Homogeneity Of Covariance Matrices Research Articles

A robust effect size measure Aw for MANOVA with non-normal and non-homogenous data

A common research question in psychology entails examining whether significant group differences (e.g. male and female) can be found in a list of numeric variables that measure the same underlying construct (e.g. intelligence). Researchers often use a multivariate analysis of variance (MANOVA), which is based on conventional null-hypothesis significance testing (NHST). Recently, a number of quantitative researchers have suggested reporting an effect size measure (ES) in this research scenario because of the perceived shortcomings of NHST. Thus, a number of MANOVA ESs have been proposed (e.g. generalized eta squared [Formula: see text], generalized omega squared [Formula: see text]), but they rely on two key assumptions—multivariate normality and homogeneity of covariance matrices—which are frequently violated in psychological research. To solve this problem we propose a non-parametric (or assumptions-free) ES ( Aw) for MANOVA. The new ES is developed on the basis of the non-parametric A in ANOVA. To test Aw we conducted a Monte-Carlo simulation. The results showed that Aw was accurate (robust) across different manipulated conditions—including non-normal distributions, unequal covariance matrices between groups, total sample sizes, sample size ratios, true ES values, and numbers of dependent variables—thereby providing empirical evidence supporting the use of Aw, particularly when key assumptions are violated. Implications of the proposed Aw for psychological research and other disciplines are also discussed.

Combination rules for homoscedastic and heteroscedastic MANOVA models from multiply imputed datasets.

Classical MANOVA tests do not pose any difficulty when the assumptions on which they are based are satisfied, while the modified Brown-Forsythe (MBF) procedure has low sensitivity to the lack of multivariate normality and homogeneity of covariance matrices. Both methods assume complete data for all subjects. In this paper, we present combination rules for the MANOVA and MBF procedures with multiply imputed datasets. These rules are illustrated by pooling the results obtained with a two-factor multivariate design after applying the two approaches to each of the imputed datasets when the covariance matrices were equal (MI-MANOVA) and when the covariance matrices were unequal (MI-MBF). A Monte-Carlo study was carried out to compare the proposed solution, in terms of type I error rates and statistical power, with the MANOVA and MBF approaches without missing data, and with listwise deletion of missing data followed by the MANOVA approach (LD-MANOVA) and listwise deletion followed by the MBF procedure (LD-MBF). Simulations showed that the type I error rates in all analyses on datasets with missing values (with or without imputation) were well controlled. We also found that the MI-MANOVA approach was substantially more powerful than LD-MANOVA. Moreover, the power of the MI-MANOVA was generally comparable to that of its complete data counterpart. Similar results were obtained for the MI-MBF procedure when covariance matrices were unequal. We conclude, based on the current evidence, that the solution presented performs well and could be of practical use. We illustrate the application of combination rules using a real dataset.

Multivariate Analysis of EEG Data: Some Aspects of Diagnostic of MANOVA Model

The main focus of this study is the multivariate analysis of Electroencephalogram data which included multivariate analysis of variance. The multivariate model diagnostics comprise checking number of assumptions of MANOVA model such as multivariate normality, homogeneity of covariance matrices. In this paper, the model X=BC+E is used and estimate the different parameters. Also by using a general form, H0: GBF=0 to test the different types of null hypothesis. Here G and F is known matrices and obtained from the hypothesis. This study gives mathematical ideas from multivariate statistical analysis to find a solution or a good approximation of a complex scientific problem.

Pemberian Umpan Balik Dalam Meningkatkan Hasil Belajar dan Minat Belajar Mahasiswa

<p>The purpose of this study is to find and analyze the effect of feedback on <br />learning outcomes in mathematics and an interest in basic statistics course. The <br />population in this study are affordable Information Technology Student cademic Year 2012/2013 Semester II Indraprasta PGRI University of South Jakarta. Sample The study sample was obtained through random sampling. This study used an experimental method to the analysis using the MANOVA test. This study has three variables, consisting of: one independent variable, namely the provision of feedback (immediate and delayed), and two dependent variable is the result of interest in the study of mathematics and basic statistics course. The data was collected for the test results to learn mathematics, and a questionnaire for the interest in basic statistics course. Collected data were analyzed using the MANOVA test. Before the data were analyzed, first performed descriptive statistical analysis and test data analysis requirements (test data normality and homogeneity of covariance matrices). The results show that the learning outcomes of interest in mathematics and basic statistics course for students who are given immediate feedback higher than students given feedback delayed. <br /><br /></p>

Distribution-free tests of mean vectors and covariance matrices for multivariate paired data

We study a permutation procedure to test the equality of mean vectors, homogeneity of covariance matrices, or simultaneous equality of both mean vectors and covariance matrices in multivariate paired data. We propose to use two test statistics for the equality of mean vectors and the homogeneity of covariance matrices, respectively, and combine them to test the simultaneous equality of both mean vectors and covariance matrices. Since the combined test has composite null hypothesis, we control its type I error probability and theoretically prove the asymptotic unbiasedness and consistency of the combined test. The new procedure requires no structural assumption on the covariances. No distributional assumption is imposed on the data, except that the permutation test for mean vector equality assumes symmetric joint distribution of the paired data. We illustrate the good performance of the proposed approach with comparison to competing methods via simulations. We apply the proposed method to testing the symmetry of tooth size in a dental study and to finding differentially expressed gene sets with dependent structures in a microarray study of prostate cancer.

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.

Robustness of asymptotic and bootstrap tests for multivariate homogeneity of covariance matrices

The present work emphasizes the importance of testing hypothesis on homogeneity of covariance matrices from multivariate k populations. The violation of the assumption of the homogeneity of covariance matrices affects the performance of the tests and the coverage probability of the confidence regions. This work intends to apply two tests of homogeneity of covariance and to evaluate type I error rates and power using Monte Carlo simulation in normal populations and robustness in non normal populations. Multivariate Bartlett's test (MBT) and its bootstrap version (MBTB) were used. Different configurations are tested combining sample sizes, number of variates, correlation and number of populations. Results show that the bootstrap test was considered superior to the asymptotic test and robust, since it controls the type error I rate.

Wald statistic for iiomogeneity of multiple parameters

The paper proposes Wald statistic for testing homogeneity of parameters of k populations and gives the asymptotic expansion of the distribution function under a null hypothesis and of power function of it under contiguous alternatives. It shows that Wald statistic is the same as Nagao's statistic (1973) for testing homogeneity of covariance matrices of multivariate normal populations.

Saddlepoint approximations for P-values of some tests of covariance matrices

Test statistics for hypotheses about covariance matrices in multivariate analysis of variance have null distributions which are rather intractable to compute. Examples include the modified likelihood ratio statistic for testing (i) homogeneity of covariance matrices across normal populations (Bartlett-Box M-statistic), (ii) homogeneity of normal populations and (iii) simultaneous sphericity of covariances across normal populations. This paper provides two saddlepoint approximations for the null distributions of each of the above mentioned test statistics. These approximations result in extremely accurate p-value computations in all settings including the more difficult unbalanced sample size case. The computations are fast and easily programmed.

Growth-curve differentiation among Japanese larch provenances

Height–age regression models were fitted to observed heights at ages 7, 11, 15, 22, and 29 years in 20 Japanese larch (Larixleptolepis (Sieb. & Zucc.) Gord.) provenances. A generalized least squares approach with maximum-likelihood estimation of the autocorrelation function and the variance function (heteroscedasticity) is proposed as a framework for growth-curve analysis. Ignoring autocorrelation and variance heteroscedasticity led to serious bias of error estimates that may invalidate statistical comparisons. Normality of log-transformed heights and homogeneity of covariance matrices were confirmed statistically. Cluster analyses followed up by likelihood-ratio tests identified three significantly different growth patterns. Ten provenances belonged to a fast-growing group, nine to a slow-growing group, and one was deemed aberrant (very slow growing). Discriminant analyses on geographic variables and clusters achieved a 95% posterior rate of correct classification. The growth-curve approach was extended to the analyses of age to age correlations and compared with observed correlations. It is concluded that a multivariate approach to growth-curve analysis is preferable to the univariate approach owing to correlations among estimated regression coefficients.

A note on homogeneity tests of covariance matrices

In this note we propose two procedures for testing homogeneity of co-variance matrices that are both extensions of Hartley's (1940) test for equality of variances. The first is a two-stage procedure where the first step is a simple test for equality of the largest eigenvalues, and corresponding eigenvectors, of the covariance matrices. The second is based on projection pursuit and seems harder to apply in practice.

PROF: A Computer Program for Analyzing Group Differences Based Upon Profile Data

Dependent variables in psychological research often take the form of test profiles. A computer program is described which provides comprehensive information on profile differences among independent groups based upon profile centroid separation, shape, elevation, and scatter. Descriptive statistics, univariate and multivariate measures of strength of association, test of homogeneity of covariance matrices, and comparative mean plots also are provided.

Reactive Inhibition Theory and Intertrial Correlations

From certain theoretical properties of Hull's reactive inhibition and from the known modest reliability of reminiscence measures, it follows that intertrial correlations among psychomotor test scores should be higher under distributed-practice conditions than under massed-practice conditions. This proposition was tested in a conventional practice distribution paradigm involving 168 Caucasian subjects. Bartlett's test for the homogeneity of covariance matrices confirmed the prediction, and a sign test for matched correlation pairs showed that correlations in the distributed practice mode were significantly more often higher than those in the massed practice mode. These results refute the prevailing view that intertrial correlations are invariant with respect to practice distribution, and they strengthen the credibility of Hull's principle of reactive inhibition.

Allometry Correction in Taximetrics

Corruccini, R. S. (Dept. of Anthropology, Univ. of California, Berkeley, Calif. 94720) 1972. Allometry Correction in Taximetrics. Syst. Zool. 21:375-383.-The process of allometry often distorts numerical results, causing size-related effects in coefficients. The study of allometry has traditionally emphasized only aspects of detection and description, rather than correction, of allometric trends. A method of allometry correction is offered for the specific purpose of adjusting metrical data prior to quantitative taxonomic analysis. [Allometry; taximetrics; hominoids.] Systematists in increasing numbers are uncritically accepting the mathematical models underlying the various methods of numerical taxonomy (Johnson, 1970). In so doing, they overlook the fact that most statistical procedures were developed for certain purposes or types of data. When obviously inaccurate classifications result, or when different techniques produce seriously divergent results using the same data, it is often claimed that this constitutes a practical detraction from the value of numerical taxonomy. However, if the assumptions underlying the techniques (e.g., linear relationship of variables, homogeneity of covariance matrices, normality of distribution, large sample size) are not satisfied, reasonable results cannot be expected. The present paper discusses the analysis of metric characters by Q-mode correlation coefficients. Shortcomings of the technique will be shown to result from failure of the data to meet the conditions necessary to the method's definition, and successful determination, of similarity. In particular. allometric distortion in the data, and the need and practicality of correcting for it, will be discussed. THE EFFECT OF ALLOMETRY ON Q-MODE CORRELATION COEFFICIENTS The debate over selection of similarity coefficients and clustering methods in numerical taxonomy is a long and continuing one. However, when dimensions (measurements) alone are used as descriptive data for organisms, there is general concurrence that shape coefficients of affinity are superior to size (distance) coefficients: (Huxley, 1932; Rohlf and Sokal, 1965; Boyce, 1964, 1965, 1969; Penrose, 1947, 1954; Burnaby, 1966; Giles, 1956; Sokal and Sneath, 1963; Gould, 1967; Stahl, 1962; Oxnard, 1968, 1969a, 1969b; White and Gould, 1965; Minkoff, 1965; Kowalski, 1972; Blackith, 1965; Mosimann, 1970; Jolicoeur and Mosimann, 1960; Blackith and Reyment, 1971). Rohlf and Sokal (1965) and Boyce (1964, 1969) specifically recommend Q-mode correlation as the best co-

Corrections: Observations on Homogeneity of Covariance Matrices in Paleontologic Biometry

Corrections: Observations on Homogeneity of Covariance Matrices in Paleontologic Biometry

Observations on Homogeneity of Covariance Matrices in Paleontologic Biometry

Some examples of heterogeneous covariance matrices in infraspecific paleontologic and neontologic material are discussed. A slightly modified form of the generalized statistical distance between two populations is suggested, which may be useful with heterogeneous samples from an interbreeding population, or otherwise on the infraspecific level. The presentation is based on material of ostracods, cephalopods and turtles.