Is categorization of random data necessary for parallel analysis on Likert-type data?

Random eigenvalues are the key elements in parallel analysis. When analyzing Likert-type data, is it necessary to convert the continuous random data to discrete type before estimating eigenvalues? The study compared the random eigenvalues obtained from continuous and categorized random data from two popular computer programs to be used as the basis for comparison in conducting parallel analysis on Likert-type data. Results indicated that categorized random data gave eigenvalues and number of factors similar to those obtained from continuous random data. It is suggested that when conducting parallel analysis on Likert-type data by the two programs, the conversion is unnecessary.     

特征值与特征向量相关问题的研究

特征值与特征向量是代数研究的中心问题之一,是两个密切相关的概念.在理论和实际应用中,特征值与特征向量都有举足轻重的地位.本文主要是对矩阵特征值与特征向量进行讨论,给出关于特征值与特征向量的相关命题.并对有关特征值与特征向量的题型做了解析以及解题的错误做出相应分析.     

On a Relation between Principal Components and Regression Analysis

A regression approach to principal component analysis is presented in this note. We provide an alternative interpretation of principal components that illustrates the relation between the extra sum of squares in regression analysis and the eigenvalues associated with the principal components.     

Finding the maximum efficiency for multistage ranked-set sampling

Multistage ranked-set sampling (MRSS) is a generalization of ranked-set sampling in which multiple stages of ranking are used. It is known that for a fixed distribution under perfect rankings, each additional stage provides a gain in efficiency when estimating the population mean. However, the maximum possible efficiency for the MRSS sample mean relative to the simple random sampling sample mean has not previously been determined. In this paper, we provide a method for computing this maximum possible efficiency under perfect rankings for any choice of the set size and the number of stages. The maximum efficiency tends to infinity as the number of stages increases, and, for large numbers of stages, the efficiency-maximizing distributions are symmetric multi-modal distributions where the number of modes matches the set size. The results in this paper correct earlier assertions in the literature that the maximum efficiency is bounded and that it is achieved when the distribution is uniform.     

Some remarks on subdesigns of symmetric designs

Let D(υ, k, λ) be a symmetric design containing a symmetric design D₁(υ₁, k₁, λ₁) (k₁ < k) and let x = υ₁(k ? k₁)/(υ ? υ₁). We show that k ≥(k₁ ? x)² + λ If equality holds, D₁ is called a tight subdesign of D. In the special case, λ₁ = λ, the inequality reduces to that of R.C. Bose and S.S. Shrikhande and tight subdesigns then correspond to their notion of Baer subdesigns. The possibilities for (7upsi;, k, λ) designs having Baer subdesigns are investigated.     

On the independence of linear and quadratic forms in normal variates

The independence of linear and quadratic forms in normal variates is very important in linear models and multivariate analysis, especially for the independence of quadratic forms. However, associated proofs in the early literature are incorrect, incomplete, or misleading. In this article, we provide simple and elegant proofs for some extended results.     

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.     

MulticolIinearity Problems in Modeling Time Series With Trading-Day Variation

This article discusses the multicollinearity problems associated with the estimation of time series models influenced by trading-day variation. An analysis of the design matrix is performed, and measures of the degree of multicollinearity are computed. The characteristics of the design matrix of a popular parameterization are also analyzed, and it is shown that in some cases use of this reparameterization significantly alleviates the multicollinearity problem.     

Two-Step Hypothesis Testing When the Number of Variables Exceeds the Sample Size

Medical images and genetic assays typically generate data with more variables than subjects. Scientists may use a two-step approach for testing hypotheses about Gaussian mean vectors. In the first step, principal components analysis (PCA) selects a set of sample components fewer in number than the sample size. In the second step, applying classical multivariate analysis of variance (MANOVA) methods to the reduced set of variables provides the desired hypothesis tests. Simulation results presented here indicate that success of the PCA in the first step requires nearly all variation to occur in population components far fewer in number than the number of subjects. In the second step, multivariate tests fail to attain reasonable power except in restrictive, favorable cases. The results encourage using other approaches discussed in the article to provide dependable hypothesis testing with high dimension, low sample size data (HDLSS).