The Statistical Analysis of Categorical Data by Professor Dr. Erling B. Andersen (auth.)

By Professor Dr. Erling B. Andersen (auth.)

This booklet is set the research of express facts with particular emphasis on purposes in economics, political technological know-how and the social sciences. The booklet provides a quick theoretical creation to log-linear modeling of express info, then offers an up to date account of versions and strategies for the statistical research of specific information, together with contemporary advancements in logistic regression versions, correspondence research and latent constitution research. additionally taken care of are the RC organization versions delivered to prominence lately by way of Leo Goodman. New statistical good points just like the use of organization graphs, residuals and regression diagnostics are rigorously defined, and the idea and techniques are broadly illustrated by way of real-life facts. The ebook introduces readers to the most recent advancements in express facts research, and are proven how genuine existence facts should be analysed, how conclusions are drawn and the way versions are converted.

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1 1 1 ß +ß z. e 0 . 1 I ß +ß z. 1/(1+e 0 I I). 1 These equations do not have explicit solutions and must be solved by numerical methods. Hence it is of interest to determine for which observed values of t l and t 2 there are solutions. 6 the likelihood equations have a unique set of solu- 50 tions, if (tl't 2) is an interior point of the convex extension of the support. In this case it is easy to derive the support. If, namely, t1=i, it follows that exactly i of the binary variables have the value 1.

Hence if z(1)Sz(2)S ... Sz(n) are the z's in order of magnitude, the minimum and maximum values of t 2 are It is further easy to see that the set obtained by connecting the 2n points {O,O}, and {i, z(n_i+l)+ ... +z(n)}' i=1, ... ,n-1 is a convex set. Hence the likelihood equations have solutions if (t 1,t 2) does not coincide i=1, ... ,10 the convex with any of the 2n boundary points. =i, 1 extension of the support is shown in fig. 2. 30 20 10 0~~~~ o __________~______________- - T 5 10 Fig. 2.

M}, and nK(T1, ... Jl J m llij ( °l,.. ·,Ok)· A discussion of estimation problems in exponential families with non-identically distributed random variables was given by Nordberg (1980). 2. 7) = tT + h1(t) - nK( T). p( 0) is the canonical parameter and t=t(xl""'xn) the sufficient statistic for T. 7) is equivalent to E[TI Tl = t, and to nK'( T) = t. ln{Eexp(tT+h1(t))}. 7) dlnHt Ir) = t - nK'( r), the theorem follows. D. In case the ML-estimator is a solution to the likelihood equation, it can thus be found by simply equating the observed value of the sufficient statistic and its mean value.

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