Oribtal energies for AuH and AgH are calculated by an all-electron relativistic self-consistent-field method using Slater type basis functions. Major relativistic effects for AgH are spin-orbit splittings and those for AuH are large shifts in orbital energies in addition to spin-orbit splittings. Relativistic effects on orbital energies in AgH and AuH imply that changes in correlation energies for relativistic calculations of AuH will be significantly larger than those of AgH, providing partial explanation for the large discrepencies in equilibrium bond length and the dissociation energy between experiments and theoretical estimates for AuH. Large relativistic effects on orbital energies indicate that relativistic contributions should be included for the correct interpretation of ionization potentials for these molecules. Relativistic effects are also evident in dipole moments for these molecules.