How to Assess a Model's Testability and Identifiability

Area of Science:

• Mathematical Modeling
• Systems Analysis
• Statistical Inference

Background:

• Models are essential in science, but their reliability depends on precise predictions and parameter ascertainment.
• Intuitive concepts of model testability and identifiability lack formal definitions, leading to potential misinterpretations.

Purpose of the Study:

• To provide formal definitions for quantitative testability, identifiability, and redundancy in models.
• To examine and refine rules of thumb for assessing model properties, particularly the Counting Rule and its generalization, the Jacobian Rule.

Main Methods:

• Formal definitions of model testability, identifiability, and redundancy are established.
• The Counting Rule and the Jacobian Rule are analyzed for their applicability to quantitatively testable and identifiable models.
• The Identifiability Rule is presented for assessing model identifiability.
• Linear and discrete-state models are used to illustrate the application of these rules.

Main Results:

• A model is quantitatively testable if its predictions are precise and narrow.
• A model is identifiable if its parameters can be determined from empirical data.
• The Counting Rule for testability is valid only for identifiable models.
• The Jacobian Rule generalizes the Counting Rule for unidentifiable models, using the rank of the Jacobian matrix.
• The Identifiability Rule uses the Jacobian matrix rank to determine if a model is identifiable.

Conclusions:

• Formal definitions and rules enhance the rigorous assessment of model quantitative testability and identifiability.
• The Jacobian Rule provides a more robust method for assessing quantitative testability, especially for unidentifiable models.
• While these rules offer strong indications, definitive conclusions require further in-depth analysis.