Grouped Dirichlet distribution

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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 θ12
Cases θ3 θ4 θ34

The GDD allows the full estimation of the cell probabilities under such aggregation conditions.[1]

Probability Distribution[edit]

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<s_1\leqslant\cdots\leqslant s_m=n. 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[edit]

  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.