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**Clustering rows and/or columns of a two-way contingency table and a related distribution theory.**
*(English)*
Zbl 1454.62022

Summary: The row-wise multiple comparison procedure proposed in [C. Hirotsu, Biometrika 70, 579–589 (1983; Zbl 0534.62036)] has been verified to be useful for clustering rows and/or columns of a contingency table in several applications. Although the method improved the preceding work there was still a gap between the squared distance between the two clusters of rows and the largest root of a Wishart matrix as a reference statistic for evaluating the significance of the clustering. In this paper we extend the squared distance to a generalized squared distance among any number of rows or clusters of rows and dissolves the loss of power in the process of the clustering procedure. If there is a natural ordering in columns we define an order sensitive squared distance and then the reference distribution becomes that of the largest root of a non-orthogonal Wishart matrix, which is very difficult to handle. We therefore propose a very nice \(\chi ^{2}\)-approximation which improves the usual normal approximation in [T. W. Anderson, An introduction to multivariate statistical analysis. 3rd ed. Hoboken, NJ: Wiley (2003; Zbl 1039.62044)] and also the first \(\chi ^{2}\)-approximation introduced in [C. Hirotsu, Biometrika 78, No. 3, 583–594 (1991; Zbl 0778.62062)]. A two-way table reported by L. Guttman [“Measurement as structural theory”, Psychometrika 36, 329–347 (1971; doi:10.1007/BF02291362)] and analyzed by M. J. Greenacre [J. Classif. 5, No. 1, 39–51 (1988; Zbl 0652.62053)] is reanalyzed and a very nice interpretation of the data has been obtained.

### MSC:

62-08 | Computational methods for problems pertaining to statistics |

62H17 | Contingency tables |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62J15 | Paired and multiple comparisons; multiple testing |

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\textit{C. Hirotsu}, Comput. Stat. Data Anal. 53, No. 12, 4508--4515 (2009; Zbl 1454.62022)

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### References:

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[2] | Anderson, T.W., An introduction to multivariate statistical analysis, (2003), Wiley Intersciences New York · Zbl 1039.62044 |

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[4] | Greenacre, M.J., Clustering the rows and columns of a contingency table, J. classification, 5, 39-51, (1988) · Zbl 0652.62053 |

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