HMP (version 2.0.1)

DM.MoM: Method-of-Moments (MoM) Estimators of the Dirichlet-Multinomial Parameters

Description

Method-of-Moments (MoM) estimators of the Dirichlet-multinomial parameters: taxa proportions and overdispersion.

Usage

DM.MoM(data)

Arguments

data

A matrix of taxonomic counts(columns) for each sample(rows).

Value

A list providing the MoM estimator for overdispersion, the MoM estimator of the RAD-probability mean vector, and the corresponding loglikelihood value for the given data set and estimated parameters.

Details

Given a set of taxa-count vectors \(\left\{\textbf{x}_{i},\ldots, \textbf{x}_{P} \right\}\), the methods of moments (MoM) estimator of the set of parameters \(\theta\) and \(\left\{\pi_{j} \right\}_{j=1}^K \) is given as follows (Mosimann, 1962; Tvedebrink, 2010): $$\hat{\pi}_{j}=\frac{\sum_{i=1}^P x_{ij}}{\sum_{i=1}^P N_{i}},$$ and $$\hat{\theta} = \sum_{j=1}^K \frac{S_{j}-G_{j}}{\sum_{j=1}^{K}\left ( S_{j}+\left ( N_{c}-1 \right )G_{j} \right )},$$ where \(N_{c}=\left( P -1 \right)^{-1} \left(\sum_{i=1}^P N_{i}-\left (\sum_{i=1}^P N_{i} \right )^{-1} \sum_{i=1}^P N_{i}^2\right)\), and \(S_{j}=\left( P -1 \right)^{-1} \sum_{i=1}^P N_{i} \left ( \hat{\pi}_{ij} -\hat{\pi}_{j} \right )^{2}\), and \(G_{j}=\left( \sum_{i=1}^P \left (N_i-1 \right ) \right)^{-1} \sum_{i=1}^P N_{i} \hat{\pi}_{ij} \left (1- \hat{\pi}_{ij}\right)\) with \(\hat{\pi}_{ij}=\frac{x_{ij}}{N_{i}}\).

References

Mosimann, J. E. (1962). On the compound multinomial distribution, the multivariate \(\beta\)-distribution, and correlations among proportions. Biometrika 49, 65-82. Tvedebrink, T. (2010). Overdispersion in allelic counts and theta-correction in forensic genetics. Theor Popul Biol 78, 200-210.

Examples

# NOT RUN {
	data(throat)
	
	fit.throat <- DM.MoM(throat)
	fit.throat
# }