Generalized Degrees of Freedom

Abstract

Two popular regression methods for the multiplicative-error model are the Minimum-Unbiased-Percent Error and Minimum-Percentage Error under the Zero-Percentage Bias methods. The Minimum-Unbiased-Percent Error method, an Iteratively Reweighted Least Squares regression, does not use any constraints, while the Minimum-Percentage Error under the Zero-Percentage Bias method requires a constraint as part of the curve-fitting process. However, Minimum-Percentage Error under the Zero-Percentage Bias users do not adjust the degrees of freedom to account for constraints included in the regression process. As a result, fit statistics for the Minimum-Percentage Error under the Zero-Percentage bias equations, e.g., the standard percent error and generalized R², can be incorrect and misleading. This results in incompatible fit statistics between Minimum-Percentage Error under the Zero-Percentage Bias and Minimum-Unbiased-Percent Error equations. This article details why degrees of freedom should be adjusted and recommends a Generalized Degrees of Freedom measure to calculate fit statistics for constraint-driven cost estimating relationships. It also explains why Minimum-Percentage Error under the Zero-Percentage Bias’s standard error underestimates the spread of the cost estimating relationship error distribution. Illustrative examples are provided. Note that this article only considers equality constraints; Generalized Degrees of Freedom for inequality constraints is another topic.

Additional information

Notes on contributors

Shu-Ping Hu

Shu-Ping Hu is the Chief Statistician at Tecolote Research, Inc. She joined Tecolote in 1984 and serves as a company expert in all statistical matters. She earned her Ph.D. in Mathematics, with an emphasis in Statistics, at the University of California, Santa Barbara. Dr. Hu has 30 years of experience designing, developing, and validating learning, uncertainty, correlation, and regression algorithms in ACE, CO$TAT, and JACS. Dr. Hu also published many technical papers. For over 20 years, she has been a regular presenter of advanced cost analysis techniques at major cost conferences.