Open access peer-reviewed chapter

Likelihood Ratio Tests in Multivariate Linear Model

By Yasunori Fujikoshi

Submitted: October 9th 2015Reviewed: January 22nd 2016Published: July 6th 2016

DOI: 10.5772/62277

Downloaded: 1540


The aim of this chapter is to review likelihood ratio test procedures in multivariate linear models, focusing on projection matrices. It is noted that the projection matrices to the spaces spanned by mean vectors in hypothesis and alternatives play an important role. Some basic properties are given for projection matrices. The models treated include multivariate regression model, discriminant analysis model, and growth curve model. The hypotheses treated involve a generalized linear hypothesis and no additional information hypothesis, in addition to a usual liner hypothesis. The test statistics are expressed in terms of both projection matrices and sums of squares and products matrices.


  • algebraic approach
  • additional information hypothesis
  • generalized linear hypothesis
  • growth curve model
  • multivariate linear model
  • lambda distribution
  • likelihood ratio criterion (LRC)
  • projection matrix

1. Introduction

In this chapter, we review statistical inference, especially likelihood ratio criterion (LRC) in multivariate linear model, focusing on matrix theory. Consider a multivariate linear model with presponse variables y1, …, ypand kexplanatory or dummy variables x1, …, xk. Suppose that y = (y1, …, yp)′ and x = (x1, …, xk)′ are measured for nsubjects, and let the observation of the ith subject be denoted by yiand xi. Then, we have the observation matrices given by


It is assumed that y1, …, ynare independent and have the same covariance matrix Σ. We express the mean of Yas follows:


A multivariate linear model is defined by requiring that


where Ωis a given subspace in the ndimensional Euclid space Rn. A typical Ωis given by


Here, ℛ[X] is the space spanned by the column vectors of X. A general theory for statistical inference on the regression parameter Θcan be seen in texts on multivariate analysis, e.g., see [18]. In this chapter, we discuss with algebraic approach in multivariate linear model.

In Section 2, we consider a multivariate regression model in which xi'sare explanatory variables and Ω = ℛ[X]. The maximum likelihood estimator (MLE)s and likelihood ratio criterion (LRC) for Θ2=Oare derived by using projection matrices. Here, Θ=Θ1Θ2.The distribution of LRC is discussed by multivariate Cochran theorem. It is pointed out that projection matrices play an important role. In Section 3, we give a summary of projection matrices. In Section 4, we consider to test an additional information hypothesis of y2 in the presence of y1, where y1 = (y1. …, yq)′ and y2 = (yq + 1. …, yp)′. In Section 5, we consider testing problems in discriminant analysis. Section 6 deals with a generalized multivariate linear model which is also called the growth curve model. Some related problems are discussed in Section 7.


2. Multivariate regression model

In this section, we consider a multivariate regression model on presponse variables and kexplanatory variables denoted by y = (y1, …, yp)′ and x = (x1, …, xk)′, respectively. Suppose that we have the observation matrices given by (1.1). A multivariate regression model is given by


where Θis a k × punknown parameter matrix. It is assumed that the rows of the error matrix Eare independently distributed as a pvariate normal distribution with mean zero and unknown covariance matrix Σ,i.e., Np0,Σ.

Let L(Θ, Σ) be the density function or the likelihood function. Then, we have


The maximum likelihood estimators (MLE) Θ^and Σ^of Θand Σare defined by the maximizers of L(Θ, Σ) or equivalently the minimizers of −2log L(Θ, Σ).

Theorem 2.1 Suppose thatYfollows the multivariate regression model in(2.1). Then, theMLEs ofΘandΣare given as


where PX = X(XX)− 1X′. Further, it holds that


Theorem 2.1 can be shown by a linear algebraic method, which is discussed in the next section. Note that PXis the projection matrix on the range space Ω=X.It is symmetric and idempotent, i.e.


Next, we consider to test the hypothesis


against K;Θ2O, where X = (X1X2), X1n × jand Θ=Θ1Θ2,Θ1;j×p.The hypothesis means that the last k − jdimensional variate x2 = (xj + 1, …, xk)′ has no additional information in the presence of the first jvariate x1 = (x1, …, xj)′. In general, the likelihood ratio criterion (LRC) is defined by


Then we can express


Using Theorem 2.1, we can expressed as


Here, Σ^Ωand Σ^ωare the maximum likelihood estimators of Σunder the model (2.1) or Kand H, respectively, which are given by




Summarizing these results, we have the following theorem.

Theorem 2.2Let λ = Λn/2 be theLRC for testing H in(2.2). Then, Λ is expressed as




andSΩandSωare given by(2.4) and(2.5), respectively.

The matrices Seand Shin the testing problem are called the sums of squares and products (SSP) matrices due to the error and the hypothesis, respectively. We consider the distribution of Λ. If a p × prandom matrix Wis expressed as


where zjNpμj,Σand z1, …, znare independent, Wis said to have a noncentral Wishart distribution with ndegrees of freedom, covariance matrix Σ,and noncentrality matrix Δ=μ1μ1++μnμn.We write that WWpn,Σ;Δ.In the special case Δ=O,Wis said to have a Wishart distribution, denoted by WWpn,Σ.

Theorem 2.3(multivariate Cochran theorem) LetY=y1yn, whereyiNpμi,Σ, i = 1, …, n andy1, …, ynare independent. LetA,A1, andA2 be n × n symmetric matrices. Then:

1. YAYWpk,Σ;ΩA2=A,trA=k,Ω=EYAEY.

2. YA1YandYA2Yare independentA1A2 = O.

For a proof of multivariate Cochran theorem, see, e.g. [3, 68]. Let Band Wbe independent random matrices following the Wishart distribution Wpq,Σand Wpn,Σrespectively, with n ≥ p. Then, the distribution of


is said to be the p-dimensional Lambda distribution with (qn)-degrees of freedom and is denoted by Λp(qn). For distributional results of Λp(qn), see [1, 3].

By using multivariate Cochran’s theorem, we have the following distributional results:

Theorem 2.4LetSeandShbe the random matrices in(2.7). Let Λ be the Λ-statistic defined by(2.6). Then,

  1. SeandShare independently distributed as a Wishart distributionWpnk,Σand a noncentral Wishart distributionWpkj,Σ;Δrespectively, where


  • Under H, the statistic Λ is distributed as a lambda distribution Λp(k − jn − k).

  • Proof.Note that PΩ = PX = X(XX)− 1X′, Pω=PX1=X1X1X11X,and PΩPω = PωPΩ. By multivariate Cochran’s theorem the first result (1) follows by checking that


    The second result (2) follows by showing that Δ0=O,where Δ0is the Δunder H. This is seen that


    since PΩX1 = PωX1 = X1.

    The matrices Seand Shin (2.7) are defined in terms of n × nmatrices PΩand Pω. It is important to give expressions useful for their numerical computations. We have the following expressions:


    Suppose that x1 is 1 for all subjects, i.e., x1 is an intercept term. Then, we can express these in terms of the SSP matrix of (y', x')′ defined by


    where y¯and x¯are the sample mean vectors. Along the partition of x=x1x2′,we partition Sas




    Here, we use the notation Syyx=SyySyxSxx1Sxy,Sy21=Sy2Sy1S111S1y, etc. These are derived in the next section by using projection matrices.

    3. Idempotent matrices and max-mini problems

    In the previous section, we have seen that idempotent matrices play an important role on statistical inference in multivariate regression model. In fact, letting EY=η=η1,,η,pconsider a model satisfying


    Then the MLE of Θis Θ^=XX1XY,and hence the MLE of ηis denoted by


    Here, PΩ = X(XX)− 1X′. Further, the residual sums of squares and products (RSSP) matrix is expressed as


    Under the hypothesis (2.2), the spaces ηi’s belong are the same and are given by ω = ℛ[X1]. Similarly, we have


    where Θ^ω=(Θ^1ω'O)and Θ^1ω=X1X11X1Y.The LR criterion is based on the following decomposition of SSP matrices;


    The degrees of freedom in the Λdistribution Λp(fhfe) are given by


    In general, an n × nmatrix Pis called idempotent if P2 = P. A symmetric and idempotent matrix is called projection matrix. Let Rnbe the ndimensional Euclid space, and Ωbe a subspace in Rn. Then, any n × 1 vector ycan be uniquely decomposed into direct sum, i.e.,


    where Ω is the orthocomplement space. Using decomposition (3.2), consider a mapping


    The mapping is linear, and hence it is expressed as a matrix. In this case, uis called the orthogonal projection of yinto Ω, and PΩis also called the orthogonal projection matrix to Ω. Then, we have the following basic properties:

    (P1) PΩis uniquely defined;

    (P2) In − PΩis the projection matrix to Ω;

    (P3) PΩis a symmetric idempotent matrix;

    (P4) ℛ[PΩ] = Ω, and dim[Ω] = trPΩ;

    Let ωbe a subset of Ω. Then, we have the following properties:

    (P5) PΩPω = PωPΩ = Pω.

    (P6) PΩPω=PωΩ,where ω is the orthocomplement space of ω.

    (P7) Let Bbe a q × nmatrix, and let N(B) = {y; By = 0}. If ω = N[B] ∩ Ω, then ω ∩ Ω = R[PΩB '].

    For more details, see, e.g. [3, 7, 9, 10].

    The MLEs and LRC in multivariate regression model are derived by using the following theorem.

    Theorem 3.1

    1. Consider a function off(Σ)=log|Σ|+trΣ1Sof p × p positive definite matrix. Then,fΣtakes uniquely the minimum atΣ=S, and the minimum value is given by


    2. LetYbe an n × p known matrix andXan n × k known matrix of rank k. Consider a function of p × p positive definite matrixΣand k × p matrixΘ=θijgiven by


    where m > 0, − ∞ < θij < ∞, for i = 1, …, k; j = 1, …, p. Then,gΘ,Σtakes the minimum at


    and the minimum value is given by mlog|Σ|^+mp.

    Proof.Let 1, …, pbe the characteristic roots of Σ1S.Note that the characteristic roots of Σ1Sand Σ1/2SΣ1/2are the same. The latter matrix is positive definite, and hence we may assume 1 ≥ ⋯ ≥ p > 0. Then


    The last inequality follows from x − 1 ≥ log x (x > 0). The equality holds if and only if 1 = ⋯ = p = 1 ⇔ Σ=S.

    Next, we prove 2. we have


    The first equality follows from that YXΘ=YXΘ^+XΘ^Θand YXΘ^XΘ^Θ=O.In the last step, the equality holds when Θ=Θ^.The required result is obtained by noting that Θ^does not depend on Σand combining this result with the first result 1.

    Theorem 3.2LetXbe an n × k matrix of rank k, and let Ω = ℛ[X] which is defined also by the set{y : y = X θ }, whereθis a k × 1 unknown parameter vector. LetCbe a c × k matrix of rank c, and define ω by the set{y : y = Xθ , C θ = 0}. Then,

    1. PΩ = X(XX)− 1X′.

    2. PΩ − Pω = X(XX)− 1C′{C(XX)− 1C}− 1C(XX)− 1X′.

    Proof.1 Let ŷ = X(XX)− 1X′ and consider a decomposition y = ŷ + (y − ŷ). Then, ŷ′(y − ŷ) = 0. Therefore, PΩy = ŷand hence PΩ = X(XX)− 1X′.

    2. Since C θ = C(XX)− 1X′ ⋅ X θ, we can write ω = N[B] ∩ Ω, where B = C(XX)− 1X′. Using (P7),


    The final result is obtained by using 1 and (P7).

    Consider a special case C = (O Ik − q). Then ω = ℛ[X1], where X = (X1X2), X1 : n × q. We have the following results:


    The expressions (2.11) for Seand Shin terms of Scan be obtained from projection matrices based on


    4. General linear hypothesis

    In this section, we consider to test a general linear hypothesis


    against alternatives Kg : CΘD ≠ Ounder a multivariate linear model given by (2.1), where Cis a c × kgiven matrix with rank cand Dis a p × dgiven matrix with rank d. When C = (O Ik − j) and D = Ip, the hypothesis Hgbecomes H : Θ2 = O.

    For the derivation of LR test of (4.1), we can use the following conventional approach: If U=YD,then the rows of Uare independent and normally distributed with the identical covariance matrix DΣD, and


    where Ξ=ΘD.The hypothesis (4.1) is expressed as


    Applying a general theory for testing Hgin (2.1), we have the LRC λ:






    Theorem 4.1The statistic Λ in(4.4) is an LR statistic for testing(4.1) under(2.1). Further, under Hg, Λ ∼ Λd(cn − k).

    Proof.Let G = (G1G2) be a p × pmatrix such that G1 = D,G1G2=O,and |G| ≠ 0. Consider a transformation from Yto UV=YG1G2.

    Then the rows of UVare independently normal with the same covariance matrix




    The conditional of Vgiven Uis normal. The rows of Vgiven Uare independently normal with the same covariance matrix Ψ112, and


    where Δ*=ΔΞΓand Γ=Ψ111Ψ12.We see that the maximum likelihood of Vgiven Udoes not depend on the hypothesis. Therefore, an LR statistic is obtained from the marginal distribution of U,which implies the results required.

    5. Additional information tests for response variables

    We consider a multivariate regression model with an intercept term x0 and kexplanatory variables x1, …, xkas follows.


    where Yand Xare the observation matrices on y = (y1, …, yp)′ and x = (x1, …, xk)′. We assume that the error matrix Ehas the same property as in (2.1), and rank (1nX) = k + 1. Our interest is to test a hypothesis H2 ⋅ 1 on no additional information of y2 = (yq + 1, …, yp)′ in presence of y1 = (y1, …, yq)′.

    Along the partition of yinto (y1′, y2′) let Y,θ,Θ,and Σpartition as


    The conditional distribution of Y2given Y1is normal with mean


    and the conditional covariance matrix is expressed as


    where Σ221=Σ22Σ21Σ111Σ12, and


    Here, for an n × pmatrix Y=y1yp,vec (Y) means an np-vector y1yp.Now we define the hypothesis H2 ⋅ 1 as


    The hypothesis H2 ⋅ 1 means that y2 after removing the effects of y1 does not depend on x. In other words, the relationship between y2 and xcan be described by the relationship between y1 and x. In this sense, y2 is redundant in the relationship between yand x.

    The LR criterion for testing the hypothesis H2 ⋅ 1 against alternatives K21:Θ˜21Ocan be obtained through the following steps.

    (D1) The density function of Y=Y1Y2can be expressed as the product of the marginal density function of Y1and the conditional density function of Y2given Y1.Note that the density functions of Y1under H2 ⋅ 1 and K2 ⋅ 1 are the same.

    (D2) The spaces spanned by each column of EY2|Y1are the same, and let the spaces under K2 ⋅ 1 and H2 ⋅ 1 denote by Ωand ω,respectively. Then


    and dim(Ω) = q + k + 1, dim(ω) = k + 1.

    (D3) The likelihood ratio criterion λis expressed as


    where SΩ=Y2InPΩY2and Sω=Y2InPωY2.

    (D4) Note that EY2|Y1PωPωEY2|Y1=Ounder H2 ⋅ 1. The conditional distribution of Λunder H2 ⋅ 1 is Λp − q(kn − q − k − 1), and hence the distribution of Λunder H2 ⋅ 1 is Λp − q(kn − q − k − 1).

    Note that the Λstatistic is defined through Y2InPΩY2and Y2PΩPωY2, which involve n × nmatrices. We try to write these statistics in terms of the SSP matrix of (y′, x′)′ defined by


    where y¯and x¯are the sample mean vectors. Along the partition of y=y1y2′,we partition Sas


    We can show that


    The first result is obtained by using


    The second result is obtained by using


    where Y˜1=InP1nY1and X˜=InP1nX.

    Summarizing the above results, we have the following theorem.

    Theorem 5.1In the multivariate regression model(5.1), consider to test the hypothesisH2 ⋅ 1 in(5.4) againstK2 ⋅ 1. Then the LR criterionλis given by


    whose null distribution isΛp − q(kn − q − k − 1).

    Note that S221can be decomposed as


    This decomposition is obtained by expressing S221xin terms of S221, S2x1, Sxx1, and Sx21by using an inverse formula


    The decomposition is expressed as


    The result may be also obtained by the following algebraic method. We have






    which gives an expression for PωΩby using Theorem 3.1 (1). This leads to (5.6).


    6. Tests in discriminant analysis

    We consider qp-variate normal populations with common covariance matrix Σand the ith population having mean vector θi. Suppose that a sample of size niis available from the ith population, and let yijbe the jth observation from the ith population. The observation matrix for all the observations is expressed as


    It is assumed that yijare independent, and


    The model is expressed as




    Here, the error matrix Ehas the same property as in (2.1).

    First, we consider to test


    against alternatives K : θi ≠ θjfor some ij. The hypothesis can be expressed as


    The tests including LRC are based on three basic statistics, the within-group SSP matrix W,the between-group SSP matrix B,and the total SSP matrix Tgiven by


    where y¯iand Siare the mean vector and sample covariance matrix of the ith population, and y¯is the total mean vector defined by 1/ni=1qniy,¯iand n=i=1qni.In general, Wand Bare independently distributed as a Wishart distribution Wpnq,Σand a noncentral Wishart distribution Wpq1,Σ;Δrespectively, where


    where θ¯=1/ni=1q+1niθi. Then, the following theorem is well known.

    Theorem 6.1Letλ = Λn/2 be theLRC for testingHin(6.4). Then,Λis expressed as


    where W, B, and Tare given in (6.6). Further, under H, the statistic Λis distributed as a lambda distribution Λp(q − 1, n − q).

    Now we shall show Theorem 6.1 by an algebraic method. It is easy to see that


    The last equality is also checked from that under H


    We have


    Further, it is easily checked that

    1. InPA2=InPA,PAP1n2=PAP1n.

    2. PAP1nPAP1n=O.

    3. fe = dim[ℛ[A]] = tr(In − PA) = n − q,


    Related to the test of H, we are interested in whether a subset of variables y1, …, ypis sufficient for discriminant analysis, or the set of remainder variables has no additional information or is redundant. Without loss of generality, we consider the sufficiency of a subvector y1 = (y1, …, yk)′ of y, or redundancy of the remainder vector y2 = (yk + 1, …, yp)′. Consider to test






    The testing problem was considered by [11]. The hypothesis can be formulated in terms of Maharanobis distance and discriminant functions. For its details, see [12, 13]. To obtain a likelihood ratio for H2 ⋅ 1, we partition the observation matrix as


    Then the conditional distribution of Y2given Y1is normal such that the rows of Y2are independently distributed with covariance matrix Σ221=Σ22Σ21Σ111Σ12, and the conditional mean is given by


    where Θ21˙=(θ1;21,,θq;21).The LRC for H2 ⋅ 1 can be obtained by use of the conditional distribution, and following the steps (D1)–(D4) in Section 5. In fact, the spaces spanned by each column of EY2|Y1are the same, and let the spaces under K2 ⋅ 1 and H2 ⋅ 1 denote by Ωand ω, respectively. Then


    dim(Ω) = q + k, and dim(ω) = q + 1. The likelihood ratio criterion λcan be expressed as


    where SΩ=Y2InPΩY2and Sω=Y2InPωY2.We express the LRC in terms of W, B, and T.Let us partition W, B, and Tas


    where W12:q×pq, B12:q×pq, and T12:q×pq.Noting that PΩ=PA+PInPAY1, we have


    Similarly, noting that Pω=P1n+PInP1nY1, we have


    Theorem 6.2Suppose that the observation matrixYin(6.1) is a set of samples fromNpθi,Σi = 1, …, q. Then the likelihood ratio criterionλfor the hypothesisH2 ⋅ 1 in(6.8) is given by


    whereWandTare given by(6.6). Further, underH2 ⋅ 1,


    Proof.We consider the conditional distributions of W221and T221given Y1by using Theorem 2.3, and see also that they do not depend on Y1.We have seen that


    It is easy to see that Q12=Q1, rankQ1=trQ1=nqk, Q1A=O, Q1X1 = O, and


    This implies that W221|Y1Wpknqk,Σ221and hence W22 ⋅ 1 ∼ Wp − k(n − q − kΣ22 ⋅ 1). For T221, we have


    and hence


    Similarly, Q2is idempotent. Using P1nPA=PAP1n=P1n, we have Q1Q2=Q2Q1=Q1, and hence


    Further, under H2 ⋅ 1,


    7. General multivariate linear model

    In this section, we consider a general multivariate linear model as follows. Let Ybe an n × pobservation matrix whose rows are independently distributed as p-variate normal distribution with a common covariance matrix Σ.Suppose that the mean of Yis given as


    where Ais an n × kgiven matrix with rank k, Xis a p × qmatrix with rank q, and Θis a k × qunknown parameter matrix. For a motivation of (7.1), consider the case when a single variable yis measured at ptime points t1, …, tp(or different conditions) on nsubjects chosen at random from a group. Suppose that we denote the variable yat time point tjby yj. Let the observations yi1, …, yipof the ith subject be denoted by


    If we consider a polynomial regression of degree q − 1 of yon the time variable t, then




    If there are kdifferent groups and each group has a polynomial regression of degree q − 1 of y, we have a model given by (7.1). From such motivation, the model (7.1) is also called a growth curve model. For its detail, see [14].

    Now, let us consider to derive LRC for a general linear hypothesis


    against alternatives Kg:CΘDO.Here, Cis a c × kgiven matrix with rank c, and Dis a q × dgiven matrix with rank d. This problem was discussed by [1517]. Here, we obtain LRC by reducing it to the problem of obtaining LRC for a general linear hypothesis in a multivariate linear model. In order to relate the model (7.1) to a multivariate linear model, consider the transformation from Yto UV:


    where G1 = X(XX)− 1, G2=X˜, and X˜are a p × (p − q) matrix satisfying X˜X=Oand X˜X˜=Ipq.Then, the rows of UVare independently distributed as p-variate normal distributions with means


    and the common covariance matrix


    This transformation can be regarded as one from y = (y1, …, yp)′ to a q-variate main variable u = (u1, …, uq)′ and a (p − q)-variate auxiliary variable v = (v1, …, vp − q)′. The model (7.1) is equivalent to the following joint model of two components:

    1. The conditional distribution of Ugiven Vis


  • The marginal distribution of Vis


  • where


    Before we obtain LRC, first we consider the MLEs in (7.1). Applying a general theory of multivariate linear model to (7.4) and (7.5), the MLEs of Ξ, Ψ112, and Ψ22are given by




    and partition Was


    Theorem 7.1For ann × pobservation matrixY, assume a general multivariate linear model given by(7.1). Then:

    1. The MLEΘ^ofΘis given by


    2. The MLEΨ^112ofΨ112is given by


    Proof.The MLE of Ξis Ξ^=A*A*1A*U.The inverse formula (see (5.5)) gives


    Therefore, we have




    we obtain 1. For a derivation of 2, let B=InPAV.Then, using PA*=PA+PB, the first expression of (1) is obtained. Similarly, the second expression of (2) is obtained.

    Theorem 7.2Letλ = Λn/2 be the LRC for testing the hypothesis(7.2) in the generalized multivariate linear model(7.1). Then,






    HereΘ^is given in Theorem7.1.1. Further, the null distribution isΛd(cn − k − (p − q)).

    Proof.The test of Hgin (7.2) against alternatives Kgis equivalent to testing


    under the conditional model (7.4), where C* = (C O). Since the distribution of Vdoes not depend on Hg, the LR test under the conditional model is the LR test under the unconditional model. Using a general result for a general linear hypothesis given in Theorem 4.1, we obtain




    By reduction similar to those of MLEs, it is seen that S˜e=Seand S˜h=Sh.This completes the proof.

    8. Concluding remarks

    In this chapter, we discuss LRC in multivariate linear model, focusing on the role of projection matrices. Testing problems considered involve the hypotheses on selection of variables or no additional information of a set of variables, in addition to a typical linear hypothesis. It may be noted that various LRCs and their distributions are obtained by algebraic methods.

    We have not discussed with LRCs for the hypothesis of selection of variables in canonical correlation analysis, and for dimensionality in multivariate linear model. Some results for these problems can be found in [3, 18].

    In multivariate analysis, there are some other test criteria such as Lawley-Hotelling trace criterion and Bartlett-Nanda-Pillai trace criterion. For the testing problems treated in this chapter, it is possible to propose such criteria as in [12].

    The LRCs for tests of no additional information of a set of variables will be useful in selection of variables. For example, it is possible to propose model selection criteria such as AIC (see [19]).


    The author wishes to thank Dr. Tetsuro Sakurai for his variable comments for the first draft. The author’s research is partially supported by the Ministry of Education, Science, Sports, and Culture, a Grant-in-Aid for Scientific Research (C), no. 25330038, 2013–2015.

    © 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    How to cite and reference

    Link to this chapter Copy to clipboard

    Cite this chapter Copy to clipboard

    Yasunori Fujikoshi (July 6th 2016). Likelihood Ratio Tests in Multivariate Linear Model, Applied Linear Algebra in Action, Vasilios N. Katsikis, IntechOpen, DOI: 10.5772/62277. Available from:

    chapter statistics

    1540total chapter downloads

    More statistics for editors and authors

    Login to your personal dashboard for more detailed statistics on your publications.

    Access personal reporting

    Related Content

    This Book

    Next chapter

    Matrices, Moments and Quadrature: Applications to Time- Dependent Partial Differential Equations

    By James V. Lambers, Alexandru Cibotarica and Elisabeth M. Palchak

    Related Book

    First chapter

    PID Control Design

    By A.B. Campo

    We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

    More About Us