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 and Begg 1976), the...
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Agresti A (2002) Categorical data analysis, 2nd edn. Wiley, New York
Aitchison J (1982) The statistical analysis of compositional data (with discussion). J R Stat Soc Ser B 44:139–177
Aitchison J (1986) The statistical analysis of compositional data. Chapman & Hall, London
Aitchison J, Begg CB (1976) Statistical diagnosis when basic cases are not classified with certainty. Biometrika 63:1–12
Aitchison J, Shen SM (1980) Logistic-normal distributions: some properties and uses. Biometrika 67:261–272
McCulloch CE, Searle SR (2001) Generalized, linear and mixed models. Wiley, New York
Williams D (1982) Extra-binomial variation in logistic linearmodels. Appl Stat 31:144–148
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Hinde, J. (2011). Logistic Normal Distribution. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_342
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