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The Coleman-Weinberg (CW) renormalization scheme for renormalization-group improvement of the effective potential is particularly valuable for CW symmetry-breaking mechanisms (including the challenging case of models with multiple scalar fields). CW mechanism is typically studied using models with classical scale invariance which not only provide a possibility for an alternative symmetry breaking mechanism but also partially address the gauge hierarchies through dimensional transmutation. As outlined in our discussion section, when the couplings are not large, models with CW symmetry-breaking mechanisms have also been shown to naturally provide the strong first-order phase transition necessary for stochastic gravitational wave signals. A full understanding of the CW-MS scheme transformation of couplings thus becomes important in the era of gravitational wave detection and precision coupling measurements. A generalized Coleman-Weinberg (GCW) renormalization scheme is formulated and methods for transforming scalar self-couplings between the GCW and MS (minimal-subtraction) renormalization schemes are developed. Scalar $λΦ^4$ theory with global $O(4)$ symmetry is explicitly studied up to six-loop order to explore the magnitude of this scheme transformation effect on the couplings. The dynamical rescaling of renormalization scales between the GCW and MS schemes can lead to significant (order of 10\%) differences in the coupling at any order, and consequently GCW-MS scheme transformation effects must be considered within precision determinations of scalar couplings in extensions of the Standard Model.