One important objective in measuring efficiency is to find the factors that cause inefficiencies so that its performance can be improved. The conventional data envelopment analysis approach is able to decompose the overall efficiency of a system into the product of the technical and scale efficiencies when the internal structure is ignored. For two-stage systems, where the inputs are supplied to the first process to produce intermediate products for the second process to produce the final outputs, the system efficiency can be decomposed into process efficiencies. This paper further decomposes each process efficiency into the product of the technical and scale efficiencies via an input-oriented model for the first process and an output-oriented one for the second. The decomposition also reveals that the overall efficiency of the two-stage system, when the operations of the two processes are considered, is still the product of the technical and scale efficiencies. The concept is illustrated using an example of 24 non-life insurance companies in Taiwan.