  This paper introduces diagnostic tests for the nature of lack of fit in
ordinary differential equation models (ODEs) proposed for data. We present a
hierarchy of three possible sources of lack of fit: unaccounted-for stochastic
variation, misspecification of functional forms in rate equations, and omission
of dynamic variables in the description of the system. We represent lack of fit
by allowing a parameter vector to vary over time, and propose generic testing
procedures that do not rely on specific alternative models. Instead, different
sources for lack of fit are characterized in terms of nonparametric
relationships among latent variables. The tests are carried out through a
combination of residual bootstrap and permutation methods. We demonstrate the
effectiveness of these tests on simulated data and on real data from laboratory
ecological experiments and electro-cardiogram data.
