Parameter identifiability and redundancy: theoretical considerations

TL;DR

This paper discusses theoretical aspects of parameter identifiability, introduces new concepts related to local identifiability, and applies these ideas to complex cancer models to assess their parameter redundancy and identifiability.

Contribution

It introduces the notions of weak local identifiability and gradient weak local identifiability, linking them to existing concepts and applying them to advanced cancer models.

Findings

Weak local identifiability is equivalent to local identifiability in exponential family models.

Parameter irredundancy and local identifiability are related through the Hessian matrix.

Applications demonstrate the practical relevance of these concepts to cancer modeling.

Abstract

In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577-591) and Catchpole and Morgan (Biometrika 84 (1997) 187-196). Within the exponential family parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111-134) and Little et al. (J Theoret Biol 254 (2008) 229-238) that generalize a large number of other recently used quasi-biological cancer models, in particular those of Armitage and Doll (Br J Cancer 8 (1954) 1-12) and the two-mutation model (Moolgavkar and Venzon Math Biosciences 47 (1979) 55-77).