Table 3 Estimates of variances (diagonals), co-variances (above the diagonal), and correlations (below the diagonal) between the random regression coefficients, along with the two eigenvalues (λ) and the relative proportions of variance that they explain (%)

Model 3 using adjusted TD from Model 1

\({\alpha }_{0}\)

1.097 (0.210)

− 0.093 (0.062)

3.584 (0.171)

− 0.349 (0.049)

2.908 (0.194)

− 0.357 (0.064)

1.169

\({\alpha }_{1}\)

− 0.345 (0.211)

0.067 (0.036)

− 0.355 (0.04)

0.270 (0.026)

− 0.228 (0.038)

0.842 (0.043)

λ (%)

1.11 (94.98)

0.06 (5.02)

3.62 (93.93)

0.23 (6.07)

2.97 (79.15)

0.78 (20.85)

Model 3 using adjusted TD from Model 2-M1

\({\alpha }_{0}\)

1.256 (0.227)

− 0.105 (0.063)

4.063 (0.185)

− 0.357 (0.048)

2.854 (0.204)

− 0.331 (0.063)

1.072

\({\alpha }_{1}\)

− 0.357 (0.195)

0.069 (0.033)

− 0.364 (0.043)

0.237 (0.023)

− 0.229 (0.040)

0.734 (0.038)

λ (%)

1.27 (95.50)

0.06 (4.50)

4.10 (95.26)

0.20(4.74)

2.90 (80.95)

0.68 (19.05)

Model 4 using adjusted TD from Model 2-M2

\({\alpha }_{0}\)

0.470 (0.105)

− 0.094 (0.048)

–

–

2.321 (0.102)

− 0.299 (0.048)

1.073

\({\alpha }_{1}\)

− 0.346 (0.158)

0.158 (0.042)

–

–

− 0.210 (0.031)

0.867 (0.043)

λ (%)

0.50 (78.98)

0.13 (21.02)

–

–

2.38 (74.64)

0.81 (25.36)

Model 3: BC*ENV in the genetic analysis

\({\alpha }_{0}\)

1.562 (0.233)

− 0.149 (0.066)

2.875 (0.151)

− 0.303 (0.043)

2.575 (0.200)

− 0.289 (0.063)

1.072

\({\alpha }_{1}\)

− 0.459 (0.184)

0.068 (0.033)

− 0.369 (0.046)

0.235 (0.023)

− 0.211 (0.043)

0.734 (0.038)

λ (%)

1.58 (96.75)

0.05 (3.25)

2.91 (93.54)

0.20 (6.46)

2.62 (79.16)

0.69 (20.84)

Model 3: BC*ENV in the phenotype analysis

\({\alpha }_{0}\)

0.642 (0.170)

− 0.091 (0.057)

1.010 (0.081)

− 0.088 (0.031)

3.034 (0.165)

− 0.293 (0.058)

1.073

\({\alpha }_{1}\)

− 0.388 (0.214)

0.085 (0.034)

− 0.182 (0.062)

0.228 (0.023)

− 0.199 (0.037)

0.719 (0.038)

λ (%)

0.66 (90.30)

0.07 (9.70)

1.02 (82.33)

0.22 (17.67)

3.07 (81.83)

0.68 (18.17)