We investigate the formal connections between “quantitative predicate abstractions” for stability analysis of hybrid systems and “continuous simulation relations”. It has been shown recently that stability is not bisimulation invariant, and hence, stronger notions which extend the classical simulation and bisimulation relations with continuity constraints have been proposed, which force preservation of stability. In another direction, a quantitative version of classical predicate abstraction has been proposed for approximation based stability analysis of certain classes of hybrid systems. In this paper, first, we present a general framework for quantitative predicate abstraction for stability analysis. We then show that this technique can be interpreted as constructing a one dimensional system which continuously simulates the original system. This induces an ordering on the class of abstract systems and hence, formalizes the notion of refinement.