  Based on the renormalization group summation method of McKeon ${\it et\;
al.}$, it is shown that the renormalization group equation, while related to
the radiatively mass scale $\mu$, would perform a summation over QCD
perturbative terms. Employing the full QCD $\beta$-function within this
summation, all logarithmic corrections can be presented as log-independent
contributions. In another step of this approach, the renormalization scheme
dependence of QCD observables, characterized by Stevenson, is required to be
examined. In this regard, two choices of renormalization scheme would be
exposed. In one of them, the QCD observable is expressed in terms of two powers
of running coupling constant. In the other one, the perturbative series
expansion is written as an infinite series in terms of the two-loop running
coupling represented by the Lambert $W$-function. In both cases, the QCD
observable involves parameters which are renormalization scheme invariant and a
coupling constant which is independent of the renormalization scale. We then
consider the approach of complete renormalization group improvement (CORGI). In
this approach, using the self consistency principle, it is possible to
reconstruct the conventional perturbative series in terms of scheme invariant
quantities and the coupling constant as a function of Lambert $W$-function. One
of the key differences between CORGI and the renormalization group summation
method of McKeon ${\it et\; al.}$ is that the later treats renormalization
scale and scheme independently while the former would encounter both
dependencies together. In continuation, numerical results of the two approaches
are compared for $R_{e^+e^-}$ ratio and the Higgs decay width to gluon-gluon.
