## Three Centuries of Categorical Data Analysis: Log-linear Models and Maximum Likelihood Estimation

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Citations: | 7 - 3 self |

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@MISC{Fienberg_threecenturies,

author = {Stephen E. Fienberg and Alessandro Rinaldo},

title = {Three Centuries of Categorical Data Analysis: Log-linear Models and Maximum Likelihood Estimation},

year = {}

}

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### Abstract

The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but it extends back at least into the 19th century. Moreover it remains an active area of research today. In this paper we give an overview of this history focussing on the development of log-linear models and their estimation via the method of maximum likelihood. S. N. Roy played a crucial role in this development with two papers co-authored with his students S. K. Mitra and Marvin Kastenbaum, at roughly the mid-point temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for loglinear models. Understanding the problem of non-existence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem. 1

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Citation Context ...ssessment of fit, model selection and interpretation. The existence of the MLE is essential for the usual derivation of large-sample chi-square approximations to numerous measures of goodness of fit (=-=Bishop et al., 1975-=-; Agresti, 2002; Cressie and Read, 1988) which are utilized to perform hypothesis tests and, most importantly, are an integral part of model selection. If the distribution of the statistic measuring t... |

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(Show Context)
Citation Context ...el selection and interpretation. The existence of the MLE is essential for the usual derivation of large-sample chi-square approximations to numerous measures of goodness of fit (Bishop et al., 1975; =-=Agresti, 2002-=-; Cressie and Read, 1988) which are utilized to perform hypothesis tests and, most importantly, are an integral part of model selection. If the distribution of the statistic measuring the goodness of ... |

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Citation Context ...tence of the MLE” to signify lack of solutions for the maximum likelihood optimization problem, in accordance with a terminology long established in the log-linear model literature (see, for example, =-=Birch, 1963-=-; Fienberg and Gilbert, 1970; Haberman, 1974). Alternatively, we can say that the MLE of the cell mean vector does not exist whenever there is no strictly positive solution to the MLE defining equatio... |

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7 |
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(Show Context)
Citation Context ...e added and subtracted to the table cells in an appropriate order. Norton (1945) extended Bartlett’s results to the case of 2 × 2 × t tables. Figure 1: Bartlett’s representation of a 2 × 2 × 2 table (=-=Bartlett, 1935-=-, page 248). Deming and Stephan (1940) proposed the method of iterative proportional fitting (IPF) for estimating the cell values in a contingency table subject to constraints coming from “known” marg... |

7 |
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(Show Context)
Citation Context ...Kastenbaum. One of his other Ph.D. students Vasant P. Bhapkar was to follow up on these ideas in a series of papers (e.g., see Bhapkar, 1961, 1966) and also in collaboration with Gary Koch (e.g., see =-=Bhapkar and Koch, 1968-=-). This work led to the paper by Grizzle et al. (1969) and a number of subsequent contributions by Koch and his students and colleagues. 2.3 The Emergence of Log-Linear Models and Methods The 1960s sa... |