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Table 3 Overview of the different models tested

From: Two approaches to account for genotype-by-environment interactions for production traits and age at first calving in South African Holstein cattle

Analysis by region

Univariate

\({\mathrm{y}}_{\mathrm{i}}=\sum f+{\mathrm{a}}_{\mathrm{i}}+{\upvarepsilon }_{\mathrm{i}}\)

Multivariate

\({\mathrm{y}}_{\mathrm{im}}=\sum f+{\mathrm{a}}_{\mathrm{im}}+{\upvarepsilon }_{\mathrm{im}}\)

Analysis including climatic variables

Univariate with heterogeneous residual

\({\mathrm{y}}_{\mathrm{im}}=\sum f+{\mathrm{a}}_{\mathrm{i}}+{\upvarepsilon }_{\mathrm{im}}\)

Reaction norm on 1 variable

\({\mathrm{y}}_{\mathrm{ir}}=\sum f+{\mathrm{a}}_{\mathrm{i}}+{\uplambda }_{\mathrm{j}}{\mathrm{b}}_{\mathrm{ij}}+{\upvarepsilon }_{\mathrm{im}}\)

Reaction norm on > 1 variables

\({\mathrm{y}}_{\mathrm{ir}}=\sum f+{\mathrm{a}}_{\mathrm{i}}+\sum {\uplambda }_{\mathrm{j}}{\mathrm{b}}_{\mathrm{ij}}+{\upvarepsilon }_{\mathrm{im}}\)

  1. \({\mathrm{y}}_{\mathrm{im}}\) performance of animal \(\mathrm{i}\) in region \(\mathrm{m}\); \(\sum f\) sum of fixed effects (herd-year, calving season and age at first calving for production traits); \({\mathrm{a}}_{\mathrm{im}}\) additive genetic value for animal \(\mathrm{i}\) in region \(\mathrm{m}\); \({\uplambda }_{\mathrm{j}}\) reaction norm coefficient for climatic variable \(\mathrm{j}\); \({\mathrm{b}}_{\mathrm{ij}}\) standardized value of climatic variable \(\mathrm{j}\) for animal \(\mathrm{i}\); \({\upvarepsilon }_{\mathrm{i}}\) (\({\upvarepsilon }_{\mathrm{im}}\)): residual for animal \(\mathrm{i}\) (in region \(\mathrm{m}\))