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.