## A Note on Multivariate Logistic Models for Contingency Tables (1997)

Venue: | Austral. J. Statist |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Kauermann97anote,

author = {Göran Kauermann},

title = {A Note on Multivariate Logistic Models for Contingency Tables},

journal = {Austral. J. Statist},

year = {1997},

volume = {39},

pages = {261--276}

}

### OpenURL

### Abstract

Log-linear models are a widely accepted tool for modeling discrete data given in a contingency table. Although their parameters reflect the interaction structure in the joint distribution of all variables, they do not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters recently introduced by Glonek & McCullagh (1995). They have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. The link between the cell probabilities and the multivariate logistic parameters is given in Glonek & McCullagh in an algebraic fashion. In this paper we focus on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff-Nielsen, 1978) is used. This also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. Further emphasis is paid to the analysis of independence hypotheses in margins of a contingency table.

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Citation Context ...2 = ; and I 1 [ I 2 = I. Moreover, special marginal independencies which can be formulated by a recursive factorization of the distribution can also be modeled by a type of a log-linear approach (see =-=Wermuth & Lauritzen, 1983-=-). ffl The multivariate logistic parameter shows marginal but not conditional independence relations. This is in contrast to the log-linear parameter (see for instance Darroch et. al., 1980). However,... |

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Citation Context ...ndence is a type of marginal independence sincesj P(J ) is calculated from the marginal table of YJ only. Moreover the pattern (16) can be interpreted in terms of asymptotic separable hypotheses (see =-=Aitchinson, 1962-=-, Lang & Agresti, 1994). 4 Example Table 1 and 2 about here The example is originally from Solomon (1961) and given in McCullagh & Nelder (1989, p. 239). The data given in Table 1 show the response (a... |

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Citation Context ...linear approach. The disadvantage of alternative parameterizations of log-linear models is that these models typically lead to complicated restricted parameter spaces (see Cox, 1972; Darroch, 1974 or =-=Darroch & Speed, 1983-=-). In particular, the modeling of marginal distributions with arbitrary multivariate logistic parameters does not generally provide a joint distribution for all variables. A familiar counter-example i... |

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