Comparison of Frailty Models for Acute Leukemia Data under Gompertz Baseline Distribution

In this article, we consider two different shared frailty regression models under the assumption of Gompertz as baseline distribution. Mostly assumption of gamma distribution is considered for frailty distribution. To compare the results with gamma frailty model, we consider the inverse Gaussian shared frailty model also. We compare these two models to a real life bivariate survival data set of acute leukemia remission times (Freireich et al., 1963 Freireich, E.J., Gehan, E., Frei, E., Schroeder, L.R., Wolman, I.J., Anbari, R., Burgert, E.O., Mills, S.D., Pinkel, D., Selawry, O.S., Moon, J.H., Gendel, B.R., Spurr, C.L., Storrs, R., Haurani, F., Hoogstraten, B., Lee, S. (1963). The effect of 6-mercaptopurine on the duration of steroid-induced remissions in acute leukemia: a model for evaluation of other potentially useful therapy. Blood 21:699–716.[Web of Science ®] , [Google Scholar]). Analysis is performed using Markov Chain Monte Carlo methods. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the acute leukemia data.     

General Saddlepoint Approximations: Application to the Anderson-Darling Test Statistic

We consider the relative merits of various saddlepoint approximations for the cumulative distribution function (cdf) of a statistic with a possibly non normal limit distribution. In addition to the usual Lugannani-Rice approximation, we also consider approximations based on higher-order expansions, including the case where the base distribution for the approximation is taken to be non normal. This extends earlier work by Wood et al. (1993 Wood , A. T. A. , Booth , J. G. , Butler , R. W. ( 1993 ). Saddlepoint approximations to the CDF of some statistics with nonnormal limit distributions . Journal of the American Statistical Association 88 : 680 – 686 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). These approximations are applied to the distribution of the Anderson-Darling test statistic. While these generalizations perform well in the middle of the distribution's support, a conventional normal-based Lugannani-Rice approximation (Giles, 2001 Giles , D. E. A. ( 2001 ). A Saddlepoint approximation to the distribution function of the Anderson-Darling test statistic . Communications in Statistics B 30 : 899 – 905 .[Taylor & Francis Online] , [Google Scholar]) is superior for conventional critical regions.     

Selection of Binary Variables and Classification by Boosting

We adopt boosting for classification and selection of high-dimensional binary variables for which classical methods based on normality and non singular sample dispersion are inapplicable. Boosting seems particularly well suited for binary variables. We present three methods of which two combine boosting with the relatively classical variable selection methods developed in Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]). Our primary interest is variable selection in classification with small misclassification error being used as validation of proposed method for variable selection. Two of the new methods perform uniformly better than Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]) in one set of simulated and three real life examples.     

Saddlepoint Approximations to the Limiting Distribution of the Modified Anderson–Darling Test Statistic

The approximation for the distribution function of test statistic is extremely important in statistics. The standard and higher-order saddlepoint approximations are considered in tails of the limiting distribution for the modified Anderson–Darling test. The saddlepoint approximations are compared with the approximation of Sinclair et al. (1990 Sinclair , C. D. , Spurr , B. D. , Ahmad , M. I. ( 1990 ). Modified Anderson Darling test . Communication Statistics—Theory and Methods 19 : 3677 – 3686 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for upper tail area. An empirical function is derived to estimate the critical values of a saddlepoint approximation.     

Population Attributable Fraction as a Measure of Heritability in Dichotomous Twin Data

In many genetic analyses of dichotomous twin data, odds ratios have been used to test hypotheses on heritability and shared common environment effects of a given disease (Lichtenstein et al., 2000 Lichtenstein , P. , Holm , N. , Verkasalo , P. , Iliadou , A. , Kaprio , J. , Koskenvuo , M. , Pukkala , E. , Skytthe , A. , Hemminki , K. ( 2000 ). Environmental and heritable factors in the causation of cancer . New England Journal of Medicine 343 : 78 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]; Ahlbom et al., 1997 Ahlbom , A. , Lichtenstein , P. , Malmström , H. , Feychting , M. , Hemminki , K. , Pedersen , N. L. ( 1997 ). Cancer in twins: genetic and nongenetic familial risk factors . Journal of the National Cancer Institute 89 : 287 – 93 . [Google Scholar]; Ramakrishnan et al., 1992 Ramakrishnan , V. , Goldberg , J. , Henderson , W. , Elsen , S. , True , W. , Lyons , M. , Tsuang , M. ( 1992 ). Elementary methods for the analysis of dichotomous outcomes in unselected samples of twins . Genetic Epidemiology 9 : 273 – 287 . [Google Scholar], 4). However, estimates of these two effects have not been dealt with in the literature. In epidemiology, the attributable fraction (AF), a function of the odds ratio and the prevalence of the risk factor has been used to describe the contribution of a risk factor to a disease in a given population (Leviton, 1973 Leviton , A. ( 1973 ). Definitions of attributable risk . American Journal of Epidemiology 98 : 231 . [Google Scholar]). In this article, we adapt the AF to quantify the heritability and the shared common environment. Twin data on cancer, gallstone disease and phobia are used to illustrate the applicability of the AF estimate as a measure of heritability.     

A Generalized Correlation Coefficient for the General Two-Treatment Crossover Design

This article extends the correlation methodology developed by Chinchilli et al. (2005 Chinchilli , V. M. , Phillips , B. R. , Mauger , D. T. , Szefler , S. J. ( 2005 ). A general class of correlation coefficients for the 2 × 2 crossover design . Biometr. J. 47 : 1 – 10 . [Google Scholar]) for the 2 × 2 crossover design to more complex crossover designs for clinical trials. We describe how the methodology can be adapted to a general type of two-treatment crossover design which includes either at least two sequences or at least two treatment periods or both. We then derive the asymptotic theory for the corresponding correlation statistics, investigate the statistical accuracy of the estimators via bootstrap analyses, and demonstrate their use with two real data examples.     

Steady-state Gibbs sampler estimation for lung cancer data

This paper is based on the application of a Bayesian model to a clinical trial study to determine a more effective treatment to lower mortality rates and consequently to increase survival times among patients with lung cancer. In this study, Qian et al. [13 J. Qian, D.K. Stangl, and S. George, A Weibull model for survival data: Using prediction to decide when to stop a clinical trial, in Bayesian Biostatistics, D. Berry and D. Stangl, eds., Marcel Dekker, New York, 1996, pp. 187–205. [Google Scholar]] strived to determine if a Weibull survival model can be used to decide whether to stop a clinical trial. The traditional Gibbs sampler was used to estimate the model parameters. This paper proposes to use the independent steady-state Gibbs sampling (ISSGS) approach, introduced by Dunbar et al. [3 M. Dunbar, H.M. Samawi, R. Vogel, and L. Yu, A more efficient Gibbs sampler estimation using steady state simulation: Application to public health studies, J. Stat. Simul. Comput. 10.1080/00949655.2013.770857.[Taylor &; Francis Online] , [Google Scholar]], to improve the original Gibbs sampler in multidimensional problems. It is demonstrated that ISSGS provides accuracy with unbiased estimation and improves the performance and convergence of the Gibbs sampler in this application.     

A Bottom-Up Dynamic Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries

In Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]), the authors introduced a Markov copula model of portfolio credit risk where pricing and hedging can be done in a sound theoretical and practical way. Further theoretical backgrounds and practical details are developed in Bielecki et al. (2014b Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014b ). A bottom-up dynamic model of portfolio credit risk - Part I: Markov copula perspective . In: Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.1844574) . [Google Scholar],c) where numerical illustrations assumed deterministic intensities and constant recoveries. In the present paper, we show how to incorporate stochastic default intensities and random recoveries in the bottom-up modeling framework of Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]) while preserving numerical tractability. These two features are of primary importance for applications like CVA computations on credit derivatives (Assefa et al., 2011 Assefa , S. , Bielecki , T. R. , Crépey , S. , Jeanblanc , M. ( 2011 ). CVA computation for counterparty risk assessment in credit portfolios . In: Bielecki , T.R. , Brigo , D. , Patras , F. , Eds., Credit Risk Frontiers . Hoboken : Wiley/Bloomberg-Press . [Google Scholar]; Bielecki et al., 2012 Bielecki , T. R. , Crépey , S. , Jeanblanc , M. , Zargari , B. ( 2012 ). Valuation and Hedging of CDS counterparty exposure in a markov copula model . Int. J. Theoret. Appl. Fin. 15 ( 1 ): 1250004 .[Crossref] , [Google Scholar]), as CVA is sensitive to the stochastic nature of credit spreads and random recoveries allow to achieve satisfactory calibration even for “badly behaved” data sets. This article is thus a complement to Bielecki et al. (2014a Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014a ). Dynamic hedging of portfolio credit risk in a markov copula model . J. Optimiz. Theor. Applic . doi: DOI 10.1007/s10957-013-0318-4 (forthcoming) .[Crossref] , [Google Scholar]), Bielecki et al. (2014b Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014b ). A bottom-up dynamic model of portfolio credit risk - Part I: Markov copula perspective . In: Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.1844574) . [Google Scholar]) and Bielecki et al. (2014c Bielecki , T. R. , Cousin , A. , Crépey , S. , Herbertsson , A. ( 2014c ). A bottom-up dynamic model of portfolio credit risk - Part II: Common-shock interpretation, calibration and hedging issues . Recent Adv. Fin. Eng. 2012 , World Scientific (preprint version available at http://dx.doi.org/10.2139/ssrn.2245130) . [Google Scholar]).     

Multivariate Generalized Distributions of Order k

Abstract

The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1988. Multivariate distributions of order k. Statistics and Probability Letters, 7(3): 207–216.  [Google Scholar]). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1990. Multivariate distributions of order k, part II. Statistics and Probability Letters, 10(1): 29–35.  [Google Scholar]). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996 Sen, K. and Jain, R. 1996. “Cluster generalized negative binomial distribution”. In Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday Edited by: Borthakur, A. C. 227–241. New Delhi: New Age International Publishers.  [Google Scholar]). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997 Sen, K. and Jain, R. 1997. A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics—Theory and Methods, 26: 871–884. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.     

Weighting variables in Kohonen competitive learning algorithms

This paper presents a new variable weight method, called the singular value decomposition (SVD) approach, for Kohonen competitive learning (KCL) algorithms based on the concept of Varshavsky et al. [18 R. Varshavsky, A. Gottlieb, M. Linial, and D. Horn, Novel unsupervised feature filtering of bilogical data, Bioinformatics 22 (2006), pp. 507–513.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Integrating the weighted fuzzy c-means (FCM) algorithm with KCL, in this paper, we propose a weighted fuzzy KCL (WFKCL) algorithm. The goal of the proposed WFKCL algorithm is to reduce the clustering error rate when data contain some noise variables. Compared with the k-means, FCM and KCL with existing variable-weight methods, the proposed WFKCL algorithm with the proposed SVD's weight method provides a better clustering performance based on the error rate criterion. Furthermore, the complexity of the proposed SVD's approach is less than Pal et al. [17 S.K. Pal, R.K. De, and J. Basak, Unsupervised feature evaluation: a neuro-fuzzy approach, IEEE. Trans. Neural Netw. 11 (2000), pp. 366–376.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]], Wang et al. [19 X.Z. Wang, Y.D. Wang, and L.J. Wang, Improving fuzzy c-means clustering based on feature-weight learning, Pattern Recognit. Lett. 25 (2004), pp. 1123–1132.[Crossref], [Web of Science ®] , [Google Scholar]] and Hung et al. [9 W. -L. Hung, M. -S. Yang, and D. -H. Chen, Bootstrapping approach to feature-weight selection in fuzzy c-means algorithms with an application in color image segmentation, Pattern Recognit. Lett. 29 (2008), pp. 1317–1325.[Crossref], [Web of Science ®] , [Google Scholar]].     

Adaptive tangent distance classifier on recognition of handwritten digits

Simard et al. [16 Simard, P. Y., LeCun, Y., Denker, J. S. and Victorri, B. 2000. Transformation invariance in pattern recognition: Tangent distance and tangent propagation. J. Imaging Syst. Technol., 11: 181–197.  [Google Scholar] 17 Sona, D., Sperduti, A. and Starita, A. 1997. A constructive learning algorithm for discriminant tangent models. Advances in Neural Information Processing Systems. 1997, Cambridge, MA. Edited by: Mozer, M. C., Jordan, M. I. and Petsche, T. Vol. 9, pp.786–792. MIT Press.  [Google Scholar]] proposed a transformation distance called “tangent distance” (TD) which can make pattern recognition be efficient. The key idea is to construct a distance measure which is invariant with respect to some chosen transformations. In this research, we provide a method using adaptive TD based on an idea inspired by “discriminant adaptive nearest neighbor” [7 Hastie, T., Tibshirani, R. and Friedman, J. 2009. The Elements of Statistical Learning, Data Mining, Inference, and Prediction, 2, New York, Berlin, Heidelberg: Springer. Available at http://www-stat.stanford.edu/ElemStatLearn [Google Scholar]]. This method is relatively easy compared with many other complicated ones. A real handwritten recognition data set is used to illustrate our new method. Our results demonstrate that the proposed method gives lower classification error rates than those by standard implementation of neural networks and support vector machines and is as good as several other complicated approaches.     

Estimating Incidence Rate on Current Status Data with Application to a Phase IV Cancer Trial

New drug discovery in the pediatrics has dramatically improved survival, but with long- term adverse events. This motivates the examination of adverse outcomes such as long-term toxicity in a phase IV trial. An ideal approach to monitor long-term toxicity is to systematically follow the survivors, which is generally not feasible. Instead, cross-sectional surveys are conducted in Hudson et al. (2007 Hudson , M. M. , Rai , S. N. , Nunez , C. , Merchant , T. E. , Marina , N. M. , Zalamea , N. , Cox , C. , Phipps , S. , Pompeu , R. , Rosenthal , D. ( 2007 ). Noninvasive evaluation of late anthracycline cardiac toxicity in childhood cancer survivors . J. Clin. Oncol. 25 : 3635 – 3643 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), with one of the objectives to estimate the cumulative incidence rates along with specific interest in fixed-term (5 or 10 year) rates. We present inference procedures based on current status data to our motivating example with very interesting findings.     

Some Properties of Semi-Folding Designs

Suppose that some information is available for the particular factor. The experimenter may apply the technique of foldover to isolate the factor and the two-factor interactions involving this factor. In fact, under some situations this can be done by the method of semi-folding. We will discuss this property in detail in this article. Furthermore, we use the computer to search the corresponding optimal semi-folding design for the given 2 ^k?p designs that are tabulated in Chen et al. (1993 Chen , J. , Sun , D. X. , Wu , C. F. J. ( 1993 ). A catalogue of two-level and three-level fractional factorial designs with small runs . International Statistical Review 61 : 131 – 145 .[Crossref], [Web of Science ®] , [Google Scholar]) research.     

Some asymptotic results for the integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975 Komlós, J., Major, P., Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32:111–131.[Crossref], [Web of Science ®] , [Google Scholar])'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial-sum process representation of the integrated empirical process.     

Bayesian estimation of sensitivity level and population proportion of a sensitive characteristic in a binary optional unrelated question RRT model

Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]) proposed an unrelated question binary optional randomized response technique (RRT) model for estimating the proportion of population that possess a sensitive characteristic and the sensitivity level of the question. In our work, decision theoretic approach has been followed to obtain Bayes estimates of the two parameters along with their corresponding minimal Bayes posterior expected losses (BPEL) using beta prior and squared error loss function (SELF). Relative losses are also examined to compare the performances of the Bayes estimates with those of the classical estimates obtained by Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]). The results obtained are illustrated with the help of real survey data using non informative prior.     

Local labour markets delineation: an approach based on evolutionary algorithms and classification methods

In this paper a methodology for the delineation of local labour markets (LLMs) using evolutionary algorithms is proposed. This procedure, based on that in Flórez-Revuelta et al. [13 F. Flórez-Revuelta, J.M. Casado-Díaz, and L. Martínez-Bernabeu, An evolutionary approach to the delineation of functional areas base on travel-to-work flows, Int. J. Autom. Comput. 5(1) (2008), pp. 10–21. doi: 10.1007/s11633-008-0010-6[Crossref] , [Google Scholar],14 F. Flórez-Revuelta, J.M. Casado-Díaz, L. Martínez-Bernabeu, and R. Gómez-Hernández, A memetic algorithm for the delineation of local labour markets, in Parallel Problem Solving from Nature X, Vol. 5199, Lecture Notes in Computer Science, G. Rudolph, T.H. Jansen, S.M. Lucas, C. Poloni, and N. Beume, eds., Springer, Berlin, 2008, pp. 1011–1020. [Google Scholar]], introduces three modifications. First, initial groups of municipalities with a minimum size requirement are built using the travel time between them. Second, a not fully random initiation algorithm is proposed. And third, as a final stage of the procedure, a contiguity step is implemented. These modifications significantly decrease the computational times of the algorithm (up to a 99%) without any deterioration of the quality of the solutions. The optimization algorithm may give a set of potential solutions with very similar values with respect to the objective function what would lead to different partitions, both in terms of number of markets and their composition. In order to capture their common aspects an algorithm based on a cluster partitioning of k-means type is presented. This stage of the procedure also provides a ranking of LLMs foci useful for planners and administrations in decision-making processes on issues related to labour activities. Finally, to evaluate the performance of the algorithm a toy example with artificial data is analysed. The full methodology is illustrated through a real commuting data set of the region of Aragón (Spain).     

Linear Subspace Spanned by Principal Points of a Mixture of Spherically Symmetric Distributions

For each positive integer k, a set of k-principal points of a distribution is the set of k points that optimally represent the distribution in terms of mean squared distance. However, explicit form of k-principal points is often difficult to obtain. Hence a theorem established by Tarpey et al. (1995 Tarpey , T. , Li , L. , Flury , B. D. ( 1995 ). Principal points and self-consistent points of elliptical distributions . Ann. Statist. 23 : 102 – 112 .[Crossref], [Web of Science ®] , [Google Scholar]) has been influential in the literature, which states that when the distribution is elliptically symmetric, any set of k-principal points is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem is called a “principal subspace theorem”. Recently, Yamamoto and Shinozaki (2000b Yamamoto , W. , Shinozaki , N. ( 2000b ). Two principal points for multivariate location mixtures of spherically symmetric distributions . J. Japan Statist. Soc. 30 : 53 – 63 .[Crossref] , [Google Scholar]) derived a principal subspace theorem for 2-principal points of a location mixture of spherically symmetric distributions. In their article, the ratio of mixture was set to be equal. This article derives a further result by considering a location mixture with unequal mixture ratio.     

A stratified unrelated question randomized response model using Neyman allocation

This paper applies stratified random sampling using Neyman allocation to Mangat et al. (1992 Mangat, N.S., Singh, R., Singh, S. (1992). An improved unrelated question randomized response strategy. Cal. Stat. Assoc. Bull. 42:277–281.[Crossref] , [Google Scholar]) unrelated question randomized response (RR) strategy for both completely truthful reporting and less than completely truthful reporting. It is shown that, for the prior information given, our new model is more efficient in terms of variance (in the case of completely truthful reporting) and mean square error (in terms of less than completely truthful reporting) than Kim and Elam's (2007 Kim, J.M., Elam, M.E. (2007). A stratified unrelated question randomized response model. Stat. Papers 48:215–233.[Crossref], [Web of Science ®] , [Google Scholar]) model. Numerical illustrations and graphs are also given in support of the present study.     

Permutation approach for ordinal preference data

Best et al. (Best, D. J., Rayner, J. C. W., O'Sullivan, M. G. (2000 Best, D. J., Rayner, J. C. W., O'Sullivan, M. G. (2000). Product maps for consumer categorical data. Food Quality and Preference 11:91–97.[Crossref], [Web of Science ®] , [Google Scholar]). Product maps for consumer categorical data. Food Quality and Preference, 11:91–97) suggested tests based on partitioning the X² statistic into relevant components of location, dispersion, and skewness effects for testing equality of each effect for ordinal preference data. It is known that the chi-square approximation requires large counts for categories. For this purpose, in this study, we investigate a permutation approach for these statistics and compare the performance of these tests with simulation study. In addition, the permutation approach can be used to produce a product map that classifies the products. We illustrate the approach with a real data example.     

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.