### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 20-40 |

Number of pages | 21 |

Journal | Journal of Mathematical Psychology |

Volume | 44 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Experimental and Cognitive Psychology

### Cite this

*Journal of Mathematical Psychology*,

*44*(1), 20-40. https://doi.org/10.1006/jmps.1999.1275

**How to assess a model's testability and identifiability.** / Bamber, Donald; Van Santen, Jan.

Research output: Contribution to journal › Article

*Journal of Mathematical Psychology*, vol. 44, no. 1, pp. 20-40. https://doi.org/10.1006/jmps.1999.1275

}

TY - JOUR

T1 - How to assess a model's testability and identifiability

AU - Bamber, Donald

AU - Van Santen, Jan

PY - 2000/3

Y1 - 2000/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0347684419&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347684419&partnerID=8YFLogxK

U2 - 10.1006/jmps.1999.1275

DO - 10.1006/jmps.1999.1275

M3 - Article

AN - SCOPUS:0347684419

VL - 44

SP - 20

EP - 40

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

SN - 0022-2496

IS - 1

ER -