By Theorem 3, the expectation of \(\sqrt{Q_i+c}\) has approximately the square root of the expectation of \(Q_i\) which establishes the first objective. Furthermore, the variance of all the \(\sqrt{Q_i+c}\)\(\)'s is approximately \(\frac{1}{4}\) establishing the second objective. Therefore, the final goal is to choose the constant \(c\) so that both approximations are reduced. The best candidates for the constant \(c\) are \(c\ =\ \frac{1}{4}\) to minimize the first order bias or \(c\ =\ \frac{3}{8}\) which would minimize the difference in variance between \(g\left(Q_i\right)\)'s. It is at this point that there is a payoff between minimizing the bias and stabilizing the variance. However, Brown \cite{Brown_2009} shows, through visual plots of bias and variance vs. lambda, that minimizing the bias at \(c\ =\ \frac{1}{4}\) outweighs the loss in stability in variance. Therefore, the transformation \(\sqrt{Q_i+\frac{1}{4}}\) is chosen.