- Probability Distribution
- References
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et al 2008.[1] The Grouped Dirichlet distribution arises in the analysis of categorical data where some observations could fall into any of a set of other 'crisp' category. For example, one may have a data set consisting of cases and controls under two different conditions. With complete data, the cross-classification of disease status forms a 2(case/control)-x-(condition/no-condition) table with cell probabilities
| Treatment | No Treatment | |
| Controls | θ1 | θ2 |
| Cases | θ3 | θ4 |
If, however, the data includes, say, non-respondents which are known to be controls or cases, then the cross-classification of disease status forms a 2-x-3 table. The probability of the last column is the sum of the probabilities of the first two columns in each row, e.g.
| Treatment | No Treatment | Missing | |
| Controls | θ1 | θ2 | θ1+θ2 |
| Cases | θ3 | θ4 | θ3+θ4 |
The GDD allows the full estimation of the cell probabilities under such aggregation conditions.[1]
Probability Distribution
Consider the closed simplex set 
and
. Writing 
for the first 
elements of a member of 
, the distribution of 
for two partitions has a density function given by
where 
is the multivariate beta function.
Ng et al[1] went on to define an m partition grouped Dirichlet distribution with density of 
given by
where 
is a vector of integers with 
. The normalizing constant given by
The authors went on to use these distributions in the context of three different applications in medical science.
References
- Vitamin D Deficiency in Australia
- Vitamin D deficiency in Australia
- Vitamin (disambiguation)
- Vitamin d response element
- Vitamin D (song)
- Vitamin E toxicity
- Vitamin (Jamiroquai song)
- Vitamin k epoxide reductase
- Vitamin K oxide reductase
- Vitamin K shot
- Vitamin L2
- Vitamin O
- Vitamins and Minerals (journal)
- Vitamins & Minerals (journal)
- Vitamin (song)
- Vitamin (South Korean show)
- Vitamin toxicity
- Vitamin (TV series)
- VitaminWater
- Vitamol
- Vitam-R
- Vitam Regum
- Vitanestii
- Vitangelo Bisceglia
- Vitanidis