Bootstrap confidence regions for correspondence analysis

A bootstrap-based method for constructing confidence regions (CRs) around row or column points projected onto a pair of axes from the correspondence analysis (CA) of a two-way contingency table is presented. These regions deal with the specific question of the sampling variation of sample row and column profile points around population row and column profile points when both are projected onto the observed axes, rather than the decomposition of the χ²-test of independence or the general question of the stability of the observed CA display which has been considered in previous work. The method therefore constructs the regions in a different way to what has been proposed before. A simulation experiment shows that the method performs well in most of the situations in which it might be used, with a few exceptions being noted. An example illustrates that the method produces conclusions which are consistent with those from detailed parametric modelling.     

Clustering for contingency tables: boxes and partitions

The correspondence analysis (CA) method appears to be an effective tool for analysis of interrelations between rows and columns in two-way contingency data. A discrete version of the method, box clustering, is developed in the paper using an approximation version of the CA model extended to the case when CA factor values are required to be Boolean. Several properties of the proposed SEFIT-BOX algorithm are proved to facilitate interpretation of its output. It is also shown that two known partitioning algorithms (applied within row or column sets only) could be considered as locally optimal algorithms for fitting the model, and extensions of these algorithms to a simultaneous row and column partitioning problem are proposed.     

A note on Whittaker–Henderson graduation: Bisymmetry of the smoother matrix

Abstract

Whittaker–Henderson (WH) graduation is a popular smoothing method that has been used for mortality table construction in the actuarial sciences and for the trend-cycle decomposition in time series econometrics. This paper proves that the smoother matrix of WH graduation is bisymmetric (i.e., symmetric centrosymmetric). This result implies, for example, that the first row of the smoother matrix is equivalent to the last row of it in reverse order. We also provide some related results.     

Vec and vech operators for matrices,with some uses in jacobians and multivariate statistics

The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣?(vecX)/?(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ?(vechX)/?(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.     

Characteristic Polynomials by Diagonal Expansion

This article describes the method of expansion by diagonal elements for the evaluation of the determinant of the sum of two square matrices, one of which is diagonal. This result is applied to the problem of obtaining the characteristic polynomial of a square matrix. For problems of interest in statistics, this technique is frequently much more direct than the usual approach of simplification (e.g., triangularization) by elementary row and column operations. Several examples are presented.     

Data‐driven choice of the smoothing parametrization for kernel density estimators

There are several levels of sophistication when specifying the bandwidth matrix H to be used in a multivariate kernel density estimator, including H to be a positive multiple of the identity matrix, a diagonal matrix with positive elements or, in its most general form, a symmetric positive‐definite matrix. In this paper, the author proposes a data‐based method for choosing the smoothing parametrization to be used in the kernel density estimator. The procedure is fully illustrated by a simulation study and some real data examples. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.     

Diagonal uniform association symmetry modelsfor cumulative probabilities in square tables

For the analysis of square contingency tables with ordered categories, Tomizawa (1991) considered the diagonal uniform association symmetry (DUS) model, which has a multiplicative form for cell probabilities and has the structure of uniform association in the tables constructed using two diagonals that are equidistant from the main diagonal. This paper proposes another DUS model which has a similar multiplicative form for cumulative probabilities. The model indicates that the odds that an observation will fall in row category i or below and column category i+k or above, instead of in column category i or below and row category i+k or above, increase (decrease) exponentially as the cutpoint i increases for a fixed k. Examples are given.     

An alternative multivariate skew Laplace distribution: properties and estimation

A special case of the multivariate exponential power distribution is considered as a multivariate extension of the univariate symmetric Laplace distribution. In this paper, we focus on this multivariate symmetric Laplace distribution, and extend it to a multivariate skew distribution. We call this skew extension of the multivariate symmetric Laplace distribution the “multivariate skew Laplace (MSL) distribution” to distinguish between the asymmetric multivariate Laplace distribution proposed by Kozubowski and Podgórski (Comput Stat 15:531–540, 2000a) Kotz et al. (The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Chap. 6. Birkhäuser, Boston, 2001) and Kotz et al. (An asymmetric multivariate Laplace Distribution, Working paper, 2003). One of the advantages of (MSL) distribution is that it can handle both heavy tails and skewness and that it has a simple form compared to other multivariate skew distributions. Some fundamental properties of the multivariate skew Laplace distribution are discussed. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of the multivariate skew Laplace distribution is given. Some examples are provided to demonstrate the modeling strength of the skew Laplace distribution.     

Marginal inhomogeneity models for square contingency tables with nominal categories

For the analysis of square contingency tables with nominal categories, this paper proposes two kinds of models that indicate the structure of marginal inhomogeneity. One model states that the absolute values of log odds of the row marginal probability to the corresponding column marginal probability for each category i are constant for every i. The other model states that, on the condition that an observation falls in one of the off-diagonal cells in the square table, the absolute values of log odds of the conditional row marginal probability to the corresponding conditional column marginal probability for each category i are constant for every i. These models are used when the marginal homogeneity model does not hold, and the values of parameters in the models are useful for seeing the degree of departure from marginal homogeneity for the data on a nominal scale. Examples are given.     

Computation of the stationary distribution of a markov chain

Eight algorithms are considered for the computation of the stationary distribution l´ of a finite Markov chain with associated probability transition matrix P. The recommended algorithm is based on solving l´(I—P+eú)=ú, where e is the column vector of ones and u´ is a row vector satisfying u´e ≠0.An error analysis is presented for any such u including the choices ú= e_jP and ú=e´_j where é_j is the jth row of the identity matrix. Computationalcomparisons between five of the algorithms are made based on twenty 8 x 8, twenty 20 x 20, and twenty 40 x 40 transition matrices. The matrix (I—P+eú)^?1 is shown to be a non-singular generalized inverse of I—P when the unit root of P is simple and úe ≠ 0. A simple closed form expression is obtained for the Moore-Penrose inverse of I—P whenI—P has nullity one     

Correspondence Analysis and Its Application in Oncology

To explore the application value of correspondence analysis in oncology, we adopted correspondence analysis method to analyze the relationship between the amount of food eaten in some cities in China and the male gastric carcinoma mortality. According to scatter plots of row and column points, there are regional differences among the male gastric carcinoma mortality in different cities of China. Southern male citizens ate more rice and salt, less wheaten food, and fewer light vegetables than northern ones. There may be some carcinogenic factors in some food.     

Some partially variance balanced structurally incomplete row-column designs

Earlier results by the authors are used to provide the intrablock analysis for row-column designs that have observations at nodes of the row-column lattice, the design being structurally incomplete when some nodes are empty. Construction, properties, and intrablock analyses of some special b× b row-column designs with b empty nodes taken along the principal diagonal of the lattice are developed. The designs discussed have m > 1 associate classes and are said to be partially variance balanced. The special designs fall in two classes and are shown to be nearly optimal in a specified class of designs. A small catalog of designs constructed is provided and they should be useful when empty nodes do not represent wasted experimental units, perhaps because the row and column assignments of treatments are sequenced.     

Some new aspects of taxicab correspondence analysis

Correspondence analysis (CA) and nonsymmetric correspondence analysis are based on generalized singular value decomposition, and, in general, they are not equivalent. Taxicab correspondence analysis (TCA) is a \(\hbox {L}_{1}\) variant of CA, and it is based on the generalized taxicab singular value decomposition (GTSVD). Our aim is to study the taxicab variant of nonsymmetric correspondence analysis. We find that for diagonal metric matrices GTSVDs of a given data set are equivalent; from which we deduce the equivalence of TCA and taxicab nonsymmetric correspondence analysis. We also attempt to show that TCA stays as close as possible to the original correspondence matrix without calculating a dissimilarity (or similarity) measure between rows or columns. Further, we discuss some new geometric and distance aspects of TCA.     

Characteristic Function‐based Semiparametric Inference for Skew‐symmetric Models

Skew‐symmetric models offer a very flexible class of distributions for modelling data. These distributions can also be viewed as selection models for the symmetric component of the specified skew‐symmetric distribution. The estimation of the location and scale parameters corresponding to the symmetric component is considered here, with the symmetric component known. Emphasis is placed on using the empirical characteristic function to estimate these parameters. This is made possible by an invariance property of the skew‐symmetric family of distributions, namely that even transformations of random variables that are skew‐symmetric have a distribution only depending on the symmetric density. A distance metric between the real components of the empirical and true characteristic functions is minimized to obtain the estimators. The method is semiparametric, in that the symmetric component is specified, but the skewing function is assumed unknown. Furthermore, the methodology is extended to hypothesis testing. Two tests for a null hypothesis of specific parameter values are considered, as well as a test for the hypothesis that the symmetric component has a specific parametric form. A resampling algorithm is described for practical implementation of these tests. The outcomes of various numerical experiments are presented.     

Solving Survo puzzles using matrix combinatorial products

We investigate combinatorial matrix problems that are related to restricted integer partitions. They arise from Survo puzzles, where the basic task is to fill an m×n table by integers 1, 2,?…?, mn, so that each number appears only once, when the column sums and the row sums are fixed. We present a new computational method for solving Survo puzzles with binary matrices that are recoded and combined using the Hadamard, Kronecker, and Khatri–Rao products. The idea of our method is based on using the matrix interpreter and other data analytic tools of Survo R, which represents the newest generation of the Survo computing environment, recently implemented as a multiplatform, open source R package. We illustrate our method with detailed examples.     

Compositional Tables Analysis in Coordinates

Compositional tables – a continuous counterpart to the contingency tables – carry relative information about relationships between row and column factors; thus, for their analysis, only ratios between cells of a table are informative. Consequently, the standard Euclidean geometry should be replaced by the Aitchison geometry on the simplex that enables decomposition of the table into its independent and interactive parts. The aim of the paper is to find interpretable coordinate representation for independent and interaction tables (in sense of balances and odds ratios of cells, respectively), where further statistical processing of compositional tables can be performed. Theoretical results are applied to real‐world problems from a health survey and in macroeconomics.     

Some sufficient conditions for establishing (M,S)-optimality

Two sufficient conditions are given for an incomplete block design to be (M,S- optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.     

Testing for Circular Reflective Symmetry about a Known Median Axis

Circular data arise in many contexts, a particularly rich source being animal orientation experiments. Often, in the analysis of such data, a fundamental question of scientific interest is whether the underlying distribution is reflectively symmetric about some specific axis. In this paper, the situation in which the axis of interest is known to be a median axis is considered and a simple, asymptotically distribution- free test for circular reflective symmetry against skew alternatives is developed. The results from a simulation study lead to a testing strategy incorporating the new test and the circular analogue of the modified runs test of Modarres & Gastwirth (1996). The application of the testing strategy is illustrated using circular data arising from two animal orientation experiments.     

投入产出系数作用新探及其实证分析

直接消耗系数是生产单位j总产出要直接消耗的i产品量，反映j产品部门与i产品部门的生产技术联系，是建立投入产出模型的最基本系数；分配系数是i产品部门分配（提供）给各个部门作生产使用和提供给社会最终使用的数量占该部门产品总量的比重。目前对这些系数的作用阐述得不够充分，文章进一步从行、列、行和、列和的角度探讨它们的作用，也探讨其它一些可以直接从投入产出表计算的系数的新作用，并用中国2002年投入产出表系数进行分析。