General-Error Regression for Deriving Cost-Estimating Relationships

Abstract

Development of cost-estimating relationships (CERs) in most cost models involving historical data is based on explicit solutions of the classical least-squares linear regression equation Y = a + bX + E, where Y is cost, X is the numerical value of a cost driver, E is a Gaussian error term whose variance does not depend on the numerical value of X, and a and b are numerical coefficients derived from the historical data. The coefficients of nonlinear forms such as Y = aX^b E are derived by taking logarithms of both sides and reducing the formulation to log(Y) = log(a) + b log(X) + log(E). This approach has a number of well-documented weaknesses in addition to the fact that the error of estimation is expressed in meaningless units (“log dollars”). A second weakness is that the analyst is forced to assume an additive-error (uniform dollar value across the board) model when historical data indicate a linear relationship between cost driver and cost, but a multiplicative-error (a percentage of the estimate) model when a nonlinear relationship is indicated. A further weakness a priori excludes from consideration certain potentially attractive nonlinear forms, such as Y = a + bX^c , because a logarithmic (or any other reasonable) transformation fails to reduce the problem to the classical linear-regression format. All known weaknesses are circumvented by applying “general-error” regression, which allows the analyst to determine the optimal coefficients for any curve shape and to choose the error model independently of the CER's shape. The optimal (error-minimizing) solution is found by sequential computer search rather than by explicit solution of simultaneous equations, as in the classical regression methods.

CERs that comprise Version 7 (August 1994) of the U.S. Air Force's Unmanned Space Vehicle Cost Model (USCM-7) have been statistically derived from historical cost data using general-error least-squares regression. In the case of USCM-7 CER development, the multiplicative-error model was selected for all CERs, whether their shape be linear or one of several nonlinear types. The optimality criterion for CER selection is minimization of percentage standard error of the estimate.