Table 2 Summary of linear mixed models and linear regression models frequently used for the estimation of realized rate of genetic gain in plant breeding programs

Era trial

\({y}_{i}=\mu +\beta {r}_{i}+{\varepsilon }_{i}\)

Phenotypic value (y)

Regression coefficient (β)

Year of release (r)

Duvick (2005), Hallauer et al. (2010), Rutkoski (2019b)

EBV^b

\({y}_{ijk}=\mu +{g}_{i}+{d}_{i}+{l}_{k}+{\varepsilon }_{ijk}\)

\({g}_{i}=\mu +{\beta r}_{i}+{\varepsilon }_{i}\)

Phenotypic value (y)

Genotype effect (g)^a

Location (L)

Year (d)

Regression coefficient (β)

Breeding cycle (r)

Garrick (2010), Rutkoski (2019b)

Variety registration trial

\({y}_{ijk}=\mu +{g}_{i}+{\beta r}_{i}+{d}_{j}+{\gamma t}_{i}+{l}_{k}+{h}_{jk}+{x}_{ij}+{z}_{ik}+{\varepsilon }_{ijk}\)

Phenotypic value (y)

Genotype (g)^a

Location (l)^a

Year (d)^a

Regression coefficients for genetic and non-genetic trends (\(\beta et \gamma\))

G x Location (z)^a

G x Year (x)^a

Location x Year (h)^a

Laidig et al. (2014), Piepho et al. (2014), Rutkoski (2019b)