Tests Concerning Equicorrelation Matrices with Grouped Normal Data

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1 Tests Concerning Equicorrelation Matrices with Grouped Normal Data Paramjit S Gill Department of Mathematics and Statistics Okanagan University College Kelowna, BC, Canada, VV V7 pgill@oucbcca Sarath G Banneheka Department of Statistics and Computer Science University of Sri Jayewardenepura Gangodawila, Nugegoda, Sri Lanka bannehek@sjpaclk Tim B Swartz Department of Statistics and Actuarial Science Simon Fraser University Burnaby, BC, Canada, V5A S6 tim@statsfuca Key Words: intraclass correlation; likelihood ratio test; maximum likelihood estimation; score test; small-sample inference ABSTRACT This paper considers three practical hypotheses involving the equicorrelation matrix for grouped normal data We obtain statistics and computing formulae for common test procedures such as the score test and the likelihood ratio test In addition, statistics and computing formulae are obtained for various small sample procedures as proposed in Skovgaard (200) The properties of the tests for each of the three hypotheses are compared using Monte Carlo simulations

2 Introduction A common data structure in applied statistics occurs when a small number of multivariate measurements are collected from a finite number of groups or classes For example, instruments known as computer-analyzed corneal topographers (CACT) are used to obtain multivariate observations on curvature points where the sample size is typically around 0 (Viana, Olkin and McMahon 993) In such situations, a parsimonious model for the covariance matrix is desired Also, in order to increase sample size, one may wish to combine data from different sources (eg instruments) Therefore, there is a need to test the equality of covariance matrices arising from different groups In this paper we consider tests of hypotheses regarding a special form of the covariance matrix called the equicorrelation or intraclass correlation matrix Such a covariance matrix is suitable when subjects are related (such as a family or a litter of animals) or when measurements are made repeatedly on the same subject More formally, suppose that we have samples of p-dimensional measurements from K groups Let n i be the number of observations from group i =, 2,, K and let n = K i= n i Let X i = x i x i2 x ip x i2 x i22 x i2p x ini x ini 2 x ini p denote the i th data matrix with x ij = (x ij, x ij2,,x ijp ) as the vector of observations on the j th sample from group i Assume x ij N p (µ i, Σ i ), i =, 2,, K, j =, 2,, n i where denotes the unit vector of dimension p and the equicorrelation or intraclass correlation matrix is given by Σ i = σ 2 i ρ i ρ i ρ i ρ i, ( p) < ρ i < ρ i ρ i 2

3 Note that for each group, the measurements share a common mean and correlation Although this is a restrictive structure, it is still appropriate in many contexts The intraclass correlation coefficient ρ i measures the degree of resemblance between the members in the i th group This covariance pattern also arises in the one-way random effects (within each group i) linear model x ijk = µ i + α ij + ε ijk where α ij is the random effect due to the j th unit from group i with E(α ij ) = 0, V ar(α ij ) = σ 2 iα, ε ijk is the random error with E(ε ijk ) = 0, V ar(ε ijk ) = σ 2 iε and Cov(α ij, ε ijk ) = 0 It is easily seen that the covariance matrix of x ij is Σ i with σ 2 i = σ 2 iα+σ 2 iε and ρ i = σ 2 iα/(σ 2 iα+σ 2 iε) (see, for example, Donner and Koval 980) We are interested in the following three practical hypotheses concerning the equicorrelation matrix Case : H 0 : ρ = = ρ K Case 2: H 02 : σ = = σ K Case 3: H 03 : ρ = = ρ K, σ = = σ K The Case problem was first addressed by Konishi and Gupta (989) using a maximum likelihood approach with combined moment estimators for the common ρ Paul and Barnwal (990) derived a score-type test based on Neyman s C(α) While estimating the intraclass correlation coefficient, Ahmed, Gupta, Khan and Nicol (200) also derived a score-type test statistic for the Case problem The Case 2 problem does not seem to have been studied For the Case 3 problem, Han (975) obtained a modified likelihood ratio test Our main concern regarding the Case, Case 2 and Case 3 problems involves data sets where the sample sizes are small For small sample sizes, asymptotic tests such as those described above may be inappropriate For example, the actual type error of a statistic may differ considerably from its nominal level In this regard, we have derived various small sample tests based on the general theory of Skovgaard (200) and we have compared these tests to the standard tests Some of these standard tests have not previously appeared in 3

4 the literature and are derived here This paper serves as a summary of the variety of tests that one might consider in connection with the Case, Case 2 and Case 3 problems Small sample tests are typically derived by considering additional terms in asymptotic expansions These corrections attempt to hasten the convergence of statistics to their asymptotic distributions The downside of such corrections is that they sometimes involve difficult calculations Well known examples of correction terms include Bartlett and Edgeworth series type corrections, saddlepoint corrections and the signed loglikelihood ratio correction Reviews of asymptotic methods in statistics are given by Barndorff-Nielsen and Cox (994), Skovgaard (200) and Reid (2003) Notably, Skovgaard (200) generalized the Barndorff-Nielsen (99) correction to the multi-parameter case; we follow Skovgaard s approach to derive small sample corrections for likelihood ratio tests involving equicorrelation matrices In section 2, we provide a general description of the modified likelihood ratio test due to Skovgaard (200) and the score test We also discuss unrestricted maximum likelihood estimation for the problem considered in the paper In sections 3, 4 and 5, Cases, 2 and 3 are discussed in detail with an emphasis on deriving asymptotic test procedures In section 6, simulation results are provided to compare the various tests under each of the three hypotheses 2 PRELIMINARIES 2 The Skovgaard Modifications Suppose that a random variable X belongs to the exponential family having a density of the form f(x; θ) = b(x) exp{θ t(x) κ(θ)} Then θ is called the canonical parameter and t is the corresponding canonical sufficient statistic Let ω = (ν, ψ ) be a (possibly nonlinear) function of θ where ψ ν = (ψ,, ψ q ) is the vector of parameters of interest and = (ν,,ν r q ) is a vector of nuisance parameters Suppose one is interested in testing the null hypothesis H 0 : ψ = = ψ q = ψ 0 (unspecified) Let ω = ( ν, ψ ) denote the maximum likelihood (ML) estimate of the full parameter vector and ω 0 =, ψ ( ν 0 ) be the 4

5 maximum likelihood estimate of the free parameters under the null hypothesis Let l( ω) and l( ω 0 ) be the corresponding maximum values of the loglikelihood Denote J νν and I νν as the observed and expected Fisher information matrices for the ν-part of the parameter vector and let J νν and Ĩνν denote the corresponding estimates obtained by replacing the unknown parameter values by their restricted ML estimates Further, let τ and Ω denote the mean and variance matrix of t; τ and Ω are the corresponding estimates when the unknown parameters are replaced by the restricted ML estimates and Ω is the estimate under unrestricted ML estimation Let w = 2(l( ω) l( ω 0 )) denote the likelihood ratio test statistics for testing H 0 Following Skovgaard (200), we define δ = {(t τ) Ω (t τ)} (q )/2 w (q )/2 (ˆθ θ) (t τ) The corrected likelihood ratio is then given by ( Ω J ) /2 νν () Ω Ĩνν w = w ( w ln δ ) 2 (2) where it is hoped that w χ 2 q is a better approximation under H 0 than the likelihood ratio test w χ 2 q Another corrected χ2 q test statistic (Skovgaard 200) is which may assume a negative value, especially when w is small w = w 2 lnδ (3) 22 The Score Test The score test is attractive for its relative simplicity of computations as we need maximum likelihood estimation only under the null hypothesis For the r-dimensional parameter vector ω = (ν, ψ ), let U = (U ν, U ψ) be the score vector where U ν = l ν, U ψ = l ψ Then under the null hypothesis H 0 : ψ = = ψ q = ψ 0, 5

6 U N r 0, T S S R, where T is a (r q) (r q) matrix with T(i, j) = E, ν i ν j H0 S is a q (r q) matrix with S(i, j) = E and R is a q q matrix with R(i, j) = E The score test statistic is then given by ψ i ν j ( 2 l ψ i ψ j ) H0 H0 ξ = U ψ(r ST S ) U ψ and under the null hypothesis, ξ is asymptotically distributed as χ 2 q Note that all the terms involved in ξ are evaluated at the ML estimates of ω under H 0 In this paper, we compare the performance of the score test with other tests using Monte Carlo simulations 23 Unrestricted Maximum Likelihood Estimation In Cases, 2 and 3, tests in this paper require maximum likelihood estimation under the alternative hypotheses We refer to this as unrestricted maximum likelihood estimation Let A be the p p matrix A = p p p p p( p) p( p) p( p) p( p) p (p ) p( p) 6

7 Define Z i = X i A, i =, 2,, K or equivalently, z ij = Ax ij, i =, 2,, K, j =, 2,, n i Then z ij N p (µ i, Σ i), where µ = i ρ i + p ( ρ i ) µ i 0 ρ i , Σ i = AΣ ia = σ 2 i 0 0 ρ i ρ i Note that z ij N (µ i, σ 2 i {ρ i + p ( ρ i )}) and z ij2,, z ijp iid N (0, σ 2 i { ρ i}), i =, 2,, K ; j =, 2,, n i Also, z ij is independent of z ij2,,z ijp The log-likelihood in terms of µ i, σ i and ρ i is l = l(µ,,µ K, σ 2,,σ 2 K, ρ,,ρ K ) = K 2 p n i ln σ 2 i K n i ln{ρ i= 2 i + p ( ρ i )} K i= 2 (p ) n i ln( ρ i ) i= K n i (z ij µ i ) 2 K 2σ 2 i {ρ i + p ( ρ i )} n i p zijk 2 2σ 2 i( ρ i ) i= j= i= j= k=2 It follows that the ML estimates ˆµ i, ˆσ 2 i and ˆρ i of µ i, σ 2 i and ρ i are given by ˆµ i = n i n i z ij j= and ˆσ 2 i = ni j=(z ij ˆµ i ) 2 ni n i p{ˆρ i + p ( ˆρ i )} + j= k=2 z2 ijk n i p( ˆρ i ) ˆρ i = p n i j=(z ij ˆµ i ) 2 (p ) n i p n i j=(z ij ˆµ i ) 2 + n i j= k=2 z2 ijk j= k=2 z2 ijk (4) 3 CASE : H 0 : ρ = = ρ K The Case problem with grouped normal data has been well studied in the literature We fill in some gaps concerning the likelihood ratio test and derive the Skovgaard (200) test For comparison purposes, we present the score test (Paul and Barnwall 990) 7

8 3 Likelihood Ratio Test Under the hypothesis H 0 : ρ = ρ 2 = = ρ K = ρ (an unspecified value), the log-likelihood is l 0 = l 0 (µ,,µ K, σ 2,,σ2 k, ρ) = 2 p K i= K n i i= j= n i ln σ 2 i 2 n ln{ρ + p ( ρ)} n(p ) ln( ρ) 2 (z ij µ i ) 2 2σ 2 i {ρ + p ( ρ)} K n i p i= j= k=2 z 2 ijk 2σ 2 i( ρ) The maximum likelihood estimates for µ i and σ 2 i are given by and σ 2 i = The score equation for ρ µ i = n i z ij = ˆµ n i i j= U(ρ) = l 0 ρ = n(p ) n(p ) + 2p{ρ + p ( ρ)} 2( ρ) ni j=(z ij µ i ) 2 ni n i p{ ρ + p ( ρ)} + j= k=2 z2 ijk (5) n i p( ρ) 2( ρ) 2 K n i p zijk 2 /σ2 i = 0 i= j= k=2 (p ) Ki= ni + j=(z ij µ i ) 2 /σ 2 i 2p{ρ + p ( ρ)} 2 does not admit an explicit solution Estimates are obtained by the scoring method ρ (m+) = ρ (m) + U(ρ(m) ), m = 0,, (6) I(ρ (m) ) where I(ρ) = E ( 2 l 0 ρ 2 ) is the Fisher information To obtain the Fisher information, we have 2 l 0 ρ = n(p ) 2 n(p ) 2 2p 2 {ρ + p ( ρ)} 2+ 2( ρ) (p )2 K ni i= j=(z ij µ i ) 2 /σ 2 i 2 p 2 {ρ + p ( ρ)} 3 Since E(z ij µ i ) 2 = σ 2 i {ρ + p ( ρ)} and E(zijk 2 ) = σ2 i ( ρ), it follows that I(ρ) = E ( 2 ) l 0 = ρ 2 n(p ) 2 n(p ) + 2p 2 {ρ + p ( ρ)} 2 2( ρ) 2 Ki= ni j= k=2 z2 ijk /σ2 i ( ρ) 3 We summarize the estimation procedure for µ,, µ K, σ 2,,σ2 K algorithm: and ρ in the following 8

9 Let m = 0 and ρ (0) = K i= ˆρ i /K where ˆρ i is given by (4) 2 Let (σ 2 i) (m) be the m th approximation to σ 2 i obtained by substituting ρ (m) for ρ in (5) 3 Calculate ρ (m+) using (6) where (σ 2 i )(m) and ρ (m) replace σ 2 i and ρ respectively 4 Let m = m + 5 If U(ρ (m) ) < 0 6 then stop Else, go to step 2 and continue In all of the simulations that we have considered, we have not experienced any difficulty with the above algorithm Let l = l( ω) = l(ˆµ,, ˆµ K, ˆσ 2,, ˆσ 2 K, ˆρ,,ˆρ K ) and l 0 = l 0 ( ω 0 ) = l 0 ( µ,, µ K, σ 2,, σ 2 K, ρ) be the maximized values of the loglikelihood Then, under the null hypothesis H 0, the likelihood ratio test statistic w = 2( l l 0 ) follows the chi-square distribution with (K ) degrees of freedom for large sample sizes 32 Modified Likelihood Ratio Tests In the following, we derive the Skovgaard (200) modifications to the likelihood ratio test statistic which is intended to provide more accurate inference in small-sample situations For our problem, the canonical parameter θ = (θ,,θ K, θ K+,,θ 2K, θ 2K+,,θ 3K ) and canonical sufficient statistic t = (t,,t K, t K+,,t 2K, t 2K+,,t 3K ) are as follows θ i = λ 2, t i = zij 2 /2, i j= θ K+i = µ n i i λ 2, t K+i = z ij, i j= θ 2K+i = n i p β 2, t 2K+i = zijk 2 /2 i j= k=2 where i =,, K, λ 2 i = σ 2 i {ρ i + p ( ρ i )} and β 2 i = σ 2 i ( ρ i) Then the terms in τ = E(t) are τ i = n i E(z 2 ij )/2 = n i(µ 2 i + λ2 i )/2, n i 9

10 τ K+i = n i E(z ij ) = n i µ i, and the elements in Ω = V ar(t) are τ 2K+i = n i (p )E(z 2 ij2)/2 = n i (p )β 2 i/2 Ω(i, i) = n i V ar(z 2 ij)/4 = n i (λ 2 i + 2µ 2 i)λ 2 i/2, Ω(K + i, K + i) = n i V ar(z ij ) = n i λ 2 i, Ω(2K + i, 2K + i) = n i (p ) V ar(z 2 ij2)/4 = n i (p )β 4 i/2, n i Ω(K + i, i) = Ω(K + i, i) = Cov where i =,,K All the other covariances are zero n i zij, 2 z ij j= j= /2 = n i µ i λ 2 i In order to obtain the Skovgaard (200) term δ in (), we need to evaluate J νν and Ĩνν for the nuisance parameter ν = (µ,,µ K, σ 2,, σ 2 K) We have J (µ i, µ i ) = 2 l n i µ 2 = i ω= ω σ 2 i { ρ + p ( ρ)}, J (µ i, σ 2 i) = 2 l ni j=(z ij µ µ i σ 2 = i ) i ω= ω σ 4 i { ρ + p ( ρ)} = 0, J (σ 2 i, σ 2 i) = 2 l σ 4 = n ni ip j=(z ij µ i ω= ω 2 σ 4 + i ) 2 ni i σ 6 i { ρ + p ( ρ)} + j= k=2 z2 ijk σ 6 = i ( ρ) In addition, J (µ i, µ j ) = J (µ i, σ 2 j ) = J (σ 2 i, σ2 j ) = 0, i j since Further, 2 l µ i µ j = 2 l σ 2 i σ 2 j = 2 l µ i σ 2 j = 0, i j n i Ĩ(µ i, µ i ) = E = µ 2 i ω= ω σ 2 i { ρ + p ( ρ)}, ( Ĩ(µ i, σ 2 2 ) i) = E = 0, µ i σ 2 i ( ω= ω Ĩ(σ 2 2 ) i, σ2 i ) = E = n ip σ 4 i ω= ω 2 σ 4, i Ĩ(µ i, µ j ) = Ĩ(µ i, σ 2 j) = Ĩ(σ2 i, σ 2 j) = 0, i j n ip 2 σ 4 i using equation (5) Note that as J νν = Ĩνν, it is not necessary to evaluate Ĩνν and J νν as they cancel out in δ 0

11 33 Score Test In order to evaluate the score test statistic ( ξ = U ψ R ST S ) Uψ we require the vector of scores U ψ = (U,,U K ) where U i = l ρ i ω= ω and the K K matrix R, the K 2K matrix S and the 2K 2K matrix T where R(i, j) = E, ρ i ρ j ( ω= ω 2 ) S(i, j) = E, ρ i µ j ( ω= ω 2 ) S(i, K + j) = E, ρ i σ 2 j ( ω= ω 2 ) T(i, j) = E, µ i µ j ( ω= ω 2 ) T(i, K + j) = T K+i j = E, µ i σ 2 j ( ω= ω 2 ) T(K + i, K + j) = E σ 2 i σ 2 j ω= ω Explicit expressions for the components of ξ are given by U i n i = 2{ρ + p ( ρ)} n i(p ) 2( ρ) (p ) ni j=(x ij x i ) 2 ni + 2pσ 2 i {ρ + p ( ρ)} j= k=2 z2 ijk, 2 2σ 2 i( ρ) 2 R = diag ( 3ni (p ) 2( ρ) n ) i(3p 2)(p ), 2 2p 2 {ρ + p ( ρ)} 2 K K S = [ ( 0 K K diag ) n i (p )ρ 2σ 2 i ( ρ){ρ+p ( ρ)} K K ], T = ( diag n i σ 2 i {ρ+p ( ρ)} ) K K 0 K K diag 0 K K ( ) n i p 2σ 4 i K K As the matrices R, S and T have a special structure, the inverses required in the score test statistic ξ can be easily computed

12 4 CASE 2: H 02 : σ = = σ K As mentioned previously, there has been considerable amount of work directed at the Case problem The Case 2 problem does not seem to have been studied It is conceivable that group correlations may differ yet group variances may be the same For example, if the same type of instruments are used across groups and the source of variability is due to measurement error, then group variances are comparable However, if additional sources of variability are at play, then H 02 may be false For the three Case 2 statistics that follow, the asymptotic distributions follow a χ 2 K distribution 4 Likelihood Ratio Test Under H 02 : σ = = σ K = σ (an unspecified value), the log-likelihood is l 0 = l 0 (µ,,µ K, σ 2, ρ,,ρ K ) = 2 np ln σ2 K n i ln{ρ 2 i + p ( ρ i )} K 2 (p ) n i ln( ρ i ) n i K σ 2 i= j= i= (z ij µ i ) 2 2{ρ i + p ( ρ i )} σ 2 K n i p i= j= k=2 i= z 2 ijk 2( ρ i ) The maximum likelihood estimates for µ,, µ K and σ 2 are given by and The equations σ 2 = np { K n i i= j= l 0 n i (p ) = ρ i 2p{ρ i + p ( ρ i )} + n i(p ) 2( ρ i ) µ i = n i z ij = ˆµ n i i j= (z ij µ i ) 2 { ρ i + p ( ρ i )} + p k=2 zijk 2 } (7) ( ρ i ) (p ) ni + j=(z ij µ i ) 2 ni 2pσ 2 {ρ i + p ( ρ i )} j= k=2 z2 ijk = 0 2 2σ 2 ( ρ i ) 2 do not lead to an explicit solution for ρ i, i =,, K To obtain the ML estimate ρ i of ρ i under H 02, our first instinct was to use the scoring method However, we experienced some convergence problems Instead, the method of bisection was successfully used on equation (8) Because of the interdependence between σ 2 and ρ i, the following algorithm was used for estimation 2 (8)

13 Let m = 0 and ρ (0) i = ˆρ i where ˆρ i is given by (4), i =,, K 2 Let (σ 2 ) (m) be the m th approximation to σ 2 obtained by substituting ρ (m) i (7) for ρ i in 3 Calculate ρ (m+) i using bisection on (8) where (σ 2 ) (m) replaces σ 2, i =,, K 4 Let m = m + 5 If l 0 ρ i < 0 6 for i =,,K then stop Else, go to step 2 and continue The likelihood ratio test statistic is given by w = 2( l l 0 ) where l is the maximized unrestricted loglikelihood and l 0 is the maximized restricted loglikelihood 42 Modified Likelihood Ratio Tests Everything except the terms Ĩνν and J νν in the δ formula () remains the same as in Case The matrices J νν and Ĩνν for the nuisance parameter ν = (µ,,µ K, ρ,,ρ K ) are as follows For i =,,K, J νν (i, i) = 2 l n i µ 2 = i ω= ω σ 2 { ρ i + p ( ρ i )} ni J νν (i, K + i) = 2 l (p ) = j=(z ij µ i ) µ i ρ i ω= ω p σ 2 { ρ i + p ( ρ i )} = 0 2 J νν (K + i, K + i) = 2 l n i (p ) 2 ρ 2 = i ω= ω 2p 2 { ρ i p ( ρ i )} n i(p ) 2 2( ρ i ) 2 + (p )2 n i j=(z ij µ i ) 2 ni p 2 σ 2 { ρ i + p ( ρ i )} + frac zijk σ 2 2 ( ρ 3 i ) 3 j= k=2 All other terms are zero since 2 l = 2 l = µ i µ j ρ i ρ j 2 l µ i ρ j = 0, i j For i =,,K, n i Ĩ νν (i, i) = E = µ 2 i ω= ω σ 2 { ρ i + p ( ρ i )} Ĩ νν (i, K + i) = E = 0 µ i ρ i ( ω= ω 2 ) n i (p ) 2 Ĩ νν (K + i, K + i) = E = ω= ω 2p 2 { ρ i + p ( ρ i )} + n i(p ) 2 2( ρ i ) 2 ρ 2 i 3

14 and all other terms are zero 43 Score Test The score vector U ψ and the matrices R, S and T are given by the following expressions where i, j =,,K U i = l σ 2 i = n ip ω= ω 2 σ 2 + ni j=(z ij µ i ) 2 ni 2 σ 4 { ρ i + p ( ρ i )} + j= k=2 z2 ijk 2 σ 4, ( ρ i ) R(i, i) = E = n ip ω= ω 2 σ 4, σ 4 i R(i, j) = E = 0, i j, σ 2 i σ 2 j ω= ω S(i, i) = E = 0, σ 2 i µ i ω= ω S(i, K + i) = E ( 2 l σ 2 i ρ i ) ω= ω = (p )n i 2p σ 2 { ρ i + p ( ρ i )} (p )n i 2 σ 2 ( ρ i ), S(i, j) = E = 0, i j, σ 2 i µ j ω= ω S(i, K + j) = E = 0, i j, σ 2 i ρ j ω= ω T(i, j) = E = ν i ν j Ĩνν(i, j) ω= ω where Ĩνν is given in Subsection 42 The score statistic for testing H 02 : σ = = σ K is then given by ξ = U ψ (R ST S ) U ψ 4

15 5 CASE 3: H 03 : σ = = σ K, ρ = = ρ K Since H 0 (Case) and H 02 (Case2) are common hypotheses of interest with respect to the grouped normal data problem, it follows that the composite hypothesis H 03 is also of interest Although there have been a number of test procedures proposed for H 03 (Han 975), we make use of the methodologies developed for Cases and 2 to derive the Skovgaard (200) modifications The likelihood ratio test and the score test are presented for comparison purposes For the three Case 3 statistics that follow, the asymptotic distributions follow a χ 2 2K 2 distribution 5 Likelihood Ratio Test Under H 03 : σ = = σ K = σ and ρ = = ρ K = ρ, the log-likelihood is l 0 = l 0 (µ,,µ K, σ 2, ρ) = 2 np ln σ2 2 n ln{ρ + p ( ρ)} n(p ) ln( ρ) 2 K n i (z 2σ 2 {ρ + p ij µ ( ρ)} i ) 2 K n i 2σ 2 ( ρ) The ML estimates are given by µ i = n i z ij = ˆµ n i, i σ 2 = np j= K n i i= j= i= j= { (zij µ i ) 2 ρ + p ( ρ) + p k=2 zijk 2 }, ( ρ) ρ = p K ni i= j=(z ij µ i ) 2 (p ) K ni i= p K ni i= j=(z ij µ i ) 2 + K ni i= j= k=2 z2 ijk 52 Modified Likelihood Ratio Tests p zijk 2 i= j= k=2 j= k=2 z2 ijk For the Skovgaard (200) modifications, the matrices J νν and Ĩνν for the nuisance parameter ν = (µ,,µ K ) are as follows J νν (i, i) = 2 l n µ 2 = i ω= ω σ 2 { ρ + p ( ρ)}, J νν (i, j) = 2 l = 0, i j, µ i µ j ω= ω 5

16 Ĩ νν (i, i) = E = ω= ω µ 2 i Ĩ νν (i, j) = E ( 2 l µ i µ j ) n σ 2 { ρ + p ( ρ)}, ω= ω = 0, i j Note that Ĩνν = J νν providing a simplification in the δ formula () 53 Score Test The score vector U ψ = (U,,U 2K ) and the matrices R and S are given by U i = l σ 2 i = n ip ω= ω 2 σ 2 + ni j=(z ij µ i ) 2 ni 2 σ 4 { ρ + p ( ρ)} + j= k=2 z2 ijk 2 σ 4, ( ρ) U K+i = l n i (p ) = ρ i ω= ω 2p{ ρ + p ( ρ)} +n i(p ) ) ni +(p j=(z ij µ i ) 2 ni 2( ρ) 2p σ 2 { ρ + p ( ρ)} j= k=2 z2 ijk 2 2 σ 2, ( ρ) 2 R(i, i) = E = n ip σ 4 i ω= ω 2 σ 4, (p )n i R(i, K + i) = R(K + i, i) = E = σ 2 i ρ i ω= ω 2p σ 2 { ρ + p ( ρ)} (p )n i 2 σ 2 ( ρ), n(p ) 2 n(p ) R(K + i, K + i) = E = + ω= ω 2p 2 { ρ + p ( ρ)} 2 2( ρ) 2 ρ 2 i and all other terms are zero since ( 2 ) ( 2 ) E = E = E = 0, i j σ 2 i σ 2 j ω= ω ρ i ρ j ω= ω σ 2 i ρ j ω= ω Further, since S = E = 0 ψ i ν j ω= ω ( 2 ) E = E = 0, σ 2 i µ i ω= ω ρ i µ i ( ω= ω 2 ) ( 2 ) E = E = 0, i j σ 2 i µ j ω= ω ρ i µ j ω= ω Then, the score statistic for testing H 03 reduces to ξ = U ψ (R ST S ) U ψ = U ψ R U ψ 6

17 6 SIMULATION RESULTS AND DISCUSSION We present the results of some Monte Carlo simulations to investigate the adequacy of the test statistics discussed in this paper The following tables show the Type I error probabilities of the likelihood ratio statistic (w), the modified likelihood ratio statistics (w and w ) and the score statistic (ξ) based on N =00,000 simulations In each case, we used the nominal value 005 and note that the results are comparable for the nominal value 00 In addition to investigating the Type I error rate, we are also interested in the adequacy of the χ 2 approximations over the entire range of the test statistics To do so, we calculate the Cramér-von Mises statistic (Anderson and Darling 954) W 2 = /(2N) + {Z (i) (2i )/(2N)} 2 where Z (i) is the i th order statistic corresponding to Z = F(Y ), F is the cumulative distribution function (cdf) of the relevant χ 2 distribution and Y is the test statistic obtained when generating from the equicorrelation model in question The Cramér-von Mises statistic is one in the wide class of discrepancy measures given by the Cramér-von Mises family Q = n (F n (y) F(y)) 2 Ψ(y)dF(y) where F and F n = F n (Y,,Y n ) are the cdf and the empirical distribution function (edf) respectively, and Ψ is a suitable function which gives weights to the squared difference (F n (y) F(y)) 2 Larger values of the statistic Q (and in our case W 2 ), provide greater evidence of the lack of fit between the data Y,,Y n and the proposed F Theory and practical issues associated with the Cramér-von Mises family of statistics are described in detail in Stephens (986) The simulation results are presented in Tables, 2 and 3 for K = 5 groups where n = = n K Before discussing the particular simulation results, we make some general comments It appears that when the sample size n i reaches 20, all four statistics are adequate as the Type I error rates are close to the nominal values Therefore, our discussion is focused on small sample problems and we choose n = = n K = 5 Also, except in a few instances, the likelihood ratio statistic w is inferior to the w, w and ξ Finally, there seems to be little difference in the results when the number of variables changes from p = 5 to p = 0 7

18 In Table, we present the simulation results with respect to the Case scenario The w and ξ statistics perform best with respect to the Type I error rate However, our recommendation is the modified likelihood ratio statistic w based on its superior goodness of fit In Table 2, we present the simulation results with respect to the Case 2 scenario We observe that the performance of all of the test statistics depends on the correlation ρ i Notably, w, w and ξ perform worse at the extreme values of ρ i Again, our recommendation is w although its performance is not as good as in Case In Table 3, we present the simulation results with respect to the Case 3 scenario In Case 3, the score test ξ has a slight edge over the other tests However, we note that the score test is anti-conservative whereas the two modified likelihood ratio tests are conservative ACKNOWLEDGMENTS Gill and Swartz were partially supported by grants from the Natural Sciences and Engineering Research Council of Canada The work was initiated during Banneheka s sabbatical visit to Simon Fraser University in 2003 Authors thank a referee for valuable comments References [] Ahmed, S E; Gupta, A K; Khan, S M; Nicol, C J Simultaneous estimation of several intraclass correlation coefficients Annals of the Institute of Statistical Mathematics 200, 53, [2] Anderson, T W; Darling, D A Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes Annals of Mathematical Statistics, 954, 23, [3] Barndorff-Nielsen, O E Modified signed loglikelihood ratio Biometrika, 99, 78, [4] Barndorff-Nielsen, O E; Cox, D R Inference and Asymptotics Chapman and Hall, London, 994 8

19 [5] Donner, A; Koval, J J The estimation of intraclass correlation in the analysis of family data Biometrics, 980, 36, 9-25 [6] Han, C -P Testing the equality of covariance matrices under intraclass correlation models Annals of the Institute of Statistical Mathematics, 975, 27, [7] Konishi, S; Gupta, A K Testing the equality of several intraclass correlation coefficients Journal of Statistical Planning and Inference, 989, 2, [8] Paul, S R; Barnwal, R K Maximum likelihood estimation and a C(α) test for a common intraclass correlation The Statistician, 990, 39, 9-24 [9] Reid, N Asymptotics and the theory of inference Annals of Statistics, 2003, 3, [0] Skovgaard, I M Likelihood asymptotics Scandinavian Journal of Statistics, 200, 28, 3-32 [] Stephens, M A Tests based on EDF statistics Goodness of Fit Techniques eds R B D Agostino and M A Stephens, Eds; Marcel Dekker, New York, 986 [2] Viana, M A G; Olkin, I; McMahon, T T Multivariate assessment of computeranalyzed corneal topographers Journal of the Optical Society of America Series A, 993, 0,

20 Table Case (H 0 : ρ = = ρ K where σ 2 = = σ2 K = 0 and K = 5) Type I error rates Cramer-von Mises statistic p = 5 ρ i w w w ξ w w w ξ (n i = 5) (n i = 20) p = 0 (n i = 5) (n i = 20)

21 Table 2 Case 2 (H 02 : σ = = σ K where K = 5) Type I error rates Cramer-von Mises statistic p = 5 ρ i w w w ξ w w w ξ (n i = 5) (n i = 20) p = 0 (n i = 5) (n i = 20)

22 Table 3 Case 3 (H 03 : σ = = σ K, ρ = = ρ K where K = 5) Type I error rates Cramer-von Mises statistic p = 5 ρ i w w w ξ w w w ξ (n i = 5) (n i = 20) p = 0 (n i = 5) (n i = 20)