Heavy rare-earth trihydrides such as GdH3, DyH3, and yttrium trihydide (YH3) usually show a hexagonal crystal structure under ambient pressure. This structure is known to transform to a face-centered cubic (FCC) one at higher pressures (at the order of GPa), and the FCC one returns to the hexagonal structure when the applied pressure is released. In this study, we investigated the structure of alkaline or alkaline-earth (A)-substituted REH3 (RE = Y, Gd, Dy; A = Li, K, or Mg) using X-ray diffraction, and measured the phase transition pressure. We found that this FCC high-pressure phase can be stabilized by 10-33 mol % A substitution for RE in the REH3. The mechanism of phase stabilization is simply explained by the ionic radius ratio between the cation and anion (r(cat)/r(ani)), as well as the stabilities of other ionic crystals such as perovskite materials. For all considered REH3 samples, the FCC phase becomes stable when r(cat)/r(ani) > 0.856, such as in the case of 10 mol % Li-substituted YH3.