where b,j ∼ N (0, (σ b ) 2 ) and t,j ∼ N (0, (σ t ) 2 ), (σ b ) 2 = 100 2 ( Rt-Rb ) 2 σ 2 b , (σ t ) 2 = 100 2 ( Rt-Rb ) 2 σ 2 t , and R b and R t are previously defined. Therefore, the models for the normalized control data follow the same model as the unnormalized ones, as long as the parameters are transformed accordingly. Using the same logic as for θc , we can directly conclude that the MLE θ ctb using the normalized data have the same estimates for x 50 and H using the unnormalized data. For R b and R c , using either one of the two approaches, it is always the case that R b = 100 Rb -Rb Rt-Rb and R t = 100 Rt-Rb Rt-Rb .