Decomposition of the main effects and interaction term by using orthogonal polynomials in multiple non symmetrical correspondence analysis

The multiple non symmetric correspondence analysis (MNSCA) is a useful technique for analyzing a two-way contingency table. In more complex cases, the predictor variables are more than one. In this paper, the MNSCA, along with the decomposition of the Gray–Williams Tau index, in main effects and interaction term, is used to analyze a contingency table with two predictor categorical variables and an ordinal response variable. The Multiple-Tau index is a measure of association that contains both main effects and interaction term. The main effects represent the change in the response variables due to the change in the level/categories of the predictor variables, considering the effects of their addition, while the interaction effect represents the combined effect of predictor categorical variables on the ordinal response variable. Moreover, for ordinal scale variables, we propose a further decomposition in order to check the existence of power components by using Emerson's orthogonal polynomials.     

The Confidence Ellipses in Decomposition Multiple Non Symmetric Correspondence Analysis

Non Symmetric Correspondence Analysis (NSCA) (D'Ambra and Lauro, 1989 D'Ambra , L. , Lauro , N. ( 1989 ). Non symmetrical analysis of three way contingency tables . In: Multiway Data Analysis , Coppi , R. , Bolasco , S. , Eds., North Holland , Amsterdam : pp. 301 – 315 . [Google Scholar]) is a useful technique for analyzing a two-way contingency table.

The key difference between the symmetrical and non symmetrical versions of correspondence analysis rests on the measure of the association used to quantify the relationship between the variables. For a two-way, or multi-way, contingency table, the Pearson chi-squared statistic is commonly used when it can be assumed that the categorical variables are symmetrically related. However, for a two-way table, it may be that one variable can be treated as a predictor variable and the second variable can be considered as a response variable.

Yet, for such a variable structure, the Pearson chi-squared statistic is not an appropriate measure of the association. Instead, one may consider the Goodman-Kruskal tau index. In the case that there are more than two cross-classified variables, multivariate versions of the Goodman-Kruskal tau index can be considered. These include Marcotorchino's index (Marcotorchino, 1985) and Gray-Williams’ index (Gray and Williams, 1975 Gray , L. N. , Williams , J. S. ( 1975 ). Goodman and Kruskals Tau B: Multiple and partial analogy. Amer. Statist. Assoc. Proc. Soc. Statist. Sec. pp. 444 – 448 . [Google Scholar]).

In this article, the Multiple non Symmetric Correspondence Analysis (MNSCA), along with the decomposition of the TAU by Gray-Williams in main effects and interaction (D'Ambra et al., 2011 D'Ambra , L. , D'Ambra , A. , Sarnacchiaro , P. ( 2011 ). Visualising main effects and interaction term in multiple non symmetric correspondence analysis. Submitted.  [Google Scholar]), is used for the evaluation of the innovative performance of the manufacturing enterprises in Campania.

Finally, to identify a category which is statistically significant, the confidence ellipses have been proposed for the Multiple Non Symmetric Correspondence Analysis starting from the ellipses suggested by Beh (2010 Beh , E. J. ( 2010 ). Elliptical confidence regions for simple correspondence analysis . J. Statisti. Plann. Infer. [Web of Science ®] , [Google Scholar]) for the symmetrical analysis.     

Studying the dependence between ordinal-nominal categorical variables via orthogonal polynomials

In situations where the structure of one of the variables of a contingency table is ordered recent theory involving the augmentation of singular vectors and orthogonal polynomials has shown to be applicable for performing symmetric and non-symmetric correspondence analysis. Such an approach has the advantage of allowing the user to identify the source of variation between the categories in terms of components that reflect linear, quadratic and higher-order trends. The purpose of this paper is to focus on the study of two asymmetrically related variables cross-classified to form a two-way contingency table where only one of the variables has an ordinal structure.     

Sweeping,alignment and the analysis of unbalanced data

The terms sweeping and alignment refer to the same process. Sweeping/alignment is used by data analysts as a technique for describing the effects of a model factor (e.g., treatments in a randomized block design) after the effects of nuisance parameters (e.g., blocks) have been removed from the data. In this paper sweeping/alignment is used as the basis for developing tests of factors in unbalanced experimental design models. Formulas are presented for treatment effects in randomized block designs with missing observations, and for interaction and main effects in unbalanced two-way factorial designs with empty cells.     

A non linear analysis of the preference for preservation of the teeth

Data resulting from behavioral dental research, usually categorical or discretized and having unknown measurement and distributional characteristics, often cannot be analyzed with classical multivariate techniques. A non linear principal components technique called multiple correspondence analysis is presented with its corresponding computer program that can handle this kind of data. The model is described as a form of multidimensional scaling. The technique Is applied in order to establish which factors are associated with an Individual's preference for preservation of the teeth.     

The analysis of dependence for three ways contingency tables with ordinal variables: A case study of patient satisfaction data

For many questionnaires and surveys in the marketing, business, and health disciplines, items often involve ordinal scales (such as the Likert scale and rating scale) that are associated in sometimes complex ways. Techniques such as classical correspondence analysis provide a simple graphical means of describing the nature of the association. However, the procedure does not allow the researcher to specify how one item may be associated with another, nor does the analysis allow for the ordinal structure of the scales to be reflected. This article presents a graphical approach that can help the researcher to study in depth the complex association of the items and reflect the structure of the items. We will demonstrate the applicability of this approach using data collected from a study that involves identifying major factors that influence the level of patient satisfaction in a Neapolitan hospital.     

TESTING SUBHYPOTHESES AND ESTIMATING σ2 IN THE NONREPLICATED THREE-WAY MULTIPLICATIVE INTERACTION MODEL

ABSTRACT

Standard statistical techniques do not provide methods for analyzing data from nonreplicated factorial experiments. Such experiments occur for several reasons. Many experimenters may prefer conducting experiments having a large number of factor levels with no replications than conducting experiments with a few factor levels with replications particularly in pilot studies. Such experiments may allow one to identify factor combinations to be used in follow-up experiments. Another possibility is when the experimenter thinks that an experiment is replicated when in fact it is not. This occurs when a naive researcher believes that sub-samples are replicates when in reality they are not. Nonreplicated two-way experiments have been extensively studied. This paper discusses the analysis of nonreplicated three-way experiments. In particular, estimation of σ² is discussed and a test is derived for testing whether three-factor interaction is absent in sub-areas of three-way data using a nonreplicated three-way multiplicative interaction model with a single multiplicative term. Approximate null distribution of the derived test statistic is studied using Monte Carlo studies and results are illustrated through an example.     

A comparison of several analysis of variance programs with unequal cell size

Various programs in statistical packages for analysis of variance with unequal cell size give different results to the same data because of nonorthogonality of the main effects and interactions. This paper explains how these programs treat the problem of analysis of variance of unbalanced data.     

Type i error rates for divisive clustering methods for grouping means in analysis of variance

Five univariate divisive clustering methods for grouping means in analysis of variance are considered.Unlike pairwise multiple comparison procedures, cluster analysis has the advantage of producing non-overlapping groups of the treatment means. Comparisonwise Type I error rates and average numbers of clusters per experiment are examined for a heterogeneous set of 20 true treatment means with 11 embedded homogenous sub-groups of one or more treatments. The results of a simulation study clearly show that observed comparisonwise error rate and number of clusters are determined to a far greater extent by the precision of the experiment (as determined by the magnitude of the standard deviation) than by either the stated significance level or the clustering method used.     

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.     

Visualizing main effects and interaction in multiple non-symmetric correspondence analysis

Non-symmetric correspondence analysis (NSCA) is a useful technique for analysing a two-way contingency table. Frequently, the predictor variables are more than one; in this paper, we consider two categorical variables as predictor variables and one response variable. Interaction represents the joint effects of predictor variables on the response variable. When interaction is present, the interpretation of the main effects is incomplete or misleading. To separate the main effects and the interaction term, we introduce a method that, starting from the coordinates of multiple NSCA and using a two-way analysis of variance without interaction, allows a better interpretation of the impact of the predictor variable on the response variable. The proposed method has been applied on a well-known three-way contingency table proposed by Bockenholt and Bockenholt in which they cross-classify subjects by person's attitude towards abortion, number of years of education and religion. We analyse the case where the variables education and religion influence a person's attitude towards abortion.     

A method of unconstrained multiple comparisons with the best

There are three types of multiple comparisons: all-pairwise multiple comparisons (MCA), multiple comparisons with the best (MCB), and multiple comparisons with a control (MCC). There are also three levels of multiple comparisons inference: confidence sets, subset comparisons, test of homogeneity. In current practice, MCA procedures dominate. In correct attempts at more efficient comparisons, in the form of employing lower level MCA procedures for higher level inference, account for the most frequent abuses in multiple comparisons. A better strategy is to choose the correct type of inference at the level of inference desired. In particular, very often the simulataneous comparisons of each treatment with the best of the other treatments (MCB) suffice. Hsu (1984b) gave simultaneous confidence intervals for θ_i ? max_j≠iθ_j having the simple form [? (Y_i ?max_j≠i Y_j ? C) (Y_i?max_j≠i Y_j + C)⁺]. Those intervals were constrained, sothat even if a treatment is inferred to be the best, no positive bound on how much it is better thatn the rest is given, a somewhat undesirable property. In this article it is shown that by employing a slightly larger critical value, the nonpositivity constraint on the lower bound is removed.     

A novel approach based on multiple correspondence analysis for monitoring social networks with categorical attributed data

In in most cases, the distribution of communications is unknown and one may summarize social network communications with categorical attributes in a contingency table. Due to the categorical nature of the data and a large number of features, there are many parameters to be considered and estimated in the model. Hence, the accuracy of estimators decreases. To overcome the problem of high dimensionality and unknown communications distribution, multiple correspondence analysis is used to reduce the number of parameters. Then the rescaled data are studied in a Dirichlet model in which the parameters should be estimated. Moreover, two control charts, Hotelling’s T² and multivariate exponentially weighted moving average (MEWMA), are developed to monitor the parameters of the Dirichlet distribution. The performance of the proposed method is evaluated through simulation studies in terms of average run length criterion. Finally, the proposed method is applied to a real case.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

Multiple imputation compared with restricted pseudo‐likelihood and generalized estimating equations for analysis of binary repeated measures in clinical studies

Non‐likelihood‐based methods for repeated measures analysis of binary data in clinical trials can result in biased estimates of treatment effects and associated standard errors when the dropout process is not completely at random. We tested the utility of a multiple imputation approach in reducing these biases. Simulations were used to compare performance of multiple imputation with generalized estimating equations and restricted pseudo‐likelihood in five representative clinical trial profiles for estimating (a) overall treatment effects and (b) treatment differences at the last scheduled visit. In clinical trials with moderate to high (40–60%) dropout rates with dropouts missing at random, multiple imputation led to less biased and more precise estimates of treatment differences for binary outcomes based on underlying continuous scores. Copyright © 2005 John Wiley & Sons, Ltd.     

A hierarchy of lorenz curves based on the generalized tukey's lambda distribution

A hierarchy of Lorenz curves based on the generalized Tukey's Lambda distribution is proposed. Representations of the corresponding distribution and density function are also provided, together with popular inequality measures. Estimation methods are suggested. Finally, a comparison with other parametric families of Lorenz curves is established.     

Sensitivity analysis in principal component analysis:influence on the subspace spanned by principal components.

The problem of detecting influential observations in principalcomponent analysis was discussed by several authors. Radhakrishnan and kshirsagar ( 1981 ), Critchley ( 1985 ), jolliffe ( 1986 )among others discussed this topicby using the influence functions I(X;θ_s)and I(X;V_s)of eigenvalues and eigenvectors, which wwere derived under the assumption that the eigenvalues of interest were simple. In this paper we propose the influence functionsI(X;∑^q _s=1θ_sV_sV_s ^T)and I(x;∑^q _s=1V_sV_s ^t)(q<p;p:number of variables) to investigate the influence onthe subspace spanned by principal components. These influence functions are applicable not only to the case where the edigenvalues of interst are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

Bayesian analysis of the generalized gamma distribution using non-informative priors

The Generalized gamma (GG) distribution plays an important role in statistical analysis. For this distribution, we derive non-informative priors using formal rules, such as Jeffreys prior, maximal data information prior and reference priors. We have shown that these most popular formal rules with natural ordering of parameters, lead to priors with improper posteriors. This problem is overcome by considering a prior averaging approach discussed in Berger et al. [Overall objective priors. Bayesian Analysis. 2015;10(1):189–221]. The obtained hybrid Jeffreys-reference prior is invariant under one-to-one transformations and yields a proper posterior distribution. We obtained good frequentist properties of the proposed prior using a detailed simulation study. Finally, an analysis of the maximum annual discharge of the river Rhine at Lobith is presented.     

Two-stage procedures for the difference of two multinormal means with covariance matrices different only by unknown scalar multipliers

We consider the problem of constructing a fixed-size confidence region for the difference of means of two multivariate normal populations It is assumed that the variance-covariance matrices of two populations are different only by unknown scalar multipliers Two-stage procedures are presented to derive such a confidence region We also discuss the asymptotic efficiency of the procedure.