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Shusaku Tsumoto - One of the best experts on this subject based on the ideXlab platform.
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Combinatorics of Information Granule in Contingency Table
International Journal of Intelligent Systems, 2013Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper focuses on the degree of freedom and number of subdeterminants in a Pearson residual in a multiway Contingency Table. The results show that multidimensional residuals are represented as linear sum of determinants of 2 × 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Geometrical interpretation of Pearson residual is investigated. Furthermore, the number of subdeterminants in a residual is equal to the degree of freedom in χ2-test statistic. Since the way of calculation of the number of subdeterminants corresponds to the construction of a statistical model for a Contingency Table, it has been found that the combinatorics of the number subdeterminants is closely related with permutation of attributes in a given Table, where symmetric group may play an important role.
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degree of freedom and numbers of subdeterminants in Contingency Table
Granular Computing, 2012Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper focuses on the degree of freedom and number of subdetermiants in a pearson residual in a multiway Contingency Table. The results show that multidimensional residuals are represented as linear sum of determinants of 2 × 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Furthermore, the number of subderminants in a residual is equal to the degree of freedom.
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GrC - Combinatorics of pearson residuals and degree of freedom in Contingency Tables
2011 IEEE International Conference on Granular Computing, 2011Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper gives further investigation on correlation between pearson residuals and degree of freedom in a Contingency Table. An interesting formula is obtained, where an independent variable in a Contingency Table will be a fundamental granule of degree of freedom. The structure of formula is also closely related with combinatorial natures of given variables.
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Contingency matrix theory statistical dependence in a Contingency Table
Information Sciences, 2009Co-Authors: Shusaku TsumotoAbstract:Chance discovery aims at understanding the meaning of functional dependency from the viewpoint of unexpected relations. One of the most important observations is that such a chance is hidden under a huge number of coocurrencies extracted from a given data. On the other hand, conventional data-mining methods are strongly dependent on frequencies and statistics rather than interestingness or unexpectedness. This paper discusses some limitations of ideas of statistical dependence, especially focusing on the formal characteristics of Simpson's paradox from the viewpoint of linear algebra. Theoretical results show that such a Simpson's paradox can be observed when a given Contingency Table as a matrix is not regular, in other words, the rank of a Contingency matrix is not full. Thus, data-ordered evidence gives some limitations, which should be compensated by human-oriented reasoning.
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statistical independence and determinants in a Contingency Table interpretation of pearson residuals based on linear algebra
Fundamenta Informaticae, 2009Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper analyzes pearson residuals, which is an important element of chi-square test statistic, in a Contingency Table from the viewpoint of matrix theory as follows. First, a given Contingency Table is viewed as a matrix and the residual of each element in a matrix are obtained as the difference bewteen observed values and expected values calculated by marginal distributions. Then, each residual σ$_{ij}$ is decomposed into the linear sum of the 2 × 2 subderminants of a original matrix, except for i-th column and j-th row. Furthermore, the number of the determinants is equal to the degree of freedom for the chi-square test statistic for a given Contingency Table. Thus, 2 × 2 subdeterminants in a Contingencymatrix determine the degree of statistical independence of two attributes as elementary granules.
Shoji Hirano - One of the best experts on this subject based on the ideXlab platform.
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Combinatorics of Information Granule in Contingency Table
International Journal of Intelligent Systems, 2013Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper focuses on the degree of freedom and number of subdeterminants in a Pearson residual in a multiway Contingency Table. The results show that multidimensional residuals are represented as linear sum of determinants of 2 × 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Geometrical interpretation of Pearson residual is investigated. Furthermore, the number of subdeterminants in a residual is equal to the degree of freedom in χ2-test statistic. Since the way of calculation of the number of subdeterminants corresponds to the construction of a statistical model for a Contingency Table, it has been found that the combinatorics of the number subdeterminants is closely related with permutation of attributes in a given Table, where symmetric group may play an important role.
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degree of freedom and numbers of subdeterminants in Contingency Table
Granular Computing, 2012Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper focuses on the degree of freedom and number of subdetermiants in a pearson residual in a multiway Contingency Table. The results show that multidimensional residuals are represented as linear sum of determinants of 2 × 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Furthermore, the number of subderminants in a residual is equal to the degree of freedom.
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GrC - Combinatorics of pearson residuals and degree of freedom in Contingency Tables
2011 IEEE International Conference on Granular Computing, 2011Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper gives further investigation on correlation between pearson residuals and degree of freedom in a Contingency Table. An interesting formula is obtained, where an independent variable in a Contingency Table will be a fundamental granule of degree of freedom. The structure of formula is also closely related with combinatorial natures of given variables.
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statistical independence and determinants in a Contingency Table interpretation of pearson residuals based on linear algebra
Fundamenta Informaticae, 2009Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper analyzes pearson residuals, which is an important element of chi-square test statistic, in a Contingency Table from the viewpoint of matrix theory as follows. First, a given Contingency Table is viewed as a matrix and the residual of each element in a matrix are obtained as the difference bewteen observed values and expected values calculated by marginal distributions. Then, each residual σ$_{ij}$ is decomposed into the linear sum of the 2 × 2 subderminants of a original matrix, except for i-th column and j-th row. Furthermore, the number of the determinants is equal to the degree of freedom for the chi-square test statistic for a given Contingency Table. Thus, 2 × 2 subdeterminants in a Contingencymatrix determine the degree of statistical independence of two attributes as elementary granules.
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Contingency Table and Granularity
NAFIPS 2007 - 2007 Annual Meeting of the North American Fuzzy Information Processing Society, 2007Co-Authors: Shusaku Tsumoto, Shoji HiranoAbstract:This paper gives a matrix-theory based approach to a Contingency Table and shows that sample size gives a strong constraints on its granularity. In the former studies, relations between degree of granularity and dependence of Contingency Tables are given from the viewpoint of determinantal divisors and sample size. The nature of determinantal divisors shows that the increase of the degree of granularity may lead to that of dependence. However, a constraint on the sample size of a Contingency Table is very strong, which leads to the evaluation formula where the increase of degree of granularity gives the decrease of dependency. This paper gives a further study of the nature of sample size effect on the degree of dependency in a Contingency matrix. The results show that sample size will restrict the nature of matrix in a combinatorial way, which suggests that the dependency is closely related with integer programming.
S P Brooks - One of the best experts on this subject based on the ideXlab platform.
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prior induction in log linear models for general Contingency Table analysis
Annals of Statistics, 2001Co-Authors: Ruth King, S P BrooksAbstract:Log-linear modelling plays an important role in many statistical applications, particularly in the analysis of Contingency Table data. With the advent of powerful new computational techniques such as reversible jump MCMC, Bayesian analyses of these models, and in particular model selection and averaging, have become feasible. Coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation. The latter is greatly improved in the case where priors can be specified on interpreTable parameters about which relevant experts can express their beliefs. In this paper, we show how the specification of a general multivariate normal prior on the log-linear parameters induces a multivariate lognormal prior on the corresponding cell counts of a Contingency Table. We derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts. We discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 2 3 Contingency Table. 1. Introduction. The analysis of general k-way Contingency Table data is of interest in a wide variety of areas of statistical application. Model selection is notoriously difficult in suchsituations, since th e number of models rises doubly exponentially withdimension. Obviously, exhaustive comparisons are impossible, but various computational techniques have been proposed for obtaining posterior model probabilities for problems of this sort, depending upon the parameters upon which the analyst wishes to express prior opinions. With the introduction of powerful new computational techniques these forms of analysis have been greatly simplified and are becoming increasingly common in the applied literature. In undertaking a Bayesian analysis of Contingency Table data, it is necessary to specify priors either for the cell counts (which could alternatively be expressed in terms of the cell probabilities and total cell count) or, equivalently, the log-linear parameters. Madigan and York (1997) choose to choose to place hyper-Dirichlet priors [Dawid and Lauritzen (1993)] on the cell probabilities which has the advantage that priors of this form allow a factorization of the likelihood through the identification of cliques within the corresponding model graph. Giudici, Green and Tarantola (1999) illustrate how this decomposition
Mark W Hall - One of the best experts on this subject based on the ideXlab platform.
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a systemic inflammation mortality risk assessment Contingency Table for severe sepsis
Pediatric Critical Care Medicine, 2017Co-Authors: Joseph A Carcillo, Katherine A Sward, Scott E Halstead, Russell Telford, Adria Jimenezbacardi, Bita Shakoory, Dennis W Simon, Mark W HallAbstract:OBJECTIVES We tested the hypothesis that a C-reactive protein and ferritin-based systemic inflammation Contingency Table can track mortality risk in pediatric severe sepsis. DESIGN Prospective cohort study. SETTING Tertiary PICU. PATIENTS Children with 100 separate admission episodes of severe sepsis were enrolled. INTERVENTIONS Blood samples were attained on day 2 of sepsis and bi-weekly for biomarker batch analysis. A 2 × 2 Contingency Table using C-reactive protein and ferritin thresholds was developed. MEASUREMENTS AND MAIN RESULTS A C-reactive protein of 4.08 mg/dL and a ferritin of 1,980 ng/mL were found to be optimal cutoffs for outcome prediction at first sampling (n = 100) using the Youden index. PICU mortality was increased in the "high-risk" C-reactive protein greater than or equal to 4.08 mg/dL and ferritin greater than or equal to 1,980 ng/mL category (6/13 [46.15%]) compared with the "intermediate-risk" C-reactive protein greater than or equal to 4.08 mg/dL and ferritin less than 1,980 ng/mL or C-reactive protein less than 4.08 mg/dL and ferritin greater than or equal to 1,980 ng/mL categories (2/43 [4.65%]), and the "low-risk" C-reactive protein less than 4.08 mg/dL and ferritin less than 1,980 ng/mL category (0/44 [0%]) (odds ratio, 36.43 [95% CI, 6.16-215.21]). The high-risk category was also associated with the development of immunoparalysis (odds ratio, 4.47 [95% CI, 1.34-14.96]) and macrophage activation syndrome (odds ratio, 24.20 [95% CI, 5.50-106.54]). Sixty-three children underwent sequential blood sampling; those who were initially in the low-risk category (n = 24) and those who subsequently migrated (n = 19) to the low-risk category all survived, whereas those who remained in the "at-risk" categories had increased mortality (7/20 [35%]; p < 0.05). CONCLUSIONS A C-reactive protein- and ferritin-based Contingency Table effectively assessed mortality risk. Reduction in systemic inflammation below a combined threshold C-reactive protein of 4.08 mg/dL and ferritin of 1,980 ng/mL appeared to be a desired response in children with severe sepsis.
Jim Albert - One of the best experts on this subject based on the ideXlab platform.
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bayesian testing and estimation of association in a two way Contingency Table
Journal of the American Statistical Association, 1997Co-Authors: Jim AlbertAbstract:Abstract In a two-way Contingency Table, one is interested in checking the goodness of fit of simple models such as independence, quasi-independence, symmetry, and constant association, and estimating parameters that describe the association structure of the Table. In a large Table, one may be interested in detecting a few outlying cells that deviate from the main association pattern in the Table. Bayesian tests of these hypotheses are described using a prior defined on the set of interaction terms of the log-linear model. These tests and associated estimation procedures have several advantages over classical fitting/estimation procedures. First, the tests can give measures of evidence in support of simple hypotheses. Second, the Bayes factors can be used to give estimates of association parameters of the Table that allow for uncertainty that the hypothesized model is true. These methods are illustrated for a number of Tables.