Xdc.sevsample {HMP}

Likelihood-Ratio-Test Statistics: Several Sample Dirichlet-Multinomial Test Comparison

Description

This routine provides the value of the Likelihood-Ratio-Test Statistics and the corresponding p-value for evaluating the several sample Dirichlet-Multinomial parameter test comparison.

Usage

Xdc.sevsample(group.data, epsilon = 10^(-4), est = "mom")

Arguments

group.data

A list where each element is a matrix of taxonomic counts(columns) for each sample(rows). (See Notes 1 and 2 in details)

epsilon

Convergence tolerance. To terminate, the difference between two succeeding log-likelihoods must be smaller than epsilon. Default value is 10^(-4).

est

The type of parameter estimator to be used with the Likelihood-ratio-test statistics, 'mle' or 'mom'. Default value is 'mom'. (See Note 3 in details)

Details

To assess whether the Dirichlet parameter vector, \mathbf{\alpha}_{\mathrm{m}}=\mathbf{\pi}_{\mathrm{m}} \frac{1-\theta_{\mathrm{m}}}{\theta_{\mathrm{m}}}(a function of the RAD probability-mean vector and overdispersion), observed in J groups of microbiome samples are equal to each other, the following hypothesis \mathrm{H}_{\mathrm{o}}: \mathbf{\alpha}_{\mathrm{1}} = \cdots =\mathbf{\alpha}_{\mathrm{m}}=\cdots= \mathbf{\alpha}_{\mathrm{J}}=\mathbf{\alpha}_{\mathrm{o}} versus \mathrm{H}_{\mathrm{a}}: \mathbf{\alpha}_{\mathrm{m}} \ne \mathbf{\alpha}_{\mathrm{o}}, m=1, \ldots, J can be tested. The null hypothesis implies that the HMP samples across groups have the same mean and overdispersion, indicating that the RAD models are identical. In particular, the likelihood-ratio test statistic is used, which is given by,

x_{\mathrm{dc}}=-2 \log\left\{\frac{L\left(\mathbf{\alpha}_{\mathrm{o}}; \mathbf{X}_{\mathrm{1}},\ldots, \mathbf{X}_{\mathrm{J}} \right)}{L\left(\mathbf{\alpha}_{\mathrm{1}},\ldots,\mathbf{\alpha}_{\mathrm{J}}; \mathbf{X}_{\mathrm{1}},\ldots, \mathbf{X}_{\mathrm{J}} \right)}\right\}.

The asymptotic null distribution of x_{\mathrm{dc}} follows a Chi-square with degrees of freedom equal to (J-1)*K, where K is the number of taxa (Wilks, 1938).

1. Note 1: The matrices in group.data must contain the same taxa, in the same order.

2. Note 2: Each taxa should be present in at least 1 sample, a column with all 0's may result in errors and/or invalid results.

3. Note 3: 'mle' will take significantly longer time and may not be optimal for small sample sizes; 'mom' will provide more conservative results in such a case.

Value

A list containing the Xdc statistics and p-value.

References

Wilks, S. S. (1938). The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. The Annals of Mathematical Statistics 9, 60-62.

Examples

	data(saliva) 
	data(throat)
	
	### Combine the data sets into a single list
	group.data <- list(saliva, throat)
	
	xdc <- Xdc.sevsample(group.data)
	xdc