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
[edit]
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
![{\displaystyle \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(\sum _{i=1}^{s}x_{i}\right)^{b_{1}}\cdot \left(\sum _{i=s+1}^{n}x_{i}\right)^{b_{2}}}{\operatorname {\mathrm {B} } \left(a_{1},\ldots ,a_{s}\right)\cdot \operatorname {\mathrm {B} } \left(a_{s+1},\ldots ,a_{n}\right)\cdot \operatorname {\mathrm {B} } \left(b_{1}+\sum _{i=1}^{s}a_{i},b_{2}+\sum _{i=s+1}^{n}a_{i}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b80e5e57206148b4dcaac8d5f860515445f7fa0)
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
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe680ea9ec66ae537a9a61cbce9ab5771bd4d2bb)
where
is a vector of integers with
. The normalizing constant given by
![{\displaystyle c_{m}=\left\{\prod _{j=1}^{m}\operatorname {\mathrm {B} } \left(a_{s_{j-1}+1},\ldots ,a_{s_{j}}\right)\right\}\cdot \operatorname {\mathrm {B} } \left(b_{1}+\sum _{k=1}^{s_{1}}a_{k},\ldots ,b_{m}+\sum _{k=s_{m-1}+1}^{s_{m}}a_{k}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dec39b54434c8ee9b480232c30445d52d303860)
The authors went on to use these distributions in the context of three different applications in medical science.
- ^ 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.