  Given two metrics of positive scalar curvature metrics on a closed spin
manifold, there is a secondary index invariant in real $K$-theory. There exist
two definitions of this invariant, one of homotopical flavour, the other one
defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and
detailed proof of the folklore result that both constructions yield the same
answer. Moreover, we generalize this to the case of two families of positive
scalar curvature metrics, parametrized by a compact space. In essence, we prove
a generalization of the classical "spectral-flow-index theorem" to the case of
families of real operators.
