1471-2288-8-24 1471-2288 Research article <p>Comparison of two dependent within subject coefficients of variation to evaluate the reproducibility of measurement devices</p> Shoukri M Mohamed shoukri@kfshrc.edu.sa Colak Dilek dkcolak@gmail.com Kaya Namik namikkaya@gmail.com Donner Allan allan.donner@schulich.uwo.ca

Department of Epidemiology and Biostatistics, Schulich School of Medicine, University of Western Ontario, London, Ontario, Canada

Department of Biostatistics and Epidemiology, King Faisal Specialist Hospital and Research Centre, Riyadh, Saudi Arabia

Department of Genetics, King Faisal Specialist Hospital and Research Centre, Riyadh, Saudi Arabia

 BMC Medical Research Methodology 1471-2288 2008 8 1 24 http://www.biomedcentral.com/1471-2288/8/24 18430244 10.1186/1471-2288-8-24 18 12 2007 22 4 2008 22 4 2008 2008 Shoukri et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

The within-subject coefficient of variation and intra-class correlation coefficient are commonly used to assess the reliability or reproducibility of interval-scale measurements. Comparison of reproducibility or reliability of measurement devices or methods on the same set of subjects comes down to comparison of dependent reliability or reproducibility parameters.

Methods

In this paper, we develop several procedures for testing the equality of two dependent within-subject coefficients of variation computed from the same sample of subjects, which is, to the best of our knowledge, has not yet been dealt with in the statistical literature. The Wald test, the likelihood ratio, and the score tests are developed. A simple regression procedure based on results due to Pitman and Morgan is constructed. Furthermore we evaluate the statistical properties of these methods via extensive Monte Carlo simulations. The methodologies are illustrated on two data sets; the first are the microarray gene expressions measured by two plat- forms; the Affymetrix and the Amersham. Because microarray experiments produce expressions for a large number of genes, one would expect that the statistical tests to be asymptotically equivalent. To explore the behaviour of the tests in small or moderate sample sizes, we illustrated the methodologies on data from computer-aided tomographic scans of 50 patients.

Results

It is shown that the relatively simple Wald's test (WT) is as powerful as the likelihood ratio test (LRT) and that both have consistently greater power than the score test. The regression test holds its empirical levels, and in some occasions is as powerful as the WT and the LRT.

Conclusion

A comparison between the reproducibility of two measuring instruments using the same set of subjects leads naturally to a comparison of two correlated indices. The presented methodology overcomes the difficulty noted by data analysts that dependence between datasets would confound any inferences one could make about the differences in measures of reliability and reproducibility. The statistical tests presented in this paper have good properties in terms of statistical power.

Background

An extensive literature has been developed on procedures for testing the equality of two or more independent coefficients of variation as measures of reproducibility 345. Their work shows that likelihood-based methods such as the likelihood ratio (LR) test, score test, and tests based on the method of generalized statistics developed by Weerahandi 6, provide efficient procedures for comparing coefficient of variations (CV) in univariate normal populations or from independent samples. However, there are situations where comparing CVs from related samples should be considered. Typical situation is when two instruments are used to measure the same set of subjects, and each subject is repeatedly measured by the same instrument. We shall explain in the methods section the reason why the within-subject coefficient of variation (WSCV) is a more appropriate measure of reproducibility than the CV. Many authors use the terms reliability and reproducibility interchangeably 789; however we believe that they are conceptually different. The reliability is the degree of closeness of the repeated observation on the same subject under the same experimental conditions, so the instrument is always the same. The Intra-class correlation coefficient (ICC) is commonly used as a measure of reliability. It is calculated as the ratio between subjects variance to the total variance. Therefore, the larger the heterogeneity among the subjects, with lower or equal random error the easier it is to differentiate among subjects. In other words, the ICC measures how distinguishable the subjects are. On the other hand, reproducibility determines the degree of closeness of the repeated observations made on the same subject either by the same instrument or different instruments. There is a wide debate among statisticians and psychometricians related to the choice of appropriate measures of reliability and reproducibility. We refer the interested reader to 1011. The main focus of our paper is on the reproducibility parameter.

An important application from molecular biology research in which correlated/dependent reproducibility coefficients are compared is when microarray technologies are compared in terms of reproducibility of gene expression measurements. DNA Microarrays are powerful technologies which make it possible to study genome-wide gene expressions and are extensively used in biological research. As the technology evolves rapidly a number of different platforms became available, which introduces some challenges for researchers to know which technology is best suited for their needs. There have been various studies that directly compared the performance of one platform with another in terms of cross-platform comparability and agreement of gene expression results. However the results of these studies are conflicting: some demonstrate concordance, others discordance between technologies 121314151617. Thus one needs to take into consideration the accuracy and reproducibility of different types of microarrays when allocating the laboratory resources for future experiments. The key factors for selecting an appropriate platform are (1) Intra-assay reproducibility, and (2) the degree of cross-platform agreement 18. The concordance among microarray platforms would allow researchers to directly compare their measurements and perform meta-analyses.

Most of the microarray reliability or reproducibility and cross-platform studies use Pearson's correlation, as an index of reproducibility or agreement. However, it has long been recognized that application of procedures such as the paired t-test and Pearson's correlation are not appropriate tools for measuring agreement between measuring devices 1920. Rather, indices such as the intra-class correlation coefficient 21 and the within- subject coefficient of variation should be used as measures of reproducibility. It has also been demonstrated that the within-subject coefficient of variation is very useful in assessing instrument reproducibility 822.

The main focus of this paper is to develop several procedures for testing the equality of two dependent within-subject coefficients of variation computed from the same sample of subjects, which is, to the best of our knowledge, has not been dealt with in the statistical literature, and to evaluate the statistical properties of these methods via extensive Monte Carlo simulation. We propose two approaches; one is likelihood based (LRT, Wald, and Score test), and the other is a regression based approach coined as PM test. After evaluating the statistical properties (power and empirical level of significance) of these tests using Monte Carlo simulation, the methodology is illustrated on data from two biomedical studies.

Methods

Likelihood based methodology

Suppose that we are interested in comparing the reproducibility of two instruments. Let xijl be the jth measurement of the ith subject by the lth instrument, j = 1,2,... ml, i = 1,2,... n, and l = 1, 2. To evaluate the WSCV we consider the one-way random effects model

xijl = μl + bi + eijl

where μl is the mean value of measurements made by the lth instrument, bi are independent random subject effects with bi ~ N(0, σb2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdkgaIbqaaiabikdaYaaaaaa@3004@), and eijl are independent N(0, σl2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdYgaSbqaaiabikdaYaaaaaa@3018@). Many authors have used the intra-class correlation coefficient (ICC), ρl defined by the ratio ρl=σb2/(σb2+σl2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyWdi3aaSbaaSqaaiabdYgaSbqabaGccqGH9aqpdaWcgaqaaiabeo8aZnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaaakeaadaqadaqaaiabeo8aZnaaDaaaleaacqWGIbGyaeaacqaIYaGmaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaemiBaWgabaGaeGOmaidaaaGccaGLOaGaayzkaaaaaaaa@3F72@ as measure of reproducibility/reliability 1823. Quan and Shih 8 argued that ρl is study-population based since it involves between-subject variation. Meaning that the more heterogeneity in the population, the larger the ρl. Alternatively, they proposed the within-subject coefficient of variation (WSCV) θl = σl/μl as a measure of reproducibility. It determines the degree of closeness of repeated measurements taken on the same subject either by the same instruments or on different occasions under the same conditions. It is clear that, the smaller the WSCV, the better the reproducibility. We distinguish the WSCV from the coefficient of variation CVl=(σb2+σl2)12/μl MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qamKaemOvay1aaSbaaSqaaiabdYgaSbqabaGccqGH9aqpjuaGdaWcgaqaamaabmaabaGaeq4Wdm3aa0baaeaacqWGIbGyaeaacqaIYaGmaaGaey4kaSIaeq4Wdm3aa0baaeaacqWGSbaBaeaacqaIYaGmaaaacaGLOaGaayzkaaWaaWbaaeqabaWaaSGaaeaacqaIXaqmaeaacqaIYaGmaaaaaaqaaiabeY7aTnaaBaaabaGaemiBaWgabeaaaaaaaa@416F@ since CVl involves σb2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdkgaIbqaaiabikdaYaaaaaa@3004@ in the numerator and similar to ρl is population based. Therefore, more heterogeneity in the population would result in a large value of CVl. For that reason we shall focus our work on the WSCV rather than the CV. We also note that there is an inverse relationship between the ICC (ρl) and the corresponding within subject variance σl2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdYgaSbqaaiabikdaYaaaaaa@3018@. Clearly, larger values of ICC (higher reliability) would be associated with smaller WSCV (better reproducibility). The focus of this paper is on aspects of statistical inference on the difference between two correlated WSCV. The inferential procedure depends on the multivariate normality of the measurements and is mainly likelihood based. The following set-up is to facilitate the construction of the likelihood function.

Let

X i = ( X i 1 , X i 2 , ....... X i m 1 , X i , m 1 + 1 , X i , m 1 + 2 , .... , X i , m 1 + m 2 ) ' MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6833@

denote the measurements on the ith subject, i = 1,2,....,n where Xi1,Xi2,....,Xim1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiwaG1aaSbaaSqaaiabdMgaPjabigdaXaqabaGccqGGSaalcqWGybawdaWgaaWcbaGaemyAaKMaeGOmaidabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabc6caUiabcYcaSiabdIfaynaaBaaaleaacqWGPbqAcqWGTbqBdaWgaaadbaGaeGymaedabeaaaSqabaaaaa@3EC6@ are the m1 measurements obtained by the first method (platform), Xim1+1,Xim1+2,....,Xim1+m2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiwaG1aaSbaaSqaaiabdMgaPjabd2gaTnaaBaaameaacqaIXaqmaeqaaSGaey4kaSIaeGymaedabeaakiabcYcaSiabdIfaynaaBaaaleaacqWGPbqAcqWGTbqBdaWgaaadbaGaeGymaedabeaaliabgUcaRiabikdaYaqabaGccqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGybawdaWgaaWcbaGaemyAaKMaemyBa02aaSbaaWqaaiabigdaXaqabaWccqGHRaWkcqWGTbqBdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@490F@ are the m2 measurements obtained the second method (platform). We assume that Xi ~ N(μ, Σ), where μT=(μ11m1T,μ21m2T) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiVd02aaWbaaSqabeaacqWGubavaaGccqGH9aqpcqGGOaakcqaH8oqBdaWgaaWcbaGaeGymaedabeaakiabigdaXmaaDaaaleaacqWGTbqBdaWgaaadbaGaeGymaedabeaaaSqaaiabdsfaubaakiabcYcaSiabeY7aTnaaBaaaleaacqaIYaGmaeqaaOGaeGymaeZaa0baaSqaaiabd2gaTnaaBaaameaacqaIYaGmaeqaaaWcbaGaemivaqfaaOGaeiykaKcaaa@420D@ and,

Σ = [ σ 1 2 I m 1 + ρ 1 1 − ρ 1 σ 1 2 J m 1 ρ 12 σ 1 σ 2 J m 1 x m 2 ρ 12 σ 1 σ 2 J m 1 x m 2 σ 2 2 I m 2 + ρ 2 1 − ρ 2 σ 2 2 J m 2 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@84D1@

In these expressions 1k is a column vector with all k elements equal to 1, Ik is a k × k identity matrix and Jk and Jkxt are k × k and k × t matrices with all the elements equal to 1. Thus the model assumes that the m1 observations taken by the first platform have common mean μ1, common variance σ12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabigdaXaqaaiabikdaYaaaaaa@2FA7@, and common intra-class correlation ρ1, whereas the m2 measurements taken by the second platform have common mean μ2, common variance σ22 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabikdaYaqaaiabikdaYaaaaaa@2FA9@, and common intra-class correlation ρ2. Moreover, ρ12 denotes the interclass correlation between any pair of measurements xij (j = 1,2,... m1) and xim1+t(t=1,2,…m2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aaSbaaSqaaiabdMgaPjabd2gaTjabigdaXiabgUcaRiabdsha0bqabaGcdaqadaqaaiabdsha0jabg2da9iabigdaXiabcYcaSiabikdaYiabcYcaSiablAciljabd2gaTnaaBaaaleaacqaIYaGmaeqaaaGccaGLOaGaayzkaaaaaa@3ED4@, and also assumed constant across all subjects in the population.

For the lth method, the WSCV, which will be denoted as θl in the remainder of the paper is defined as

θl = σl/μl,   l = 1, 2.

Our primary aim is to develop and evaluate methods of testing H0:θ1 = θ2 taking into account dependencies induced by a positive value of ρ12. We restrict our evaluation to reproducibility studies having m1 = m2 = m.

Methods for testing the null hypothesis

Wald test (WT)

If X1, X2,.... Xn is a sample from the above multivariate normal distribution, then the log-likelihood function l, as a function of ψ = (μ1, μ2, σ12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabigdaXaqaaiabikdaYaaaaaa@2FA7@, σ22 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabikdaYaqaaiabikdaYaaaaaa@2FA9@, ρ1, ρ2, ρ12) is given by:

− 2 L = Q + n m log ⁡ ( σ 1 2 σ 2 2 ) − n log ⁡ ( ( 1 − ρ 1 ) ( 1 − ρ 2 ) ) + n log ⁡ w MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5ECE@

where,

w = u1u2 - m2 ρ122 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyWdi3aa0baaSqaaiabigdaXiabikdaYaqaaiabikdaYaaaaaa@3096@,

ul = 1 + (m - 1)ρl, l = 1, 2 and,

Q = S 1 2 σ 1 2 + m ( 1 − ρ 1 ) u 2 w σ 1 2 ∑ i = 1 n ( x ¯ i 1 − μ 1 ) 2 + S 2 2 σ 2 2 + m ( 1 − ρ 2 ) u 1 w σ 2 2 ∑ i = 1 n ( x ¯ i 2 − μ 2 ) 2 − 2 m 2 ρ 12 w σ 1 σ 2 ( ( 1 − ρ 1 ) ( 1 − ρ 2 ) ) 1 / 2 ∑ i = 1 n ( x ¯ i 1 − μ 1 ) ( x ¯ i 2 − μ 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@BD4D@

From 24 the conditions {1 + (m - 1)ρ1}{1 + (m - 1)ρ2} > m2 ρ122 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyWdi3aa0baaSqaaiabigdaXiabikdaYaqaaiabikdaYaaaaaa@3096@ and -1/(m - 1) <ρl < 1 must be satisfied for the likelihood function to be a sample from a non-singular multivariate normal distribution.

The summary statistics given in (3) are defined as:

x ¯ i j = ∑ k = 1 m x i j k / m i = 1 , 2 , … n ; j = 1 , 2 S j 2 = ∑ i = 1 n ∑ k = 1 m ( x i j k − x ¯ i j ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6FFD@

The maximum likelihood estimates (MLE) for μl and σl2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdYgaSbqaaiabikdaYaaaaaa@3018@ are given respectively by μ⌢l=x¯l,σ⌢l2=Sl2/n(m−1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiVd0MbambadaWgaaWcbaGaemiBaWgabeaakiabg2da9iqbdIha4zaaraWaaSbaaSqaaiabdYgaSbqabaGccqGGSaalcuaHdpWCgaWeamaaDaaaleaacqWGSbaBaeaacqaIYaGmaaGccqGH9aqpcqWGtbWudaqhaaWcbaGaemiBaWgabaGaeGOmaidaaOGaei4la8IaemOBa42aaeWaaeaacqWGTbqBcqGHsislcqaIXaqmaiaawIcacaGLPaaaaaa@4484@, where x¯l=1n∑i=1nx¯ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaemiBaWgabeaakiabg2da9KqbaoaalaaabaGaeGymaedabaGaemOBa4gaaOWaaabCaeaacuWG4baEgaqeamaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aaaa@3E62@ and l = 1, 2. Clearly, σ^l2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4WdmNbaKaadaqhaaWcbaGaemiBaWgabaGaeGOmaidaaaaa@3028@ exists for values of m > 1. Therefore we shall assume that m > 1 throughout this paper. From 24, we obtain ρ^1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqyWdiNbaKaadaWgaaWcbaGaeGymaedabeaaaaa@2EC1@ and ρ^2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqyWdiNbaKaadaWgaaWcbaGaeGOmaidabeaaaaa@2EC3@ by computing Pearson's product-moment correlation over all possible pairs of measurements that can be constructed within platforms 1 and 2 respectively, with ρ^12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqyWdiNbaKaadaWgaaWcbaGaeGymaeJaeGOmaidabeaaaaa@2FB3@ similarly obtained by computing this correlation over the nm2 pairs (xij, xi,m+l).

The WT of H0:θ1 = θ2 requires the evaluation of variance of θ⌢l MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiUdeNbambadaWgaaWcbaGaemiBaWgabeaaaaa@2F32@, l = 1, 2, and cov⁡(θ⌢1,θ⌢2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGagi4yamMaei4Ba8MaeiODay3aaeWaaeaacuaH4oqCgaWeamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIafqiUdeNbambadaWgaaWcbaGaeGOmaidabeaaaOGaayjkaiaawMcaaaaa@3856@. To obtain these values we use elements of Fisher's information matrix, along with the delta method 2627. On writing:

ψ = (ψ1, ψ2)',ψ1 = (μ1, μ2)', and ψ2=(σ12,σ22,ρ1,ρ2,ρ12)′ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiYdK3aaSbaaSqaaiabikdaYaqabaGccqGH9aqpdaqadaqaaiabeo8aZnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaGccqGGSaalcqaHdpWCdaqhaaWcbaGaeGOmaidabaGaeGOmaidaaOGaeiilaWIaeqyWdi3aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqaHbpGCdaWgaaWcbaGaeGOmaidabeaakiabcYcaSiabeg8aYnaaBaaaleaacqaIXaqmcqaIYaGmaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaakiada6SHYaIOaaaaaa@4947@, the Fisher's information matrix I = -E⌊∂2l/∂ψ∂ψ'⌋ has the following structure:

I = [ I 11 O O I 22 ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemysaKKaeyypa0ZaamWaaeaafaqabeGacaaabaGaemysaK0aaSbaaSqaaiabigdaXiabigdaXaqabaaakeaacqWGpbWtaeaacqWGpbWtaeaacqWGjbqsdaWgaaWcbaGaeGOmaiJaeGOmaidabeaaaaaakiaawUfacaGLDbaacqGGUaGlaaa@39DE@

This is based on a result from 26 (page 239) indicating that, I12 = I21' MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemysaK0aa0baaSqaaiabikdaYiabigdaXaqaaiabcEcaNaaaaaa@2FD5@ = -E(∂2l/∂ψ1∂ψ'2) = 0. Therefore, from the asymptotic theory of maximum likelihood estimation we have:

I 11 − 1 = [ var ⁡ ( μ ⌢ 1 ) cov ⁡ ( μ ⌢ 1 , μ ⌢ 2 ) cov ⁡ ( μ ⌢ 1 , μ ⌢ 2 ) var ⁡ ( μ ⌢ 2 ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5E94@

And the elements of I22 are given in the Appendix.

The elements of I22−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemysaK0aa0baaSqaaiabikdaYiabikdaYaqaaiabgkHiTiabigdaXaaaaaa@30DE@ are the asymptotic variance- covariance matrix of the maximum likelihood estimators of the covariance parameters. Inverting Fisher's information matrices we get:

var ⁡ ( μ ⌢ l ) = σ l 2 n m ( 1 − ρ l ) [ 1 + ( m − 1 ) ρ l ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGagiODayNaeiyyaeMaeiOCai3aaeWaaeaacuaH8oqBgaWeamaaBaaaleaacqWGSbaBaeqaaaGccaGLOaGaayzkaaGaeyypa0tcfa4aaSaaaeaacqaHdpWCdaqhaaqaaiabdYgaSbqaaiabikdaYaaaaeaacqWGUbGBcqWGTbqBdaqadaqaaiabigdaXiabgkHiTiabeg8aYnaaBaaabaGaemiBaWgabeaaaiaawIcacaGLPaaaaaGcdaWadaqaaiabigdaXiabgUcaRmaabmaabaGaemyBa0MaeyOeI0IaeGymaedacaGLOaGaayzkaaGaeqyWdi3aaSbaaSqaaiabdYgaSbqabaaakiaawUfacaGLDbaacqGGUaGlaaa@515D@

Applying the delta method 27, we can show, to the first order of approximation that:

var ⁡ ( σ ⌢ l ) ≈ σ l 2 / 2 n ( m − 1 ) . l = 1 , 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiGbcAha2jabcggaHjabckhaYnaabmaabaGafq4WdmNbambadaWgaaWcbaGaemiBaWgabeaaaOGaayjkaiaawMcaaiabgIKi7kabeo8aZnaaDaaaleaacqWGSbaBaeaacqaIYaGmaaGccqGGVaWlcqaIYaGmcqWGUbGBdaqadaqaaiabd2gaTjabgkHiTiabigdaXaGaayjkaiaawMcaaiabc6caUaqaaiabdYgaSjabg2da9iabigdaXiabcYcaSiabikdaYaaaaaa@496A@

The maximum likelihood estimator of θl is θ^l=μ^lσ^l MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiUdeNbaKaadaWgaaWcbaGaemiBaWgabeaakiabg2da9KqbaoaalaaabaGafqiVd0MbaKaadaWgaaqaaiabdYgaSbqabaaabaGafq4WdmNbaKaadaWgaaqaaiabdYgaSbqabaaaaaaa@3773@. Again, by application of the delta method, we can show to the first order of approximation that:

var ⁡ ( θ ⌢ l ) ≈ θ l 4 [ 1 + ( m − 1 ) ρ l ] n m ( 1 − ρ l ) + θ l 2 2 n ( m − 1 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5EBB@

as was shown by Quan and Shih 8.

Again using the delta method we show approximately that:

cov ⁡ ( θ ⌢ 1 , θ ⌢ 2 ) ≈ 2 θ 1 2 θ 2 2 ρ 12 n ( 1 − ρ 1 ) ( 1 − ρ 2 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@55F1@

From 28 we apply the large sample theory of maximum likelihood to establish that: