# Log Linear Models and Logistic Regression by Ronald Christensen

By Ronald Christensen

This e-book examines statistical versions for frequency info. the first concentration is on log-linear types for contingency tables,but during this moment edition,greater emphasis has been put on logistic regression. themes akin to logistic discrimination and generalized linear types also are explored. The therapy is designed for college students with earlier wisdom of study of variance and regression. It builds upon the relationships among those simple types for non-stop info and the analogous log- linear and logistic regression versions for discrete information. whereas emphasizing similarities among equipment for discrete and non-stop data,this ebook additionally conscientiously examines the variations in version interpretations and evaluate that take place because of the discrete nature of the information. pattern instructions are given for analyses in SAS,BMFP,and GLIM. a variety of information units from fields as varied as engineering, education,sociology,and medication are used to demonstrate strategies and supply exer! cises. This booklet features a variety of cutting edge good points. It starts off with an in depth dialogue of odds and odds ratios in addition to concrete illustrations of the fundamental independence versions for contingency tables. After constructing a legitimate utilized and theoretical foundation for the types considered,the publication provides designated discussions of using graphical types and of types choice strategies. It then explores types with quantitative components and generalized linear models,after which the elemental effects are reexamined utilizing strong matrix equipment. eventually, the booklet offers an intensive therapy of Bayesian strategies for interpreting logistic regression and different regression versions for binomial information.

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28 2. Two-Dimensional Tables and Simple Logistic Regression (0) If H0 is true, then nij and m ˆ ij should be near each other, and the terms (0) (0) (nij − m ˆ ij )2 should be reasonably small. If H0 is not true, then the m ˆ ij ’s, which are estimates based on the assumption that H0 is true, should do a (0) ˆ ij )2 should be larger poor job of predicting the nij ’s. The terms (nij − m when H0 is not true. (0) Note that a prediction m ˆ ij that is, say, three away from the observed value nij , can be either a good prediction or a bad prediction depending (0) on how large the value in the cell should be.

4 The Multinomial Distribution The multinomial distribution is a generalization of the binomial to more than two categories. Suppose we have N independent identical trials. On each trial, we check to see which of q events occurs. In such a situation, we assume that on each trial, one of the q events must occur. Let ni , i = 1, . . , q, be the number of times that the ith event occurs. Let pi be the probability that the ith event occurs on any trial. Note that the pi ’s must satisfy p1 +p2 +· · ·+pq = 1.

J. Proof. a) Equality of probabilities across rows implies that the odds ratios equal one. By substitution, pij pij pij pi j = = 1. pij pi j pij pij b) All odds ratios equal to one implies equality of probabilities across rows. Recall that pi· = 1 for all i = 1, . . , I, so that p·· = I. In addition, pij pi j /pij pi j = 1 implies pij pi j = pij pi j . 3 I × J Tables = = = = 1 I 1 I 1 I 39 J I pij pi j i =1 j =1 J I pij j =1 pi j i =1 J pij p·j j =1 1 p·j I J pij j =1 1 p·j pi· I = p·j /I . = Because this holds for any i and j, p·j /I = p1j = p2j = · · · = pIj for j = 1, .