Correspondence Analysis of Cumulative Frequencies Using a Decomposition of Taguchi's Statistic

Taguchi's statistic has long been known to be a more appropriate measure of association for ordinal variables than the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic for performing correspondence analysis when a two-way contingency table consists of one ordinal categorical variable. This article will explore the development of correspondence analysis using a decomposition of Taguchi's statistic.     

Catanova for Two-Way Contingency Tables with Ordinal Variables Using Orthogonal Polynomials

ABSTRACT

The analysis of variance of cross-classified (categorical) data (CATANOVA) is a technique designed to identify the variation between treatments of interest to the researcher. There are well-established links between CATANOVA and the Goodman and Kruskal tau statistic as well as the Light and Margolin R ² for the purposes of the graphical identification of this variation.

The aim of this article is to present a partition of the numerator of the tau statistic, or equivalently, the BSS measure in the CATANOVA framework, into location, dispersion, and higher order components. Even if a CATANOVA identifies an overall lack of variation, by considering this partition and calculations derived from them, it is possible to identify hidden, but statistically significant, sources of variation.     

Cumulative correspondence analysis of ordered categorical data from industrial experiments

Most studies of quality improvement deal with ordered categorical data from industrial experiments. Accounting for the ordering of such data plays an important role in effectively determining the optimal factor level of combination. This paper utilizes the correspondence analysis to develop a procedure to improve the ordered categorical response in a multifactor state system based on Taguchi's statistic. Users may find the proposed procedure in this paper to be attractive because we suggest a simple and also popular statistical tool for graphically identifying the really important factors and determining the levels to improve process quality. A case study for optimizing the polysilicon deposition process in a very large-scale integrated circuit is provided to demonstrate the effectiveness of the proposed procedure.     

A Perturbation Approach to Outlier Detection in Two-Way Contingency Tables

In order to identify outliers in contingency tables, we evaluate the derivatives of the perturbation-formed surface of the Pearson goodness-of-fit statistic. The resulting diagnostics are shown to be less susceptible to masking and swamping problems than residual-based measures. A Monte Carlo study further confirms the effectiveness of the proposed diagnostics.     

Tables for the Friedman rank test

This note presents tables for Friedman's test for two-way analysis of variance by ranks. These tables are more accurate than those that are presented in the literature. After intensive simulations, we have found for particular critical values some discrepancies with tables published earlier. The tables are also more extensive than those previously available.     

A Test of Independence in Two-Way Contingency Tables Based on Maximal Correlation

Maximal correlation has several desirable properties as a measure of dependence, including the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. We focus on two-dimensional contingency tables and discuss a procedure for estimating maximal correlation, which we use for constructing a test of independence. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorrelated, we point out some cases for which the new test is more powerful.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

Accuracy of Power-Divergence Statistics for Testing Independence and Homogeneity in Two-Way Contingency Tables

The small-sample accuracy of seven members of the family of power-divergence statistics for testing independence or homogeneity in contingency tables was studied via simulation. The likelihood ratio statistic G ² and Pearson's X ² statistic are among these seven members, whose behavior was studied at nominal test sizes of.01 and.05 with marginal distributions that could be uniform or skewed and with a set of sample sizes that included sparseness conditions as measured through table density (i.e., the ratio of sample size to number of cells). The likelihood ratio statistic G ² rejected the null hypothesis too often even with large table density, whereas Pearson's X ² was sufficiently accurate and only presented a minor misbehavior when table density was less than two observations/cell. None of the other five statistics outperformed Pearson's X ². A nonasymptotic variant of X ² solved the minor inaccuracies of Pearson's X ² and turned out to be the most accurate statistic for testing independence or homogeneity, even with table densities of one observation/cell. These results clearly advise against the use of the likelihood ratio statistic G ².     

Sequential Comparison of d Populations and Related Tables

The paper deals with the numerical solution of the likelihood equations for incomplete data from exponential families, that is for data being a function of exponential family data. Illustrative examples especially studied in this paper concern grouped and censored normal samples and normal mixtures. A simple iterative method of solution is proposed and studied. It is shown that the sequence of iterates converges to a relative maximum of the likelihood function, and that the convergence is geometric with a factor of convergence which for large samples equals the maxi-mal relative loss of Fisher information due to the incompleteness of data. This large sample factor of convergence is illustrated diagrammaticaily for the examples mentioned above. Experiences of practical application are mentioned.     

Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j (> i ) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.     

Tests of Symmetry in Three-dimensional Contingency Tables Based on Phi-divergence Statistics

In this paper we introduce a family of test statistics for testing complete symmetry in three-dimensional contingency tables based on phi- divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistics as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to show that some new statistics offer an attractive alternative to the classical Pearson and likelihood ratio test statistics for this problem of complete symmetry.     

Analysis of Contingency Tables with Clustered Observations

A new approach is presented for testing independence in contingency tables with clustered observations. The approach is based on the framework of generalized linear mixed models. Under the multinomial logistic link function, the category counts are modelled with random cluster effects and a modified likelihood ratio statistic is used for testing independence. The method is applicable to multi-way tables, and can accommodate multiple levels of clustering. It is illustrated using a benchmark dataset.     

Cumulative correspondence analysis using orthogonal polynomials

Taguchi's statistic has long been known to be a more appropriate measure of association of the dependence for ordinal variables compared to the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic in the correspondence analysis context when a two-way contingency table consists at least of an ordinal categorical variable. The aim of this paper, considering the contingency table with two ordinal categorical variables, is to show a decomposition of Taguchi's index into linear, quadratic and higher-order components. This decomposition has been developed using Emerson's orthogonal polynomials. Moreover, two case studies to explain the methodology have been analyzed.     

Analysis of binary data in the randomized complete block design

The technique proposed by Shah and Claypool (1984) is extended here for the randomized complete block design with one binary observation per cell. In addition, it provides an al- ternative derivation of the distribution of Cochran's Q sta- tistic which is straightforward.     

Tests of Multivariate Independence for Ordinal Data

Population and sample versions of Kendall and Spearman measures of association suitable for multivariate ordinal data are defined. The latter generalize the indices of dependence of Ruymgaart and van Zuijlen (1978 Ruymgaart , F. H. , van Zuijlen , M. C. A. ( 1978 ). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case . Ann. Statist. 6 : 588 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]), Joe (1990 Joe , H. ( 1990 ). Multivariate concordance . J. Multivariate Anal. 35 : 12 – 30 .[Crossref], [Web of Science ®] , [Google Scholar]), and Schmid and Schmidt (2007 Schmid , F. , Schmidt , R. ( 2007 ). Multivariate extensions of Spearman's rho and related statistics . Statist. Probab. Lett. 77 : 407 – 416 .[Crossref], [Web of Science ®] , [Google Scholar]) by allowing atoms in the underlying distribution. The representation of the proposed empirical measures as U-statistics enables to establish their asymptotic normality under general distributions. A special attention is given to tests of independence for multivariate ordinal data, where the power of the new methodologies are investigated under fixed and contiguous alternatives.     

属性数据的多变量对应分析模型及应用

属性数据的建模和分析是市场研究活动中最重要的一类应用问题,学术界和市场研究行业一般使用列联表及统计检验的方法进行分析。故将法国统计学家J.P.Beozecri提出的对应分析法导入市场研究实证过程,使问卷调查中常见的多变量属性数据分析建模过程呈现出稳健的统计结论和直观的二维图示,并推广到广义多变量对应分析场合,其结论可靠,有应用创新。     

对应分析统计检验体系探讨

对应分析因其结果的易读性,近些年得到了越来越广泛的应用。为了更好地应用对应分析,提出建立对应分析统计检验体系,包括对应分析适用性的统计检验以及对应分析效果的检验,同时还提出应用对应分析时应注意的其它问题。     

New Tests of Spatial Segregation Based on Nearest Neighbour Contingency Tables

Abstract.  The spatial clustering of points from two or more classes (or species) has important implications in many fields and may cause segregation or association, which are two major types of spatial patterns between the classes. These patterns can be studied using a nearest neighbour contingency table (NNCT) which is constructed using the frequencies of nearest neighbour types. Three new multivariate clustering tests are proposed based on NNCTs using the appropriate sampling distribution of the cell counts in a NNCT. The null patterns considered are random labelling (RL) and complete spatial randomness (CSR) of points from two or more classes. The finite sample performance of these tests are compared with other tests in terms of empirical size and power. It is demonstrated that the newly proposed NNCT tests perform relatively well compared with their competitors and the tests are illustrated using two example data sets.     

Measures of Association and Visualization of Log Odds Ratio Structure for a Two Way Contingency Table

The odds ratio (OR) is a measure of association used for analysing an I × J contingency table. The total number of ORs to check grows with I and J. Several statistical methods have been developed for summarising them. These methods begin from two different starting points, the I × J contingency table and the two‐way table composed by the ORs. In this paper we focus our attention on the relationship between these methods and point out that, for an exhaustive analysis of association through log ORs, it is necessary to consider all the outcomes of these methods. We also introduce some new methodological and graphical features. In order to illustrate previously used methodologies, we consider a data table of the cross‐classification of the colour of eyes and hair of 5387 children from Scotland. We point out how, through the log OR analysis, it is possible to extract useful information about the association between variables.