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Admissible Tests in Multivariate Analysis of Variance
... For this Gl -invariant testing problem, Hotelling's T 2 -test is well-known to be the uniformly most powerful test (Simaika [14]), and so is admissible. Schwartz [13] employed the Birnbaum-Stein method (Birnbaum [2], Stein [15]) to study the admissibility of fully Gl -invariant tests in the multivariate analysis of variance setting. Problem (1.2) is not Gl -invariant, although it is G-invariant. ...
... Marden and Perlman ( [6], p. 49) pointed out that, "by utilizing the exponential structure of the distribution of (X, S), the method of Stein [15] and Schwartz [13] can be applied to reveal the overall T 2 test is admissible for problem (1.2). Based on the logarithm of the joint density of (X, S), Marden and Perlman ( [6], pages 49-50) claimed that, according to the theorem of Stein [15], the set (in our notation) ...
... is an admissible acceptance region in problem (1.2) for any subset However, Marden and Perlman [6] did not offer an analytical proof for their assertion. Note that the problems considered by Stein [15] and Schwartz [13] are fully G 0 -invariant. For the G 0 -invariant models considered by Stein [15], Θ 1 = { (θ, Σ) | θ = 0, Σ p.d.} for the problem of testing H u 0 : θ = 0 against the global alternative H u 1 : θ = 0. Note thatX and S are independent, and µ and Σ are orthogonal. ...
For a multinormal distribution with a p-dimensional mean vector {\mbtheta} and an arbitrary unknown dispersion matrix {\mbSigma}, Rao ([9], [10]) proposed two tests for the problem of testing H_{0}:{\mbtheta}_{1} = {\bf 0}, {\mbtheta}_{2} = {\bf 0}, {\mbSigma}~ \hbox{unspecified},~\hbox{versus}~H_{1}:{\mbtheta}_{1} \ne {\bf 0}, {\mbtheta}_{2} ={\bf 0}, {\mbSigma}~\hbox{unspecified}, where {\mbtheta}^{'}=({\mbtheta}^{'}_{1},{\mbtheta}^{'}_{2}). These tests are referred to as Rao's W-test (likelihood ratio test) and Rao's U-test (union-intersection test), respectively. This work is inspired by the well-known work of Marden and Perlman [6] who claimed that Hotelling's -test is admissible while Rao's U-test is inadmissible. Both Rao's U-test and Hotelling's -test can be constructed by applying the union-intersection principle that incorporates the information {\mbtheta}_{2}={\bf 0} for Rao's U-test statistic but does not incorporate it for Hotelling's -test statistic. Rao's U-test is believed to exhibit some optimal properties. Rao's U-test is shown to be admissible by fully incorporating the information {\mbtheta}_{2}={\bf 0}, but Hotelling's -test is inadmissible.
... For this Gl -invariant testing problem, Hotelling's T 2 -test is well-known to be the uniformly most powerful test (Simaika [14]), and so is admissible. Schwartz [13] employed the Birnbaum-Stein method (Birnbaum [2], Stein [15]) to study the admissibility of fully Gl -invariant tests in the multivariate analysis of variance setting. Problem (1.2) is not Gl -invariant, although it is G-invariant. ...
... Marden and Perlman ( [6], p. 49) pointed out that, "by utilizing the exponential structure of the distribution of (X, S), the method of Stein [15] and Schwartz [13] can be applied to reveal the overall T 2 test is admissible for problem (1.2). Based on the logarithm of the joint density of (X, S), Marden and Perlman ( [6], pages 49-50) claimed that, according to the theorem of Stein [15], the set (in our notation) ...
... is an admissible acceptance region in problem (1.2) for any subset However, Marden and Perlman [6] did not offer an analytical proof for their assertion. Note that the problems considered by Stein [15] and Schwartz [13] are fully G 0 -invariant. For the G 0 -invariant models considered by Stein [15], Θ 1 = { (θ, Σ) | θ = 0, Σ p.d.} for the problem of testing H u 0 : θ = 0 against the global alternative H u 1 : θ = 0. Note thatX and S are independent, and µ and Σ are orthogonal. ...
For a multinormal distribution with a p-dimensional mean vector {\mbtheta} and an arbitrary unknown dispersion matrix {\mbSigma}, Rao ([9], [10]) proposed two tests for the problem of testing H_{0}:{\mbtheta}_{1} = {\bf 0}, {\mbtheta}_{2} = {\bf 0}, {\mbSigma}~ \hbox{unspecified},~\hbox{versus}~H_{1}:{\mbtheta}_{1} \ne {\bf 0}, {\mbtheta}_{2} ={\bf 0}, {\mbSigma}~\hbox{unspecified}, where {\mbtheta}^{'}=({\mbtheta}^{'}_{1},{\mbtheta}^{'}_{2}). These tests are referred to as Rao's W-test (likelihood ratio test) and Rao's U-test (union-intersection test), respectively. This work is inspired by the well-known work of Marden and Perlman [6] who claimed that Hotelling's -test is admissible while Rao's U-test is inadmissible. Both Rao's U-test and Hotelling's -test can be constructed by applying the union-intersection principle that incorporates the information {\mbtheta}_{2}={\bf 0} for Rao's U-test statistic but does not incorporate it for Hotelling's -test statistic. Rao's U-test is believed to exhibit some optimal properties. Rao's U-test is shown to be admissible by fully incorporating the information {\mbtheta}_{2}={\bf 0}, but Hotelling's -test is inadmissible.
... All the four tests are invariant, unbiased (Bradley & Russell, 1998;Bradley & Orfaly, 1999;Das Gupta, Anderson, & Mudholkar (1964) and admissible (Schwartz, 1967). These tests are used when different individuals receive different treatments and hence have different mean vectors. ...
Previously a general method to analyze a nested repeated measure model was developed when the covariance matrix had a certain pattern. A case of being a number of sub-individuals of a particular individual, such as sub-field or other types of offsprings, receives several treatments. As a consequence, the observations are correlated with certain covariance matrix pattern and such a model is known as nested repeated measures model (NRMM). In this paper, a weaker assumption is used when the covariance matrix is arbitrary and has no specific pattern. Independent normally distributed individuals are taken with their own mean and common positive definite covariance matrix. It is aimed to test hypotheses about the mean. Two techniques are used for testing. The first is based on the multivariate one sample model (MOSM), when each individual receives the same treatments and hence has the same mean vector, whilst the second is based on the multivariate linear model (MLM). Different individuals receive different treatments and hence have different mean vectors. For each technique a uniformly most powerful (UMP) invariant size test is found.
... Now, we observe the MLM, ~, ( , , Σ), ∈ , Σ > 0 This test is unbiased and invariant ( [17] and [18]) as well as being admissible [19]. ...
A general method for analyzing repeated measure models obtained, whenever the covariance matrix has a certain patterned structure. In this paper, two cases are considered with assuming a weaker assumption on the covariance matrix. In order to make a test around the mean for these two cases, which the first one is the case of being a particular individual receives two levels of treatments with their interaction and the case of three fixed factors when a number of individuals receive only one level of the first treatment and every pair of combination of the second and third treatment levels. Two techniques are handled. For the former case, a multivariate one sample model (MOSM) is used, while multivariate linear model (MLM) is used for the latter one. A uniformly most powerful (UMP) invariant size α test is obtained.
... d l' ... ,dr>. In fact, Schwartz (1967b) bas shown that every admissible invariant test for (1.5) =(1.13) musthave a monotone acceptance region A in terms of c or (equivalently) d. That is (in terms of d). if d =(d1" .. ,d t ) E A ~ D, and d'ss(d{, .. ,d/)E Dtissucb that d'Sd(i.e.,d{Sd 1 , @BULLET@BULLET@BULLET ,d/-Sd t ) , then d' EA. ...
Key words and phrases: MA1NOV,A, inVllrhmt test, consistency, power, monotone acceptance region, noncentral Wishartmatrix, characteristic roots. Two notions of cotlsist:etlCY ofitlvariant tests for theM.A;N0VA.testing problems are<examjnedandcpmPa.re¢<sample s~e C()tlsistetlcY.(theclassical notion} and parameter consistency, which requires that formed sample size, the power of the test approaches.one for any sequence of alternatives whose distance from the nun hypothesis approaches infinity. The Roy, Lawley-Hotelling, and likelihood ratio (s Wilks) tests are consistent in both senses.whereas the Bartlett Nanda-Pillai trace test, although samplesize consistent, is not parameter consistent unless thesignificance levela or the errordegrees of freedom n is sufficiently large.
... The former method has been used in Kiefer and Schwartz [16], Nishida [19], [20], [21], [22], [24]. The latter method has been seen in Ghosh [13], Birnbaum [9], Stein [33], Schwartz [29], Anderson and Takemura [5], etc. In this paper we use the former method. ...
... Since W = W 1 ¢q W 2 for W 1 ~ ~1 and Wz ~ ~, and since Awl n w+ = Awl f3 Aw+ , it suffices to show this convexity separately for Aw~ and for Aw2. The following representation is a simplified version of a result established by Schwartz (1967) with D b = Diag(b 1 ..... bp). Set H = (Db)l/~F, and apply the identity Thus for fixed (su.2, s22) ~vi X~p~, tr(DbFsF') is a convex function of s12, so the sections of K(b; F) are convex in sle. ...
In multivariate statistical analysis, orthogonally invariant sets of real positive definite p × p matrices occur as acceptance regions for tests of invariant hypotheses concerning the covariance matrix Σ of a multivariate normal distribution. Equivalently, orthogonally invariant acceptance regions can be expressed in terms of the eigenvalues l1(S), …, lp(S) of a random Wishart matrix S ∼ Wp(n, Σ) with n degrees of freedom and expectation nΣ. The probabilities of such regions depend on Σ only though λ1(Σ), …, λp(Σ), the eigenvalues of Σ. In this paper, the behavior of these probabilities is studied when some λi increase while others decrease. Our results will be expressed in terms of the majorization ordering applied to the vector μ ≡ (μ1(Σ), …, μp(Σ)), where μi(Σ) = log λi(Σ), and have implications for the unbiasedness and monotonicity of the power functions of orthogonally invariant tests.
This chapter focuses on the properties of test procedures. A statistical assessment of the different test procedures for the customary null hypothesis may be made in terms of their power functions. The power properties are of two kinds, namely, intrinsic and comparative. Statistics that are used for purposes of tests of significance in uniresponse problems, for example, a mean square or a t-statistic, have important value, apart from their formal use, as interpretable summaries of facets of the structure underlying a body of data. By contrast, in the multiresponse situation, the statistics proposed for carrying out significance tests, for example, various functions of certain latent roots, are complex derivatives from the data that do not generally provide a direct insight into the latent structure of the data. For the multivariate analysis of variance (MANOVA), even with the imposition of the requirement of affine invariance, one is led to a class of statistics whose members are not in general statistically equivalent and each statistic is a function of certain latent roots that are the basic invariants.
In an errors-in-variables model, the predicting variables are observed with errors. Traditionally, the errors are assumed to be additive. In this article, I consider the case in which the error is multiplicative, a situation that arises when analyzing some recent data released by the U.S. Department of Energy. A consistent estimator is provided for regression coefficients β by correcting the asymptotic bias of the least squares estimate. It is shown that , after being normalized by an estimate of its covariance, is asymptotically normally distributed. This can, therefore, be used to construct approximate tests and confidence sets for β. The results are essentially nonparametric.
This chapter discusses the monotonicity and unbiasedness properties of univariate analysis of variance (ANOVA) and multivariate analysis of variance (MANOVA) tests. The four tests discussed in the chapter are likelihood-ratio test, Roy's maximum-root test, Lawley–Hotelling's trace test, and the Bartlett–Nanda–Pitlai trace test. The conditions under which the power function of an invariant test increases monotonically are also discussed in the chapter. It also presents a minor extension of Anderson's result of the theorem of the multivariate case, the monotonicity of the power functions of UMP invariant tests in two special cases, mathematical preliminaries, study on monotonicity in the general case, general MANOVA models, and various theorem and lemmas.
We state some necessary and sufficient conditions for admissibility of tests for a simple and a composite null hypotheses against ”one-sided” alternatives on multivariate exponential distributions with discrete support. Admissibility of the maximum likelihood test for “one –sided” alternatives and z χtest for the independence hypothesis in r× scontingency tables is deduced among others.
Theorem 1, stated below, includes a number of statistical problems in which it may be desirable to consider conditional procedures. It is our aim to obtain a complete class theorem covering such conditional procedures. We will first state the assumptions needed, then Theorem 1.1. We conclude this section with some discussion. Section 2 contains a proof of Theorem 1.1, and Section 3 contains applications to the study of invariant tests. We assume X and Yare complete separable metric spaces with .~x the a-algebra of Borel subsets of X, and ,~r the a-algebra of Borel subsets of Y. We assume/~ is a regular totally a-finite measure on .~x, and that if coE(2, 2o, is a probability measure on ,~r dominated by the probability measure 2. Given is a set {fo, co~(2} of conditional density functions on X x Y. It is assumed that if co~2 then fo is jointly measurable. It is assumed that if y~ Y then fo(., y)~LI(X, ~x, #) and ~ f~(x, y) #(dx)= 1. Of the parameter and decision spaces it is assumed that (2 is a separable metric space and that the decision space ~) is separable locally compact metric space. We require that both spaces be complete in their respective metrics. Loss will be measured by a continuous function W: ~ x f2--* [0, oe). We assume that if coEf2 then lim W(t, co)= oc. We assume there exists a partition of f2 into subsets
In this paper, the authors considered various procedures for testing for the independence of two multivariate regression equations with different design matrices. Asymptotic null distributions as well as nonnull distributions under local alternatives of the test statistics associated with the above procedures are also derived.
An overview of hypothesis testing for the common mean of independent normal distributions is given. The case of two populations is studied in detail. A number of different types of tests are studied. Among them are a test based on the maximum of the two available t-tests, Fisher's combined test, a test based on Graybill–Deal's estimator, an approximation to the likelihood ratio test, and some tests derived using some Bayesian considerations for improper priors along with intuitive considerations. Based on some theoretical findings and mostly based on a Monte Carlo study the conclusions are that for the most part the Bayes-intuitive type tests are superior and can be recommended. When the variances of the populations are close the approximate likelihood ratio test does best.
The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.
Let and be normally distributed random matrices whose columns are mutually independent with common covariance matrix, and . It is desired to test vs. , where . Let denote the characteristic roots of . It is shown that any test with monotone acceptance region in , i.e., a region of the form where g is nondecreasing in each argument, is unbiased. Similar results hold for the problems of testing independence of two sets of variates, for the generalized MANOVA (growth curves) model, and for analogous problems involving the complex multivariate normal distribution. A partial monotonicity property of the power functions of such tests is also given.
We consider the problem of testing a hypothesis about the means of a subset of the components of a multivariate normal distribution with unknown covariance matrix, when the means of a second subset (the covariates) are known. Because of the possible correlation between the two subsets, information provided by the second subset can be useful for inferences about the means of the first subset. In this paper attention is restricted to the class of procedures invariant under the largest group of linear transformations which leaves the problem invariant. The family of tests which are admissible within this class is characterized. This family excludes several well-known tests, thereby proving them to be inadmissible (among all tests), while the admissibility (among invariant tests) of other tests is demonstrated. The powers of the likelihood ratio test LRT, the test, and the overall test are compared numerically; the LRT is deemed preferable on the basis of power and simplicity.
The notion of Bahadur efficiency is used to compare multivariate linear hypothesis tests based on six criteria: (1) Roy's largest root, (2) the likelihood ratio test, (3) the Lawley-Hotelling trace, (4) Pillai's trace, (5) Wilks' U, and (6) Olson's statistic. Bahadur exact slope is computed for each statistic as a function of noncentrality parameters using results for probabilities of large deviations. The likelihood ratio test is shown to be asymptotically optimal in the sense that its slope attains the optimal information value, and the remaining tests are shown not to be asymptotically optimal. Inequalities are derived for the slopes showing order of preference.
This paper treats a generalization of the classical MANOVA testing problem. The problem is reduced via invariance considerations and a new test statistic is proposed. This new test is shown to be a unique locally best invariant test and locally minimax.
The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose has a normal distribution, and the rows of X are independent, each with unknown covariance matrix . Let have a Wishart distribution independent of X. If K is the acceptance region of an invariant test for the null hypothesis , let denote the power function of K, where and are the t largest characteristic roots of . A main result is THEOREM. If K is a convex set (in , then is a Schur-convex function of . Standard tests to which the above theorem can be applied include the Roy maximum root test and the Lawley-Hotelling trace test.
We consider unrestricted (unordered) parametric hypotheses for multivariate or multiparameter distributions and review some optimality aspects, both exact and asymptotic, for testing of hypotheses possibly in the presence of nuisance parameters. The aim is not to provide an exhaustive review but to represent the widely used classical approaches, expose some promising recent ones and present some interesting practical problems requiring the development of new methods.
It is shown that for the MANOVA problem the power function of the test based on the trace of a multivariate beta matrix is monotonically increasing in each noncentrality parameter provided that the cutoff point is not too large. This result is also true for the problem of testing independence of two sets of variates.
The authors consider various procedures for testing the hypotheses of independence of two sets of variables and certain regression coefficients are zero under multivariate regression model. Various properties of these procedures and the asymptotic distributions associated with these procedures are also considered.
A new proof of admissibility of tests in MANOVA is given using Stein's theorem [7]. The convexity condition of Stein's theorem is proved directly by means of majorization rather than by the supporting hyperplane approach. This makes the geometrical meaning of the admissibility result clearer.
In Section 3 we shall prove a theorem based on a method of A. Birnbaum [1] and E. Lehmann concerning the admissibility of certain tests of simple hypotheses in multivariate exponential families. In Section 4 we compute the supporting hyperplanes of the convex acceptance region in some of the most common applications of Hotelling's -test and show that the theorem of Section 3 implies the admissibility of this test. In Section 5 we point out some of the limitations of the method of this paper.
In the first nontrivial case, dimension and sample size , it is proved that Hotelling's test of level maximizes, among all level tests, the minimum power on each of the usual contours where the test has constant power. A corollary is that the test is most stringent of level in this case.
This paper contains details of the results announced in the abstract by the authors (1962). Techniques are developed for proving local minimax and "type D" properties and asymptotic (that is, far in distance from the null hypothesis) minimax properties in complex testing problems where exact minimax results seem difficult to obtain. The techniques are illustrated in the settings where Hotelling's test and the test based on the squared sample multiple correlation coefficient are customarily employed.
Using Stein's [7] generalisation of a method of Lehmann and A. Birnbaum (A. Birnbaum [2]), we show that two of the tests commonly used in multivariate analysis of variance are admissible.
Cover title. Thesis--Columbia University. "Reprinted from the Annals of mathematical statistics, vol. 26, no. 1, March, 1955." Bibliography: p. 36.