In this paper, following a very recent and new approach of [1] and [2] we further generalize recently introduced summability methods in [11] and introduce new notions, namely, $\cal I$-statistical convergence of order $\alpha$ and $\cal I$-lacunary statistical convergence of order $\alpha$, where $0<\alpha\leq1$ for sequences of sets. We mainly study their relationship and also make some observations about these classes and in the way try to give a proof of theorem which is not proved in [31]. The study leaves a lot of interesting open problems.