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)A matrix of taxonomic counts(columns) for each sample(rows).
A list containing the \(C(\alpha)\)-optimal test statistic and p-value.
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,
$$T = \sum_{j=1}^{K} \sum_{i=1}^{P} \frac{1}{\sum_{i=1}^{P} x_{ij}}\left (x_{ij}-\frac{N_{i}\sum_{i=1}^{P} x_{ij}}{N_{\mathrm{g}}} \right )^2$$
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.
Kim, B. S., and Margolin, B. H. (1992). Testing Goodness of Fit of a Multinomial Model Against Overdispersed Alternatives. Biometrics 48, 711-719.
# NOT RUN {
data(saliva)
calpha <- C.alpha.multinomial(saliva)
calpha
# }
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