C.alpha.multinomial function

$C(\alpha)$ - Optimal Test for Assessing Multinomial Goodness of Fit Versus Dirichlet-Multinomial Alternative

A function to compute the $C(\alpha)$-optimal test statistics of Kim and Margolin (1992) for evaluating the Goodness-of-Fit of a Multinomial distribution (null hypothesis) versus a Dirichlet-Multinomial distribution (alternative hypothesis).

C.alpha.multinomial(data)

Arguments

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

Returns

A list containing the $C(\alpha)$-optimal test statistic and p-value.

Details

In order to test if a set of ranked-abundance distribution(RAD) from microbiome samples can be modeled better using a multinomial or Dirichlet-Multinomial distribution, we test the hypothesis $\mathrm{H}: \theta = 0$ versus $\mathrm{H}: \theta \ne 0$, where the null hypothesis implies a multinomial distribution and the alternative hypothesis implies a DM distribution. Kim and Margolin (Kim and Margolin, 1992) proposed a $C(\alpha)$-optimal test- statistics given by,

Where $K$ is the number of taxa, $P$ is the number of samples, $x_{ij}$ is the taxon $j$, $j = 1,\ldots,K$ from sample $i$, $i=1,\ldots,P$, $N_{i}$ is the number of reads in sample $i$, and $N_{\mathrm{g}}$ is the total number of reads across samples.

As the number of reads increases, the distribution of the $T$ statistic converges to a Chi-square with degrees of freedom equal to $(P-1)(K-1)$, when the number of sequence reads is the same in all samples. When the number of reads is not the same in all samples, the distribution becomes a weighted Chi-square with a modified degree of freedom (see (Kim and Margolin, 1992) for more details).

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

References

Kim, B. S., and Margolin, B. H. (1992). Testing Goodness of Fit of a Multinomial Model Against Overdispersed Alternatives. Biometrics 48, 711-719.

Examples

data(saliva)
        
        calpha <- C.alpha.multinomial(saliva)
        calpha