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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Multivariate analysis of variance ： ウィキペディア英語版 Multivariate analysis of variance In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is typically followed by significance tests involving individual dependent variables separately. It helps to answer 〔Stevens, J. P. (2002). ''Applied multivariate statistics for the social sciences.'' Mahwah, NJ: Lawrence Erblaum.〕 # Do changes in the independent variable(s) have significant effects on the dependent variables? # What are the relationships among the dependent variables? # What are the relationships among the independent variables? ==Relationship with ANOVA== MANOVA is a generalized form of univariate analysis of variance (ANOVA),〔 although, unlike univariate ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean differences. Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution. MANOVA is based on the product of model variance matrix, $\backslash Sigma\_$ and inverse of the error variance matrix, $\backslash Sigma\_^$, or $A=\backslash Sigma\_\; \backslash times\; \backslash Sigma\_^$. The hypothesis that $\backslash Sigma\_\; =\; \backslash Sigma\_$ implies that the product $A\; \backslash sim\; I$. Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis. The most common〔(【引用サイトリンク】title=Stata Annotated Output -- MANOVA )〕 statistics are summaries based on the roots (or eigenvalues) $\backslash lambda\_p$ of the $A$ matrix: * Samuel Stanley Wilks' $\backslash Lambda\_\; =\; \backslash prod\; \_(1/(1\; +\; \backslash lambda\_))\; =\; \backslash det(I\; +\; A)^\; =\; \backslash det(\backslash Sigma\_)/\backslash det(\backslash Sigma\_\; +\; \backslash Sigma\_)$ distributed as lambda (Λ) * the Pillai-M. S. Bartlett trace, $\backslash Lambda\_\; =\; \backslash sum\; \_(\backslash lambda\_/(1\; +\; \backslash lambda\_))\; =\; \backslash mathrm((I\; +\; A)^)$〔http://www.real-statistics.com/multivariate-statistics/multivariate-analysis-of-variance-manova/manova-basic-concepts/〕 * the Lawley-Hotelling trace, $\backslash Lambda\_\; =\; \backslash sum\; \_(\backslash lambda\_)\; =\; \backslash mathrm(A)$ * Roy's greatest root (also called ''Roy's largest root''), $\backslash Lambda\_\; =\; max\_p(\backslash lambda\_p)\; =\; \backslash |A\backslash |\_$ Discussion continues over the merits of each,〔 although the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases. 〔 Camo http://www.camo.com/multivariate_analysis.html 〕The best-known approximation for Wilks' lambda was derived by C. R. Rao. In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multivariate analysis of variance」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.