How to assess a model's testability and identifiability

Formal definitions are given of the following intuitive concepts: (a) A model is quantitatively testable if its predictions are highly precise and narrow. (b) A model is identifiable if the values of its parameters can be ascertained from empirical observations. (c) A model is redundant if the values of some parameters can be deduced from others or if the values of some observables can be deduced from others. Various rules of thumb for nonredundant models are examined. The Counting Rule states that a model is quantitatively testable if and only if it has fewer parameters than observables. This rule can be safely applied only to identifiable models. If a model is unidentifiable, one must apply a generalization of the Counting Rule known as the Jacobian Rule. This rule states that a model is quantitatively testable if and only if the maximum rank (i.e., the number of linearly independent columns) of its Jacobian matrix (i.e., the matrix of partial derivatives of the function that maps parameter values to the predicted values of observables) is smaller than the number of observables. The Identifiability Rule states that a model is identifiable if and only if the maximum rank of its Jacobian matrix equals the number of parameters. The conclusions provided by these rules are only presumptive. To reach definitive conclusions, additional analyses must be performed. To illustrate the foregoing, the quantitative testability and identifiability of linear models and of discrete-state models are analyzed.