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A Coordinate-Free Approach to Finding Optimal Procedures for Repeated Measures Designs
Abstract
A repeated measures design occurs in analysis of variance when a particular individual receives several treatments. Let be the vector of observations on the ith individual. It is assumed that the are independently normally distributed with mean and common covariance . The researcher wants to test hypotheses about the . Let . For this paper, in order to get powerful tests, the simplifying assumption that the are exchangeable is made. We assume that the design is given and use a coordinate-free approach to find optimal (i.e., UMP invariant, UMP unbiased, most stringent, etc.) procedures for testing a large class of hypotheses about the .
... The crossed repeated measures models (CRMM) is one of the most widely used models in experimental design, especially in biological, agriculture, education and psychological research (see Lehman, 1959;Cox, 1992;Hoshmand, 2006). Arnold (1979) has developed a general method to analyze repeated measures model (RMM), when each of m independent individuals receives several treatments and assuming that all measurements have the same 2 and every pair of measurements that comes from the same individual have covariance 2 and each individual is normally distributed. Gabbara (1985) has extended the RMM of Arnold (1979) to (i) nested repeated measures models (NRMM), (ii) generalized nested repeated measures models (GNRMM), (iii) crossed repeated measures models (CRMM), (iv) crossed-nested repeated measures models (CNRMM). ...
... Arnold (1979) has developed a general method to analyze repeated measures model (RMM), when each of m independent individuals receives several treatments and assuming that all measurements have the same 2 and every pair of measurements that comes from the same individual have covariance 2 and each individual is normally distributed. Gabbara (1985) has extended the RMM of Arnold (1979) to (i) nested repeated measures models (NRMM), (ii) generalized nested repeated measures models (GNRMM), (iii) crossed repeated measures models (CRMM), (iv) crossed-nested repeated measures models (CNRMM). Rhonda, and et al (2016) considered covariance models to account for NRM and simultaneously address mean profile estimation with penalized splines via semi parametric regression with application to a prospective study of 24-hour ambulatory blood pressure and the impact of surgical intervention on obstructive sleep apnea. 2 In this paper, we have generalized the work of Arnold (1979) to a more complicated situation occurring in the analysis of variance (ANOVA) when a particular individual receives every pair of treatment levels, in which observations cannot be assumed independent as they are assumed in the usual independent RMM. ...
... Gabbara (1985) has extended the RMM of Arnold (1979) to (i) nested repeated measures models (NRMM), (ii) generalized nested repeated measures models (GNRMM), (iii) crossed repeated measures models (CRMM), (iv) crossed-nested repeated measures models (CNRMM). Rhonda, and et al (2016) considered covariance models to account for NRM and simultaneously address mean profile estimation with penalized splines via semi parametric regression with application to a prospective study of 24-hour ambulatory blood pressure and the impact of surgical intervention on obstructive sleep apnea. 2 In this paper, we have generalized the work of Arnold (1979) to a more complicated situation occurring in the analysis of variance (ANOVA) when a particular individual receives every pair of treatment levels, in which observations cannot be assumed independent as they are assumed in the usual independent RMM. Let ijk Y be the observations of the th k ) , ( treatment on the calf from the th j cow and the th i bull, where ...
In this paper, we develop a general method to analyze several different kinds of certain crossed repeated measures models (CRMM) which represent many situations occurring in repeated measurements on the same experimental units (individuals). Let Yi=(Y1111,,Yidrc) be the vector of observations of the ith individuals. It is assumed that the Yi are jointly normally distributed with mean µi . We want to test hypotheses about µi . In order to get powerful tests we make the simplifying assumptions that all measurements have the same variance σ2 and every pair of measurements that comes from (i) different bulls and different cows (ii) different bulls but with the same cow (iii) the same bull with different cows; have covariance’s 0, σ 2 ρ 1 , σ 2ρ2 respectively. And every pair of measurements that comes from the same bull and the same cow with treatments of (a) different columns and different rows (b) the same column but different rows (c) different columns but the same row have covariance’s σ 2ρ 3 , σ 2ρ 4 and σ 2 ρ 5 , respectively. The results of this model can be used to analyze certain 4-way balanced mixed and/or random effects models. This procedure is also useful to analyze any of the mentioned 4-way models by adding any number of fixed effects to the model as long as those added effects do not interact with any random effects already in these models.
... To conducting test hypotheses about the mean, two different forms are considered for the GRMM, which can be seen in [16]. ...
... µí±12 )(í µí± í µí±í µí±í µí±í µí± − í µí± ̅ .í µí±í µí±í µí± ) í µí± From theorem (19.6) of Arnold [16], the nonzero root of (42) ...
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.
... We assume throughout this work that each individual has the same number d , of sub-individuals and each sub-individual receives the same number Arnold (1979) and Gabbara (1985) developed a general method to analyze a nested repeated measures model (NRMM) when the covariance matrix has the pattern ...
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.
... He gave a necessary and sufficient condition for BLUE to be equal to the ordinary least squares estimators. This condition became the foundation for the alternative forms, as well as their extensions, proved in [1], [2], [3], [4], [5], [6], [9], [11], [15]. ...
The existence of the best linear unbiased estimators (BLUE) for parametric es-timable functions in linear models is treated in a coordinate-free approach, and some relations between the general linear model and certain particular models are derived. A class of the BLUE is constructed using the maximal covariance operator. It is shown that a BLUE exists for each parametric estimable function and is unique in this class.
... Experimental techniques which consider the response of an individual over a period of time (or over different doses of some medicine) are generally named growth curve experiments. Their representation by growth curve models have been studied extensively in the literature because of their general aplicability (see [1] [6] [8] [10] [11]). The MANOVA models include the multivariate growth curve models but also the profile analysis models. ...
The purpose of this article is to build a class of the best linear unbiased estimators (BLUE) of the linear parametric functions, to prove some necessary and sufficient conditions for their existence and to derive them from the corresponding normal equations, when a family of multivariate growth curve models is considered. It is shown that the classical BLUE known for this family of models is the element of a particular class of BLUE built in the proposed manner. The results are expressed in a convenient computational form by using the coordinate-free approach and the usual parametric representations. El propósito del artículo es construir una clase de estimadores lineales insesgados óptimos (BLUE) de funciones paramétricas lineales para demostrar algunas condiciones necesarias y suficientes para su existencia y deducirlas de las correspondientes ecuaciones normales, cuando se considera una familia de modelos con curva de crecimiento multivariante. Se demuestra que la clase de los BLUE conocidos para esta familia de modelos es un elemento de una clase particular de los BLUE que se construyen de esta manera. Los resultados se presentan en un formato computacional adecuado usando un enfoque que es independiente de las coordenadas y las representaciones paramétricas usuales.
... The problem of the equality between OLSE and BLUE (or Gauss-Markov estimators) of the expected mean in linear regression models is treated in a coordinate-free approach. Significant contributions in solving this problem were brought by Haberman [12], Arnold [1], Baksalary and van Eijnsbergen [3], Puntanen and Styan [16], Puntanen, Styan and Tian [17] among many others, but Zyskind [23] proved the first general theorem for OLSE to be BLUE when the covariance matrix is nonnegative definite and the design matrix of the model does not necessarily have full column rank. Zyskind and Martin [24] and Seely [21] [22] obtained significant general results on unbiased estimation restricted to finite dimensional linear spaces. ...
The necessary and sufficient condition for the ordinary least squares estimators (OLSE) to be the best linear unbiased estimators (BLUE) of the expected mean in the general univariate linear regression model was given by Kruskal (1968) using a coordinate-free approach. The purpose of this article is to present in the same manner some alternative forms of this condition and to prove two of the Haberman's equivalent conditions in a different and simpler way. The results obtained in the general univariate linear regression model are applied to a family of multivariate growth curve models for which the problem of the equality between OLSE and BLUE is treated in a coordinate-free approach.
... Alternative proofs of some of the earlier result as Presentado well as their extensions were derived, using algebrical methods, by: Baksalary and Kala [4], Baksalary [5], Puntanen and Styan [29], McAleer [24], Qin and Lawless [30], Kurata and Kariya [22]. The results of this problem expressed in geometrical form were obtained by: Kruskal [21], Eaton [18], Haberman [19], Arnold [3]. Some alternative proofs using also a coordinate-free approach were given by Drygas [15,16] and Beganu [10,11]. ...
It is well known that there were proved several necessary and sufficient conditions for the ordinary least squares estimators (OLSE) to be the best linear unbiased estimators (BLUE) of the fixed effects in general linear models. The purpose of this article is to verify one of these conditions given by Zyskind [39, 40]: there exists a matrix Q such that X = XQ, where X and are the design matrix and the covariance matrix, respectively. It will be shown the accessibility of this condition in some multivariate growth-curve models, establishing the known result regarding the equality between OLSE and BLUE in this type of linear models.
... Influences of the coordinate-free approach on other writers can be found for example, in Arnold (1979Arnold ( , 1981 and Christensen (2002). The paucity of published works by Kruskal in this area makes it problematic to trace his impact in the development of linear model theory and methodology. ...
Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]
The sphericity test for repeated measures model is studied. The likelihood ratio criterion of this test is obtained.
The protein-precipitating capacities of tanniferous extracts from immature and mature leaves of threeShorea spp. (Dipterocarpaceae) seedlings were measured by an adaptation of Goldstein and Swain's -glucosidase precipitation assay. Protein precipitation by the extracts was not correlated with total phenolics (Folin-Denis assay) or proanthocyanidin content (BuOH-HCl assay) as measured in an earlier study. Extracts ofS. maxwelliana mature leaves had much lower protein-precipitating capacity than those ofS. acuminata andS. leprosula, but fewer insect species feed on and cause less damage to the foliage ofS. maxwelliana compared with the other species' foliage. Immature leaf extracts ofS. leprosula andS. acuminata had substantial protein-precipitating capacities which in the latter species exceeded that of its mature leaf extracts. Leaf extracts precipitated less protein when initial protein concentration was reduced, although not limiting, but no effect or the reverse effect occurred with quebracho tannin and tannic acid. Problems in the characterization of foliage astringency and the interpretation of its role as a potential antiherbivore defense are discussed.
For a generalized normal linear model in which the covariance matrix [Sigma] is positive definite symmetric, UMP invariant test procedures for some kinds of linear hypotheses are derived by transforming the model by an orthogonal matrix L, consisting of orthonormal eigenvectors of [Sigma] as the columns vectors. Here it is assumed that [Sigma] contains unknown elements but has a certain structure making all the elements of L known. A sufficient condition for this assumption is also obtained to examine whether the covariance matrix [Sigma] has such a form.
Fairly general sufficient conditions are given to guarantee that invariant tests about means in the multivariate linear model and the repeated measures model have the correct asymptotic size when the normal assumption under which the tests are derived is relaxed. These conditions are the same as Huber's condition which guarantees asymptotic validity of the size of the F-test for the univariate linear model.
This paper describes a general method for deriving optimal procedures for problems where the covariance matrices are patterned under both null and alternative hypotheses. The pattern considered in this paper was first suggested by Olkin (1970) and is a generalization of the intraclass correlation model of Wilks (1946) and arises in the study of interchangeable random variables. We prove a theorem showing how we can transform most such problems to products of problems where the covariance matrices are unpatterned. This theorem is applied to two problems, the multivariate analysis of variance problem and the multivariate classification problem where in both cases the covariance matrix is assumed patterned. We use theorems about products to derive optimal procedures for these problems. We then look at Olkin's pattern for the mean vector, and show that most problems where both the mean vector and covariance matrix are patterned can be transformed to a product of problems, one of which is trivial. The same two examples are studied where now both mean vectors and covariance matrices are assumed patterned. We also consider the problem of testing that the mean vector is patterned when we know the covariance matrix is.