Table 1.

Details of the mathematical models of ComBat and ComBat-Seq

ComBat model	ComBat-Seq model
$y_{\mathit{gij}}\sim Normal({\mu }_{\mathit{gi}};{\varphi }_{\mathit{gi}})$	$y_{\mathit{gij}}\sim NegativeBinomial({\mu }_{\mathit{gij}};{\varphi }_{\mathit{gi}})$
${\mu }_{\mathit{gi}}={\alpha }_g+X_j{\beta }_g+{\gamma }_{\mathit{gi}}$	$log{\mu }_{\mathit{gij}}={\alpha }_g+X_j{\beta }_g+{\gamma }_{\mathit{gi}}$
$Var(y_{\mathit{gij}})={\varphi }_{\mathit{gi}}$	$Var(y_{\mathit{gij}})={\mu }_{\mathit{gij}}+{\varphi }_{\mathit{gi}}\mu {}_{\mathit{gij}}^2$

$y_{\mathit{gij}}$ denotes the expression value for gene $g$ in sample $j$ from batch $i$, ${\alpha }_g$ denotes a gene-wise baseline expression level, $X_j$ is a vector of covariates for sample $j$, ${\beta }_g$ is the vector of corresponding regression coefficients for gene $g$, ${\gamma }_{\mathit{gi}}$ is the additive (impacting the mean) batch effect and ${\varphi }_{\mathit{gi}}$ is the multiplicative (impacting the variance) batch effect