# Grouped Dirichlet distribution

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 $\mathcal{T}_n=\left\{\left(x_1,\ldots x_n\right)\left|x_i\geq 0, i=1,\cdots,n, \sum_{i=1}^n x_n =1\right.\right\}$ and $\mathbf{x}\in\mathcal{T}_n$. Writing $\mathbf{x}_{-n}=\left(x_1,\ldots,x_{n-1}\right)$ for the first $n-1$ elements of a member of $\mathcal{T}_n$, the distribution of $\mathbf{x}$ for two partitions has a density function given by

$\operatorname{GD}_{n,2,s}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a},\mathbf{b}\right)= \frac{ \left(\prod_{i=1} ^n x_i^{a_i-1}\right)\cdot \left(\prod_{i=1} ^s x_i \right)^{b_1}\cdot \left(\prod_{i=s+1}^n x_i \right)^{b_2} }{ B\left(a_1,\ldots,a_s\right)\cdot B\left(a_{s+1},\ldots,a_n\right)\cdot B\left(b_1+\sum_{i=1}^sa_i,b_2+\sum_{i=s+1}^n a_i\right) }$

where $B\left(\mathbf{a}\right)$ is the multivariate beta function.

Ng et al[1] went on to define an m partition grouped Dirichlet distribution with density of $\mathbf{x}_{-n}$ given by

$\operatorname{GD}_{n,m,\mathbf{s}}\left(\left.\mathbf{x}_{-n}\right|\mathbf{a},\mathbf{b}\right) = c_m^{-1}\cdot \left(\prod_{i=1}^n x_i^{a_i-1}\right)\cdot \prod_{j=1}^m\left(\sum_{k=s_{j-1}+1}^{s_j}x_k\right)^{b_j}$

where $\mathbf{s} = \left(s_1,\ldots,s_m\right)$ is a vector of integers with $0=s_0. The normalizing constant given by

$c_m=\left\{\prod_{j=1}^mB\left(a_{s_{j-1}+1},\ldots,a_{s_j}\right)\right\}\cdot B\left(b_1+\sum_{k=1}^{s_1},\ldots,b_m+\sum_{k=s_{m-1}+1}^{s_m}a_k\right)$

The authors went on to use these distributions in the context of three different applications in medical science.

## References

1. ^ a b c Ng, Kai Wang (2008). "Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis". Journal of Multivariate Analysis 99: 490–509.