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I'm reading "Microbiome, Metagenomics, and High-Dimensional Compositional Data Analysis" by Li (Annual Review of Statistics and Its Application, 2015). There is a part on using Dirichlet-multinomial to model joint-count data that I don't quite follow.

In addition to scalar summary statistics, such as $\alpha$-and $\beta$-diversity, one can directly model observed taxa count data using Dirichlet multinomial regression (La Rosa et al. 2012, Chen & Li 2013b, Holmes et al. 2012). One advantage of this approach is that it automatically accounts for measurement errors and other uncertainties associated with the counts. Suppose we have $p$ bacterial taxa, and their counts $ n = ( {n}_{1}, {n}_{2},..., {n}_{p} ) $ follow a multinomial distribution with probability mass function $$ {f}_{M}( {n}_{1}, \dots, {n}_{p}; \phi) = \binom{ {n}_{+} }{ n } \prod_{j=1}^{p}{ {\phi}_{j}^{ {n}_{j} } }, $$ where $ {n}_{+} = \sum_{ j=1 }^{ p }{ {n}_{j} } $ and $\phi = ({\phi}_{1}, {\phi}_{2}, \dots, {\phi}_{p} )$ are underlying species proportions for which $\sum_{ j=1 }^{ p }{ {\phi}_{j} } = 1$.

Okay, here I don't get what is $n$ in the binomial coefficient. Moving further...

For microbiome composition data, the observed variation is usually larger than what would be predicted by the multinomial model. This increased variation results from the heterogeneity of the microbiome samples and from variation among samples in the underlying proportions. To account for the extra variation or overdispersion, we assume the underlying proportions are themselves positive random variables $ ( {\Phi}_{1}, {\Phi}_{2}, \dots, {\Phi}_{p} ) $, subject to the constraint $ \sum_{ j=1 }^{ p }{ {\Phi}_{j} } = 1$. This assumption implies that the underlying proportions follow a Dirichlet distribution $ Dir( {\gamma}_{1}, {\gamma}_{2}, \dots, {\gamma}_{p} ) $. The counts then marginally follow a Dirichlet-multinomial (DM) distribution, $$ {f}_{DM}( {n}_{1}, \dots, {n}_{p}; \gamma ) = \binom{ {n}_{+} }{ \mathbf{y} } \frac{ \Gamma( {n}_{+} + 1 ) \Gamma( {\gamma}_{+} ) }{ \Gamma( {n}_{+} + {\gamma}_{+} ) } \prod_{ j=1 }^{ p }{ \frac{ \Gamma( {n}_{j} + {\gamma}_{j} ) }{ \Gamma( {\gamma}_{j} ) \Gamma( {n}_{j} + 1 ) } } $$ To relate a $q$-dimensional covariate vector $ \mathbf{Z} = ( {z}_{1}, {z}_{2}, \dots, {z}_{q} ) $ to taxa composition, we assume that the parameters $ {\gamma}_{j} $ in the DM model depend on the covariates via a log-linear model $$ {\gamma}_{j} (\mathbf{Z}) = \exp \left ( {\alpha}_{j} + \sum_{k=1}^{q}{ {\beta}_{jk} {z}_{k} } \right ), $$ where $ {\beta}_{jk} $ measures the effect on the $j$-th taxon of the $k$-th covariate.

Here I don't get what is $ \mathbf{y} $. I suppose $ {\gamma}_{+} $ is the sum over $ \gamma $, though the author doesn't state it explicitly. The author says nothing about $\alpha$ either. In both cases I suppose that the author uses the notation for binomial coefficient to actually represent multidimensional slices of $p-1$-dimensional Pascal simplex. Am I right?

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