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A key problem in making precise perturbative QCD predictions is to set theproper renormalization scale of the running coupling. The extendedrenormalization group equations, which express the invariance of physicalobservables under both the renormalization scale- and scheme-parametertransformations, provide a convenient way for estimating the scale- andscheme-dependence of the physical process. In this paper, we present a solutionfor the scale-equation of the extended renormalization group equations at thefour-loop level. Using the principle of maximum conformality (PMC) /Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all non-conformal$\{\beta_i\}$ terms in the perturbative expansion series can be summed into therunning coupling, and the resulting scale-fixed predictions are independent ofthe renormalization scheme. Different schemes lead to different effectivePMC/BLM scales, but the final results are scheme independent. Conversely, fromthe requirement of scheme independence, one not only can obtainscheme-independent commensurate scale relations among different observables,but also determine the scale displacements among the PMC/BLM scales which arederived under different schemes. In principle, the PMC/BLM scales can be fixedorder-by-order, and as a useful reference, we present a systematic andscheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicitapplication for determining the scale setting of $R_{e^{+}e^-}(Q)$ up to fourloops is presented. By using the world average $\alpha^{\bar{MS}}_s(M_Z)=0.1184 \pm 0.0007$, we obtain the asymptotic scale for the 't Hooft associatedwith the $\bar{MS}$ scheme, $\Lambda^{'tH}_{\bar{MS}}= 245^{+9}_{-10}$ MeV, andthe asymptotic scale for the conventional $\bar{MS}$ scheme,$\Lambda_{\bar{MS}}= 213^{+19}_{-8}$ MeV.

Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops