Grouped Dirichlet distribution explained

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.

Probability Distribution

Consider the closed simplex set

l{T} _{n=\left\{\left(x} _{1,\ldots} x _{n\right)\left|x} _{i\geq} 0 , i =1 , … ,n, \sum ^{n} _{i =1} x _{n} =1 \right.\right\} and x \in l{T} _{n} . Writing x _{-n} = \left(x _{1,\ldots,x} _{n-1} \right) for the first n -1 elements of a member of l{T} _{n} , the distribution of x for two partitions has a density function given by\operatorname{GD} _{n,2,s} \left(\left. x _{-n} \right|a,b\right) = (\left(\prod _{i=1} ^{n} x ^{a _{i-1}} _{i} \right) ⋅ \left(\sum _{i=1} ^{s} x _{i} \right) ^{b _{1}} ⋅ \left(\sum _{i=s+1} ^{n} x _{i} \right) ^{b _{2}})/(\operatorname{\Beta) \left(a _{1,\ldots,a} _{s\right) ⋅} \operatorname{\Beta}\left(a _{s+1} ,\ldots,a _{n\right) ⋅} \operatorname{\Beta}\left(b _{1+\sum} ^{sa} ^{i,b} _{2+\sum} ^{n} ^{i = s +1} a _{i\right) }} where \operatorname{\Beta}\left( a \right) is the Multivariate beta function.

Ng et al. went on to define an m partition grouped Dirichlet distribution with density of

x _{-n} given by\operatorname{GD} _{n,m,s} \left(\left. x _{-n} \right| a , b \right) = c ^{-1} _{m} ⋅ \left(\prod ^{n} _{i =1} x ^{a _{i-1}} _{i} \right) ⋅ \prod ^{s _{j}} _{k = s _{j-1} +1} x ^{b _{j}} _{k\right)} where s = \left(s _{1,\ldots,s} _{m\right)} is a vector of integers with 0= s _{0<s} _{1\leqslant … \leqslant} s _{m=n} . The normalizing constant given byc _{m=\left\{\prod} ^{m\operatorname{\Beta}\left(a} ^{s _{j-1} +1} ,\ldots,a _{s _{j}} \right)\right\} ⋅ \operatorname{\Beta}\left(b _{1+\sum} ^{s _{1}} ^{k =1} a _{k,\ldots,b} _{m+\sum} ^{s _{m}} ^{k = s _{m-1} +1} a _{k\right)}

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

Notes and References

1. Ng. Kai Wang. Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis. Journal of Multivariate Analysis. 2008. 99. 490–509.