Table 2. Structure of RRMs used to test for I*E and G*E for Soay sheep traits.

Model	Question	Structure	Description
3.1	NA	Traitit ∼ μ + (Ageit*Sexi + bPDt) + indi + yrt + mm + byi + eit	LMM with heterogeneous residual structure
3.2	Is there I*E?	Traitit ∼ μ + (Ageit*Sexi + bPDt) + indi + indi*PDit + yrt + mm + byi + eit	Random regression model with heterogeneous residual structure
3.3	Is there VA for the trait?	Traitit ∼ μ + (Ageit*Sexi + bPDt) + αi + pei + yrt + mm + byi + eit	Animal model with heterogeneous residual structure
3.4	Is there I*E and VA?	Traitit ∼ μ + (Ageit*Sexi + bPDt) + αi + pei + indi*PDit + yrt + mm + byi + eit	Random regression animal model with individual-by-density
3.5	Is there G*E?	Traitit ∼ μ + (Ageit*Sexi + bPDt) + αi + pei + αi*PDit + pei*PDit + yrt + mm + byi + eit	As 3.4, with additive G*E

All model terms are described previously for models 1.1 or 2.1 except for: in models 3.1–3.5, e_it is the residual variance in the trait estimated in each of the four quartiles of population density; in model 3.2 and model 3.4, ind_i*PD_it is the random regression of a first-order polynomial of the individual effect in the trait ind_i as a function of PD; and in model 3.5, α_i*PD_it is the random regression term for the additive genetic effect and pe_i*PD_it is the random regression term for the permanent environment effect. Note that model 3.3 is the same as model 2.1, except that model 3.3 has a heterogeneous residual structure, with the residual variance varying across four quartiles of population density. NA, not applicable; PD, population density; I*E, individual-by-environment interactions; V_A, additive genetic variance; G*E, genotype-by-environment interactions.