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Logistic Normal Distribution

The logistic-normal distribution arises by assuming that the logit (or logistic transformation) of a proportion has a normal distribution, with an obvious extension to a vector of proportions through taking a logistic transformation of a multivariate normal distribution, see Aitchison and Shen (1980). In the univariate case, this provides a family of distributions on (0, 1) that is distinct from the beta distribution , while the multivariate version is an alternative to the Dirichlet distribution. Note that in the multivariate case there is no unique way to define the set of logits for the multinomial proportions (just as in multinomial logit models, see Agresti 2002) and different formulations may be appropriate in particular applications (Aitchison 1982). The univariate distribution has been used, often implicitly, in random effects models for binary data and the multivariate version was pioneered by Aitchison for statistical diagnosis/discrimination (Aitchison