  This work focuses on the question of how identifiability of a mathematical
model, that is, whether parameters can be recovered from data, is related to
identifiability of its submodels. We look specifically at linear compartmental
models and investigate when identifiability is preserved after adding or
removing model components. In particular, we examine whether identifiability is
preserved when an input, output, edge, or leak is added or deleted. Our
approach, via differential algebra, is to analyze specific input-output
equations of a model and the Jacobian of the associated coefficient map. We
clarify a prior determinantal formula for these equations, and then use it to
prove that, under some hypotheses, a model's input-output equations can be
understood in terms of certain submodels we call "output-reachable". Our proofs
use algebraic and combinatorial techniques.
