Table 1 Estimates of variance components and parameters for rank and its variability for the homogeneous (HO) and heterogeneous (HE) models with additional random variables of rider (R), rider by horse (RH), and permanent environment (P)

Genetic parameters for score

Additive

0.090 (0.064, 0.116)

0.096 (0.069, 0.126)

0.082 (0.055, 0.111)

0.089 (0.064, 0.116)

0.096 (0.068, 0.124)

0.081 (0.055, 0.110)

Additional

0.102 (0.073, 0.131)

0.093 (0.065, 0.122)

0.077 (0.052, 0.105)

0.098 (0.072, 0.126)

0.087 (0.059, 0.113)

0.078 (0.052, 0.105)

Residual^a

0.453 (0.426, 0.480)

0.448 (0.420, 0.476)

0.482 (0.455, 0.510)

0.483 (0.401, 0.563)

0.472 (0.397, 0.552)

0.509 (0.430, 0.595)

Phenotypic

0.644 (0.606, 0.683)

0.637 (0.600, 0.673)

0.644 (0.604, 0.679)

0.669 (0.586, 0.754)

0.654 (0.572, 0.734)

0.668 (0.587, 0.759

Heritability

0.139 (0.102, 0.177)

0.151 (0.112, 0.194)

0.128 (0.088, 0.170)

0.133 (0.095, 0.173)

0.146 (0.106, 0.191)

0.122 (0.081, 0.164)

Genetic parameters for variability

Additive

0.117 (0.066, 0.174)

0.130 (0.072, 0.195)

0.121 (0.068, 0.182)

Additional

0.144 (0.077, 0.210)

0.143 (0.071, 0.215)

0.116 (0.065, 0.173)

cov(u,u*)

− 0.016 (− 0.052, 0.023)

− 0.016 (− 0.052, 0.023)

− 0.013 (− 0.053, 0.030)

r(u,u*)

− 0.157 (− 0.505, 0.244)

− 0.116 (− 0.487, 0.258)

− 0.133 (− 0.548, 0.304)

DIC

1023

1053

1121

863

895

1006