Fig. 1

Experimental design for cross-validation study. Pigs in orange = pigs fed a conventional diet; pigs in green = pigs fed a high-fiber diet. \({\mathbf{y}}\) is the vector of phenotypes for a given trait. \({\mathbf{X}}\) is the incidence matrix relating observations to fixed effects. \({{\varvec{\upbeta}}}\) is the vector of fixed effects described in the next section. \({\mathbf{Z}}\) is the incidence matrix for the genetic effect of the individual, and \({\mathbf{u}}\sim {\text{N}}\left( {{\mathbf{0}},{\mathbf{G}}\sigma_{u}^{2} } \right)\) is the random vector of additive genetic effects for the considered trait, with \(\sigma_{u}^{2}\) is the additive genetic variance. \({\mathbf{W}}\) is the incidence matrix for the microbiota effects, and \({\mathbf{m}}\sim {\text{N}}\left( {{\mathbf{0}},{\mathbf{M}}\sigma_{m}^{2} } \right)\) is the vector of microbiota effects for the considered trait, with \(\sigma_{m}^{2}\) the microbiota variance. Finally, \({\mathbf{e}}\sim {\text{N}}\left( {{\mathbf{0}},{\mathbf{I}}\sigma_{e}^{2} } \right)\) is the vector of residual random effect, with \({\mathbf{I}}\) the identity matrix, and \(\sigma_{e}^{2}\) the residual variance