pyComBat, a Python tool for batch effects correction in high-throughput molecular data using empirical Bayes methods

Table 1 Details of the mathematical models of ComBat and ComBat-Seq

ComBat model	ComBat-Seq model
\({y}_{gij} \sim Normal({\mu }_{gi}; {\phi }_{gi})\)	\({y}_{gij} \sim NegativeBinomial({\mu }_{gij}; {\phi }_{gi})\)
\({\mu }_{gi} = {\alpha }_{g} + {X}_{j} {\beta }_{g} + {\gamma }_{gi}\)	\(log {\mu }_{gij} = {\alpha }_{g} + {X}_{j} {\beta }_{g} + {\gamma }_{gi}\)
\(Var({y}_{gij}) = {\phi }_{gi}\)	\(Var({y}_{gij}) = {\mu }_{gij} + {\phi }_{gi} \mu {}_{gij}^{2}\)

\({y}_{gij}\) 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 \(g\), \({\gamma }_{gi}\) is the additive (impacting the mean) batch effect and \({\phi }_{gi}\) is the multiplicative (impacting the variance) batch effect

ISSN: 1471-2105