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 and Begg 1976), the...
References and Further Reading
- Agresti A (2002) Categorical data analysis, 2nd edn. Wiley, New YorkMATHGoogle Scholar
- Aitchison J (1982) The statistical analysis of compositional data (with discussion). J R Stat Soc Ser B 44:139–177MATHMathSciNetGoogle Scholar
- Aitchison J (1986) The statistical analysis of compositional data. Chapman & Hall, LondonMATHGoogle Scholar
- Aitchison J, Begg CB (1976) Statistical diagnosis when basic cases are not classified with certainty. Biometrika 63:1–12MATHMathSciNetGoogle Scholar
- Aitchison J, Shen SM (1980) Logistic-normal distributions: some properties and uses. Biometrika 67:261–272MATHMathSciNetGoogle Scholar
- McCulloch CE, Searle SR (2001) Generalized, linear and mixed models. Wiley, New YorkMATHGoogle Scholar
- Williams D (1982) Extra-binomial variation in logistic linearmodels. Appl Stat 31:144–148MATHGoogle Scholar