Boundary groupoids can be used to model many analysis problems on singular spaces. In this talk, we first show that the $\eta$-term vanishes for elliptic differential operators on renormalizable boundary groupoids,which is based on the method of renormalized trace similar to that of Moroianu and Nistor. Next we introduce the notion of deformation from the pair groupoid. Under the assumption that a deformation from the pair groupoid of M exists for Lie groupoid G with the unit space M,we construct explicitly a deformation index map relating the analytic index on G and the index on the pair groupoid of M. We apply this map to boundary groupoids with two orbits to obtain index formulae for (fully) elliptic (pseudo)-differential operators with the aid of the index formula by M. J. Pflaum, H. Posthuma, and X. Tang. It is joint work with Bing Kwan So (Jilin University).