Fourier Amplitude Sensitivity Test (FAST) and Random Balance Design (RBD) are popular methods of estimating variance-based sensitivity indices. We revisit them in light of the discrete Fourier transform (DFT) on finite subgroups of the torus and randomized orthogonal array sampling. We then study the estimation error of both these methods. This allows to improve FAST and to derive explicit rates of convergence of its estimators by using the framework of lattice rules. We also give a natural generalization of the classic RBD by using randomized orthogonal arrays having any parameters, and we provide a bias correction method for its estimators.