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A collaborator of mine is using ComBat for some RNA-seq data. I would like to understand what it's doing, and I have a specific question about the structure of the model. Equation 2.1 of the paper reads

$$Y_{ijg} = \alpha_g + X\beta_g + \gamma_{ig} + \delta_{ig}\epsilon_{ijg}$$

where $i$ is the sample, $j$ is the batch, $g$ is a gene, $Y$ is observed expression, $\alpha$ is an intercept term, $\beta$ is a vector of coefficients, $\gamma$ is an additive batch effect, $\delta$ is a multiplicative batch effect, and $X$ is a design matrix.

Why is $X$ not indexed by $i$? If there are $n$ samples, $m$ genes, and $d$ variables in the analysis, what are the dimensions of $X$ (or $X_i$) and $\beta_g$?

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