Table 1 Notation of parameters and effects

i, j

Focal individual, group mate of focal individual

P _i

Phenotype of individual i

DGE, IGE

Direct genetic effect, indirect genetic effect

T, n_g, n, \(\bar{n}\)

Total number of individuals, number of groups, group size, average group size

\(P_{{{\text{D}}_{i} }} ,P_{{{\text{I}}_{i} }}\)

Direct phenotypic effect of i, indirect effect of i

\(a_{{{\text{D}}_{i} }} ,a_{{{\text{I}}_{i} }}\)

Direct genetic effect of i, indirect genetic effect of i

\(E_{{{\text{D}}_{i} }} ,E_{{{\text{I}}_{i} }}\)

Direct non-genetic effect of i, indirect non-genetic effect of i

\(\sigma_{{a_{\text{D}} }}^{2} ,\sigma_{{a_{\text{I}} }}^{2}\)

Variance of DGE among individuals, variance of IGE among individuals

\(\sigma_{{E_{\text{D}} }}^{2} ,\sigma_{{E_{\text{I}} }}^{2}\)

Direct non-genetic variance, indirect non-genetic variance

\(\sigma_{{a_{\text{DI}} }} ,r_{{a_{\text{DI}} }}\)

Covariance between DGE and IGE, genetic correlation between direct and indirect effects

\(\sigma_{{E_{\text{DI}} }} ,r_{{E_{\text{DI}} }}\)

Non-genetic direct–indirect covariance, non-genetic correlation between direct and indirect effects

\(\sigma_{{P_{\text{D}} }}^{2} ,\sigma_{{P_{\text{I}} }}^{2}\)

Direct phenotypic variance, indirect phenotypic variance

\(\sigma_{\text{TBV}}^{2}\)

Total heritable variance

\(h_{\text{D}}^{2} ,h_{\text{I}}^{2}\)

Direct heritability, indirect heritability

SE

Standard error

d

Dilution

CV

Coefficient of variation