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

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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

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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{α}_{\mathrm{m}}=\mathbf{π}_{\mathrm{m}} \frac{1-θ_{\mathrm{m}}}{θ_{\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{α}_{\mathrm{1}} = \cdots =\mathbf{α}_{\mathrm{m}}=\cdots= \mathbf{α}_{\mathrm{J}}=\mathbf{α}_{\mathrm{o}} versus \mathrm{H}_{\mathrm{a}}: \mathbf{α}_{\mathrm{m}} \ne \mathbf{α}_{\mathrm{o}}, m=1, …, 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≤ft\{\frac{L≤ft(\mathbf{α}_{\mathrm{o}}; \mathbf{X}_{\mathrm{1}},…, \mathbf{X}_{\mathrm{J}} \right)}{L≤ft(\mathbf{α}_{\mathrm{1}},…,\mathbf{α}_{\mathrm{J}}; \mathbf{X}_{\mathrm{1}},…, \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.

Author(s)

Patricio S. La Rosa, Elena Deych, Berkley Shands, William D. Shannon

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

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	data(saliva) 
	data(throat)
	
	### A test where we use the default values for initscalar and epsilon
	xdc.sev_check <- Xdc.sevsample(list(saliva, throat))
	xdc.sev_check

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