)vec(M0)

′)⊗ Ij.

8

According to Theorem 1 the condition (Λ|j = 0) is equivalent to b = 0. This implies, by (20),

Λ|j | (Λ|j = 0) = tr(M1)IN + tr (M2)1/2 H + tr (M3)P + tr (M4)Q + X1M5X

′1 + X1M6X

′2

+X2M7X′1 + X2M8X

′2,

where X1∼Normalj×m2(0, Σ), X2∼Normalj×m2(0, Ijm2) and

Σ = Im2×m2 − tr(M2)−1vec(M0)vec(M0)

′, (21)

which concludes the proofThe next theorem is the main result of this section. The EC densities are obtained only for

j = 1, 2, 3, which are the most important cases in practical applications.

Theorem 3 The EC densities ρj (λ) , j = 1, 2, 3 for the random field Λ are given by

ρ1 (λ) = (2π)−1/2θ0 (λ) ϕ1(λ),

ρ2 (λ) = (2π)−1θ0(λ)ϕ2(λ)

ρ3 (λ) = (2π)−3/2θ0 (λ) ϕ3(λ),

where ϕj(λ) are expressions of the type

ϕj(λ) = E(fj(B,W) | Λ(B,W) = λ),

for certain scalar values fj(B,W), and θ0 (λ) denotes the probability density function of Λ.

Proof. We shall evaluate the expectations in (5) by first conditioning on Λ = λ, W and B, andthen taking expectations over W,B conditional on Λ(B,W) = λ.This is,

ρj (λ) = E(E(Λ+j | Λ|j−1 = 0,Λ = λ;B.W)E(det(−Λ|j−1) | Λ|j−1 = 0,Λ = λ;B,W) (22)

× θ|j−1(0; g,B,W) | Λ(B,W) = λ) θ0(λ),

where the corresponding derivatives should be rewritten according to Lemma 2. From Lemma 5.3.3in Adler (1981) and Theorem 2 we have

E(Λ+j | Λ|j−1 = 0,Λ = λ;B,W) = (2π)−1/2tr(M2)

1/2, (23)

and the density of Λ|j−1 at zero conditional on Λ = λ and B,W given by

θ|j−1 (0; λ,B,W) = (2πtr(M2))− 1

2(j−1), (24)

According to Proposition 2, Lemma 7 and Lemma 8 we obtain

E(det(−Λ|0) | Λ|0 = 0,Λ = λ;B,W) = 1, (25)

E(det(−Λ|1) | Λ|1 = 0,Λ = λ;B,W) = −((ν −m)tr (M3) + (η −m)tr (M4) (26)

+tr(M1 + M5Σ + M8)),

E(det(−Λ|2) | Λ|2 = 0,Λ = λ;B,W) =(

ν−m2

)tr (M3)

2 +(

η−m2

)tr (M4)

2 +tr(M1)2 (27)

+(ν −m)(η −m)tr (M3) tr(M4)

+[(ν −m)tr (M3) + (η −m)tr(M4)]tr(M1) + tr(M5Σ + K8)2

+[(ν −m)tr (M3) + (η −m)tr(M4) + tr(M1)]tr(M5Σ + M8)

−tr(M5ΣM5Σ + M28 + M6M7Σ + M7M6Σ + M2),

9

respectively, with Σ given by (21). Using (23), (24) and (25)-(27) we finally obtain

ρ1(λ) = (2π)−1/2θ0 (λ) ϕ1(λ),

ρ2 (λ) = (2π)−1θ0(λ)ϕ2(λ)

ρ3 (λ) = (2π)−3/2θ0 (λ) ϕ3(λ),

where

ϕ1(λ) = E(tr(M2)1/2),

ϕ2(λ) = −E((ν −m)tr (M3) + (η −m)tr (M4) + tr(M1) + tr(M5Σ + M8)),

ϕ3(λ) = E(tr(M2)−1/2(

(ν−m

2

)tr (M3)

2 +(

η−m2

)tr (M4)

2 +(ν −m)(η −m)tr (M3) tr (M4)

+tr(M1)2 + tr(M5Σ + M8)

2 + ((ν −m)tr (M3) + (η −m)tr(M4))tr(M1)

+((ν −m)tr (M3) + (η −m)tr (M4) + tr(M1))tr(M5Σ + M8)− tr(M5 ΣM5Σ

+M82 + M6M7Σ + M7M6Σ + M2))),

where all expectations above have to be taken by conditioning on Λ(B,W) = λ.

5 Wilks’s Λ Random Field

In this section we shall apply the previous results for obtaining the EC densities of the Wilks’s Λrandom field.

Definition 4 Let U(t),V(t) be two independent m− dimensional Wishart RFs with ν and η degreesof freedom, respectively. The Wilk’s Λ RF is then defined as

Λ(t) =det(U(t))

det(U(t) + V(t)).

Note that although the Wilks’s Λ statistic is well defined at a point t provided ν+ η ≥ m,this may not be true for all points inside a compact subset of RN . The reason is that both thenumerator and denominator if Wilks’s Λ could be zero. For the particular case of m = 1 it wasshown in Worsley (1994) that ν+η ≥ N to avoid this, with probability one. For general m, followingthe same ideas in Cao & Worsley (1999a), we obtain ν +η ≥ m+N−1 as a necessary and sufficientcondition for having Λ well defined. When used in hypothesis testing problems, one rejects nullhypotheses based on small values of the Wilk’s Λ statistics. So, in what follows, we shall work withthe RF Y (t) = − log(Λ(t)).

It is well-known that for any symmetric matrix function U(t) the identities

˙det(U)j = det(U)tr(U−1Uj),

¨det(U)kj = det(U)−1 ˙det(U)k˙det(U)j − det(U)tr(U−1UkU

−1Uj + U−1Ukj)

hold, so that Theorem 1 givesY = 2A1vec(M

1/20 )

Y = 2(tr(M0)1/2H− tr(M0)P + tr(M4)Q + A1M5A

′1 −A1M6A

′2 −A2M7A

′1),

10

where

M0 = (B−1 − Im)W−1,

M4 = W−1,

M5 = (M1/20 ⊗M

1/20 ),

M6 = (M1/20 ⊗W−1/2)′,

M7 = M′6.

Theorem 3 yields

ρ1(y) = (2π)−1/2θ0(y)E(tr(M0)1/2 | det(B) = e−y), (28)

ρ2(y) = 2(2π)−1θ0(y)E((ν −m)tr(M0)− (η −m)tr (M4)− tr(M5Σ) | det(B) = e−y), (29)

ρ3(y) = 4(2π)−3/2θ0(y)E(tr(M0)−1/2(

(ν−m

2

)tr (M0)

2 +(

η−m2

)tr (M4)

2 + tr(M5Σ)2 (30)

−(ν −m)(η −m)tr(M0)tr(M4) + ((η −m)tr(M4)− (ν −m)tr(M0))tr(M5Σ)

−tr(M5ΣM5Σ+2M6M7Σ)− tr(M0))) | det(B) = e−y),

whereΣ = Im2×m2 − tr(M0)

−1vec(M1/20 )vec(M

1/20 )′.

There is no closed form expression for θ0(y), the density of Y = − log(Λ). A host of approximationsare available (Anderson 1984), but it is easier numerically to use Fourier methods as follows. Wecan write

Y =m∑

i=1

Yi,

whereexp(−Yi) ∼ Beta1(

12(ν + 1− i), 1

2η)

independently, i = 1, . . . ,m (Anderson 1984). Then the product of the Fourier transforms of thedensities of Yi is the Fourier transform of the density of Y . Inverting gives the desired density θ0(y)of Y .

5.1 EC densities

Providing closed expressions for the expectations that appear in the formulas (28-30) is a verydifficult task. In this subsection we propose a practical way to approximate such expectations.Specifically, we shall use expansion series for either negative or fractional powers of tr(M0) (seeShapiro & Gross (1981)), that is,

tr(M0)r ≈ E(tr(M0))

r + rE(tr(M0))r−1(tr(M0)− E(tr(M0)))

+r(r − 1)E(tr(M0))r−2(tr(M0)− E(tr(M0)))

2 + . . .

However, for simplicity in the exposition, we just describe how to deal with the simplest approxi-mation, namely, tr(U−1)r ≈ E(tr(U−1))r.

For the EC density ρ1(y) we have

ρ1(y) = (2π)−1/2θ0 (y)E(tr(M0) | A)1/2.

11

where A is the event det(B) = e−y. According to Letac & Massam (2001),

E(tr(M0) | A) =E(tr(B−1 − Im) | A

q,

where q = ν + η −m− 1. Now, by expressing tr(B−1) and det(B) in terms of the eigenvalues of Bwe obtain

E(tr(B−1) | A) = E(tr(B−1∗ )) + E(det(B∗))ey,

for some matrix B∗ ∼ Betam−1(12ν, 1

2η), namely, one of the principal minors of order m − 1 of B.

Then, by (34) and (35),

c1(y) := E(tr(B−1) | A) =(m− 1)(q + 1)

(ν −m)+ pm−1(1, 0)ey,

where

pm(a, b) =

{Γm( 1

2ν+a)Γm( 1

2(ν+η)+b)

Γm( 12ν+b)Γm( 1

2(ν+η)+a)

, m ≥ 1

0, m ≤ 0.

Thereforeρ1(y) = (2π)−1/2θ0(y)k1(y)1/2,

where

k1(y) =c1(y)−m

q.

In a similar way, we use the approximation

E(tr(M5Σ)) ≈ E(tr(M0))− E(tr(M0))−1 E(tr(M2

0))

to obtain

ρ2(y) = 2(2π)−1θ0(y)((ν −m− 1)k1(y)− (η −m)mq−1 + k1(y)−1k2(y)),

where

k2(y) =c3(y)− 2mc1(y) + m2

q3 − q+

c2(y)− 2c1(y) + m

q2 − 1,

c2(y) = pm−1(2, 0)e2y +(q + 1)(q + 3)

(ν −m)(ν −m + 2)C(2)(Im−1)− (q + 1)(q + 2)

2(ν −m)(ν −m + 1)C(1,1)(Im−1),

c3(y) = pm−1(2, 0)e2y + (m− 1)pm−2(1, 0)ey +(q + 1)(q + 3)

(ν −m)(ν −m + 2)C(2)(Im−1)

+(q + 1)(q + 2)

(ν −m)(ν −m + 1)C(1,1)(Im−1).

Note that we have used (Letac & Massam 2001)

E(tr(M20) | A) =

E(tr(B−1 − Im)2 | A)

q3 − q+E(tr((B−1 − Im)2) | A)

q2 − 1

and

c2(y) := E(tr(B−2) | A) = E(tr(B−2∗ )) + E(det(B∗)2)e2y,

c3(y) := E(tr(B−1)2 | A) = E(tr(B−1∗ )2) + 2E( tr(B−1

∗ ) det(B∗))ey + E(det(B∗)2)e2y,

12

where the expressions for c2(y) and c3(y) are obtained from (4), (34) and (35). Finally, from Letac& Massam (2001) and Graczyk, Letac & Massam (2003) we have

E(tr(M0)2 | A) =

E(tr(B−1 − Im)2 | A)

q2 − 1+E(tr((B−1 − Im)2) | A)

q3 − q,

k5(y) := E(tr (M4) tr (M0) | A) =mq + 1

q3 − qE(tr(B−1 − Im) | A),

k6(y) := E(tr (M4) tr(M20) | A)

=(mq + 2)E(tr(B−1 − Im)2 | A) + (mq2 + 2q − 2m)E(tr((B−1 − Im)2) | A)

q(q2 − 1)(q2 − 4),

k7(y) := E(tr(M20M4) | A)

=2(q + m)E(tr(B−1 − Im)2 | A)

q(q2 − 1)(q2 − 4)+

(q + m)E(tr((B−1 − Im)2) | A)

(q2 − 1)(q2 − 4),

which implies

ρ3(y) = 4(2π)−3/2θ0(y)k1(y)−1/2{(ν−m2

)k3(y) +

(η−m

2

)k4 − (ν −m)(η −m)k5(y)

+(ν −m)[k3(y)− k2(y)]− (η −m)[k5(y)− k1(y)−1k6(y)] + [k3(y)− k2(y)]

−2[k5(y)− k1(y)−1k7(y)]− k1(y)},where

k3(y) =c3(y)− 2mc1(y) + m2

q2 − 1+

c2(y)− 2c1(y) + m

q3 − q,

k4 =m2

q2 − 1+

m

q3 − q,

k5(y) =(mq + 1)(c1(y)−m)

q3 − q,

k6(y) =(mq + 2)(c3(y)− 2mc1(y) + m2) + (mq2 + 2q − 2m)(c2(y)− 2c1(y) + m)

q(q2 − 1)(q2 − 4),

k7(y) =2(q + m)(c3(y)− 2mc1(y) + m2)

q(q2 − 1)(q2 − 4)+

(q + m)(c2(y)− 2c1(y) + m)

(q2 − 1)(q2 − 4).

5.2 Simulation Results

In this subsection we shall show some simulation results for validating the previous formulae. Wegenerated 200 Y = − log(Λ) RFs on a 323 voxel rectilinear lattice as follows. We first generatedlattices of independent standard Gaussian random variables then smoothed them with a Gaussianshaped filter of standard deviation σ = 3.2 voxels, which gives v = 1/(

√2σ) = 0.22 as the standard

deviation of the spatial derivative (Worsley, Marrett, Neelin, Vandal, Friston & Evans 1996).). TheWilks’s Λ RF was generated by

Y = − log

(det(Z′1Z1)

det(Z′1Z1 + Z′2Z2)

), (31)

where Z1 and Z2 are ν ×m and η ×m matrices of independent smoothed Gaussian random fieldsas above. The variance of the derivative of the component random fields is The EC χ(Ay) of the

13

0 1 2 3 4 5 6 7−4

−2

0

2

4

6

8

10

12SimulatedTheoretical

Exp

ecte

d E

ule

r C

har

acte

rist

ic

Threshold

Figure 1: Euler Characteristic of the excursion set of Y = − log(Λ) for ν = 7, η = 4 and m = 2sampled on a 323 lattice smoothed by Gaussian filter with standard deviation 6.37 voxels.

excursion sets Ay was calculated for each of the 200 Y RFs at y = 0, 0.1, 0.2, . . . , 6.0. Then, theaverage of the measurements χ(Ay) was used as an estimator of E(χ(Ay)). This value and thetheoretical approximation obtained in the subsection above are plotted in Figure 1 for ν = 7, η = 4,and m = 2. Note that for thresholds greater that 2 both the theoretical approximation and thesimulated values are very close.

The P-value approximation (6) based on the expected EC, the (uncorrected) P-value at a singlevoxel, and the Bonferroni P-value are plotted in Figure 2. The parameter values were chosen tomatch the example in Section 6: m = 3, ν = 31 and η = 1, 2, 3, 4. The parameter set S was a 3Dball of radius with radius 68mm sampled on a 2mm voxel lattice. The data was smoothed by aGaussian filter with standard deviation 5.65mm to give v = 0.125. In the first plot with m = 1,Wilks’s Λ is equivalent to Hotelling’s T 2 = ν(exp(Y ) − 1). Exact results for the expected EC ofHotelling’s T 2 from Cao & Worsley (1999a) are added to the plot. We can see that these are inreasonable agreement with our Wilks’s Λ approximation, which appears to be too liberal by a factorof 3 to 4.

14

0 0.5 1 1.5 210

−3

10−2

10−1

m = 3, ν = 31, η = 1

y = −log Wilks’s Λ

P(m

ax Y

> y

)

ECBon1 voxT2

0 0.5 1 1.5 210

−3

10−2

10−1

m = 3, ν = 31, η = 2

y = −log Wilks’s ΛP

(max

Y >

y)

0 0.5 1 1.5 210

−3

10−2

10−1

m = 3, ν = 31, η = 3

y = −log Wilks’s Λ

P(m

ax Y

> y

)

0 0.5 1 1.5 210

−3

10−2

10−1

m = 3, ν = 31, η = 4

y = −log Wilks’s Λ

P(m

ax Y

> y

)

Figure 2: P-values of the maximum of Y = − log(Λ) over the brain, approximated by a 3D ballwith radius 68mm sampled on a 2mm voxel lattice. The data was smoothed by a Gaussian filterwith standard deviation 5.65mm. EC = Euler Characteristic approximation (6), Bon = Bonferroniupper bound, 1 vox = uncorrected P-value at a single voxel (lower bound), T 2 = exact expectedEC for equivalent Hotelling’s T 2 (η = 1 only).

15

6 Application

6.1 Multivariate Linear Models

We apply our results to a multivariate linear model. Suppose we have n independent RFs ofmultivariate observations yi(t) ∈ Rm, i = 1, . . . , n, where t ∈ S ⊂ RN , and a multivariate linearmodel (Cao & Worsley 1999a):

yi(t)′ = x′iβ(t) + εi(t)

′Σ(t)1/2, (32)

where xi is a p-vector of known regressors, and β(t) is an unknown p×m coefficient matrix. Theerror εi(t) is a m-vector of independent zero mean, unit variance isotropic Gaussian componentswith the same spatial correlation structure, and Var(yi(t)) = Σ(t) is an unknown m ×m matrix.We can now detect how the regressors are related to the multivariate data at point t by testingcontrasts in the rows of β(t). Classical multivariate test statistics evaluated at each point t thenform a random field.

Let β(t) be the usual least-squares estimate of β(t). The Wilks’s Λ random field is defined interms of two independent Wishart random fields. The first is the error sum of squares matrix

U(t) =n∑

i=1

(yi(t)′ − x′iβ(t))′(yi(t)

′ − x′iβ(t)) ∼ Wishartm(Σ(t), ν)

where ν = n− p. Let X = (x′1, . . . ,x′n)′ be the design matrix. The second is the regression sum of

squares matrix for a η × p matrix of contrasts C

V(t) = (Cβ(t))′(C(X′X)−1C′)−1(Cβ(t)) ∼ Wishartm(Σ(t), η)

under the null hypothesis of no effect, Cβ = 0. In Wilks’s Λ (31), Σ(t) will cancel so under thenull hypothesis, (31) will be a Wilks’s Λ random field.

6.2 Brain Shape Analysis

As an illustration of the methods, we apply the P-value approximations for Wilks’s Λ to a dataset on non-missile trauma (Tomaiuolo, Worsley, Lerch, Di Paulo, Carlesimo, Bonanni, Caltagirone& Paus 2005) that was analysed in a similar way in Worsley et al. (2004). The subjects were 17patients with non-missile brain trauma who were in a coma for 3-14 days. MRI images were takenafter the trauma, and the multivariate data were the N = 3 component vector deformations neededto warp the MRI images to an atlas standard sampled on a 2mm voxel lattice. The same data werealso collected on a group of 19 age and sex matched controls.

Damage is expected in white mater areas, so the search region S was defined as the voxels wheresmoothed average control subject white matter density exceeded 5%. For calculating the intrinsicvolumes, this was approximated by a sphere with the same volume, 1.31cc, which is slightly liberalfor a non-spherical search region. The effective v from (7), averaged over the search region, was0.125.

The first analysis was to look for brain damage by comparing the deformations of the 17 traumapatients with the 19 controls, so the sample size is n = 36. We are looking at a single contrast, thedifference between trauma and controls, so η = 1 and the residual degrees of freedom is ν = 34. Inthis case Wilks’s Λ is Hotelling’s T 2. The P = 0.05 threshold, found using the EC densities in Cao &Worsley (1999a) and by equating (8) to 0.05 and solving for y, was y = 54.0. The thresholded data,

16

together with the estimated contrast (mean trauma - control deformations) is shown in Figure 3(a).A large region near the corpus callosum seems to be damaged. The nature of the damage, judgedby the direction of the arrows, is away from the center (see Figure 3(b)). This can be interpreted asexpansion of the ventricles, or more likely, atrophy of the surrounding white matter, which causesthe ventricle/white matter boundary to move outwards.

We might also be interested in functional anatomical connectivity: are there any regions of thebrain whose shape (as measured by the deformations) is correlated with shape at a reference point?In other words, the explanatory variables are the deformations at a pre-selected reference point,and the test statistic is Wilks’s Λ. We chose as the reference the point with maximum Hotelling’sT 2 for damage, marked by axis lines in Figure 3. Figure 3(c) shows the resulting -log Wilks’s Λfield above the P = 0.05 threshold of 1.54 for the combined trauma and control data sets removingseparate means for both groups (ν = 31, η = 3 ). Obviously there is strong correlation with pointsnear the reference, due to the smoothness of the data. The main feature is the strong correlationwith contra-lateral points, indicating that brain anatomy tends to be symmetric.

A more interesting question is whether the correlations observed in the control subjects aremodified by the trauma (Friston, Buchel, Fink, Morris, Rolls & Dolan 1997). In other words, isthere any evidence for an interaction between group and reference vector deformations? To do this,we simply add another three covariates to the linear model whose values are the reference vectordeformations for the trauma patients, and the negative of the reference vector deformations forthe control subjects. The resulting -log Wilks’s Λ field for testing for these three extra covariates,thresholded at 1.72 (P = 0.05, η = 3, ν = 28) is shown in Figure 3(d). Apart from changes in theneighbourhood of the reference point, there is some evidence of a change in correlation at a locationin the contraleteral side, slightly anterior. Looking at the maximum canonical correlations in thetwo groups separately, we find that correlation has increased at this location from 0.719 to 0.927,perhaps indicating that the damage is strongly bilateral. These results are in close agreement withthose in Worsley et al. (2004) which used Roy’s maximum root rather than Wilks’s Λ.

A Appendix

Lemma 5 (Lemma A.2 in Worsley (1994)) Let H ∼Normalj×j(0,M(Ij)), let h be a fixed scalar,and let A be a fixed symmetric j × j matrix. Then

E(det(A + hH)) =

b j2c∑

i=0

(−1)i(2i)!

2ii!h2idetrj−2i(A).

Lemma 6 Let

P ∼ Wishartj(ν, Ij), Q ∼ Wishartj(η, Ij),

X1 ∼ Normalj×m(0,Σ⊗ Ij), X2∼Normalj×m(0, Ijm)

independently, a, b, c be fixed scalars, and C1,C2,C3,C4 be fixed m×m matrices. Then

E(detri(aP + bQ + cIj + X1C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2))

=i∑

k=0

j!

(j − i + k)!

i−k∑r=0

(ν

i− k − r

)(η

r

)ai−k−rbr

k∑

l=0

(j − l

k − l

)ck−lE(detrl(X1C1X1

′

+X1C2X2′ + X2C3X1

′ + X2C4X2′)).

17

(a) (b)

(c) (d)

Figure 3: Shape analysis of non-missile brain trauma data. (a) Trauma minus control averagedeformations (arrows and color bar), sampled every 6mm inside the brain, with Hotelling’s T 2 fieldfor significant group differences (threshold y = 54.0, P = 0.05). The reference point of maximumHotelling’s T 2 is marked by the intersection of the three axes. (b) Closeup of (a) showing thatthe damage is an outward movement of the anatomy, either due to swelling of the ventricles oratrophy of the surrounding white matter. (c) Regions of effective anatomical connectivity withthe reference point, assessed by the Wilks’s Λ field (threshold y = 1.54, P = 0.05). The referencepoint is connected with its neighbours (due to smoothness) and with contralateral regions (due tosymmetry). (d) Regions where the connectivity is different between trauma and control groups,assessed by the Wilks’s Λ field (threshold y = 1.72, P = 0.05 ). The small region in the contralateralhemisphere (right) is more correlated in the trauma group than the control group.

18

Proof. Holding X1 and X2 fixed and using Lemma in Worsley (1994) we obtain

E(detri(aP + bQ + cIj + X1C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2))

=i∑

k=0

E(detri−k(aP + bQ))detrk(cIj + X1C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2)

=i∑

k=0

j!

(j − i + k)!

i−k∑r=0

(ν

i− k − r

)(η

r

)ai−k−rbrdetrk(cIj + X1C1X

′1 + X1C2X

′2 + X2C3X

′1

+X2C4X′2).

Taking expectations over X1,X2 in the equality above we have

E(detri(aP + bQ + cIj + X1C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2))

=i∑

k=0

j!

(j − i + k)!

i−k∑r=0

(ν

i− k − r

)(η

r

)ai−k−rbr

k∑

l=0

(j − l

k − l

)ck−lE(detrl(X1C1X

′1

+X1C2X′2 + X2C3X

′1 + X2C4X

′2)),

which ends the proof.

Lemma 7 Let

P ∼ Wishartj(ν, Ij), Q ∼ Wishartj(η, Ij),

X1 ∼ Normalj×m(0,Σ⊗ Ij), X2∼Normalj×m(0, Ijm),

H ∼ Normalj×j(0, M(Ij))

independently, a, b, c, h be fixed scalars, and C1,C2,C3,C4 be fixed m×m matrices. Then

E(det(aP + bQ + cIj + X1 C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2 + hH))

=

[ j2]∑

i=0

(−1)i(2i)!

2ii!h2i

j−2i∑

k=0

j!

(2i + k)!

j−2i−k∑r=0

(ν

j − 2i− k − r

)(η

r

)aj−2i−k−rbr

k∑

l=0

(j − l

k − l

)ck−l

× E(detrl(X1C1X1′ + X1C2X2

′ + X2C3X1′ + X2C4X2

′)).

Proof. Holding P,Q,X1,X2 fixed and using Lemma 5 we obtain

E(det(aP + bQ + cIj + X1 C1X′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2 + hH))

=

[ j2]∑

i=0

(−1)i(2i)!

2ii!h2idetrj−2i(aP + bQ + cIj + X1C1X

′1 + X1C2X

′2 + X2C3X

′1 + X2C4X

′2).

Now, taking expectations and using Lemma 6 we have the desired result.

Lemma 8 LetX1∼Normalj×m(0,Σ⊗ Ij), X2∼Normalj×m(0, Ijm),

and C1,C2,C3,C4 be fixed symmetric m×m matrices.. Then

E(det(X1C1X1′ + X1C2X2

′ + X2C3X1′ + X2C4X2

′)) =

tr(C1Σ + C4), j = 1,tr(C1Σ + C4)

2 − tr(C1ΣC1Σ+C2

4 + C2C3Σ + C3C2Σ), j = 2.

19

Proof. It follows from the fact that the rows of X1 and X2 are independent vectors from Normalm(0,Σ)and Normalm(0, Im), respectively, and using the well-known identity

E(xixjxkxl) = σijσkl + σikσjl + σilσjk,

i, j, k, l = 1, . . . , r, where σij = E(xixj).

Lemma 9 If κ = (k1, . . . , km), |κ| = k then for a > k1 + (m + 1)/2,

∫

0<X<Im

det(X)a−(m+1)/2 det(Im −X)b−(m+1)/2Cκ(X−1) dX (33)

=(−1)k(−(a + b) + m+1

2)κ

(−a + m+12

)κ

Γm(a)Γm(b)

Γm(a + b)Cκ(Im),

where

Cκ(Im) = 22kk!(m

2)κ

p∏i<j

(2ki − 2kj − i + j)

p∏i=1

(2ki + p− i)!

,

and p denotes the nonzero indices in κ.

Proof. It is proved by using following similar ideas to that of Theorem 7.2.10 and Theorem 7.2.13in Muirhead (1982) .

As a particular case we have that if B ∼ Betam(12ν, 1

2η) then for ν > 2k1 + m + 1,

E(Cκ(B−1)) =

(−1)k(−ν+η2

+ m+12

)κ

(−ν2

+ m+12

)κ

Cκ(Im). (34)

Lemma 10 (Corollary 10.5.2 in Muirhead (1982)) If B ∼ Betam(12ν, 1

2η) then

E(det(B)h) =Γm(1

2ν + h)Γm(1

2(ν + η))

Γm(12ν)Γm(1

2(ν + η) + h)

. (35)

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