The purpose of this paper is to give a necessary and sufficient condition on the existence of associated splittings (defined in this paper) and to consider some applications to associated quasi-connections on fibred manifolds and vector bundles, using the idea and extending Theorem 1 from . In Section 1, a general condition on the existence of associated splittings is given. In Section 2, the basic constructions concerning q.c.s. used in the next Section are briefly described following ; they extend the q.c.s. of Wang [8, 1, 2]. In Section 3 there are proved two theorems on associated q.c.s. using essentially the main theorem from Section 1.