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transform or modify the variables, as shown in Equations 12.25 and 12.27. In this way a nonlinear quadratic model is converted to a model that linear in a modified set of variables. K47 Quadratic Model Transformations The quadratic function Y = B + BX + BX + 6 (12.26) can be transformed into a linear multiple regression model by defining new variables: 21 = x Z2 = x and then specifying the model as y = Bot Bizi + Brzit & (12.27) which is linear in the transformed variables. Transformed quadratic variables can be combined with other variables in a multiple regression model. Thus, we can fit a multiple quadratic regression using transformed variables. The goal is to find models that are linear in other mathematical forms of a variable. By transforming the variables, we can estimate a linear multiple regression model and use the results as a nonlinear model. Inference procedures for transformed quadratic mod- els are the same as those that we have previously developed for linear models. In this way we avoid confusion that would result if different statistical procedures were used for linear versus quadratic models. The coefficients must be combined for interpretation. Thus, if we have a quadratic model, then the effect of a variable, X, is indicated by the coefficients of both the linear and the quadratic terms. We can also perform a simple hypothesis test to de termine if a quadratic model is an improvement over a linear model. The Z2 or X; variable is merely an additional variable whose coefficient can be tested-Ho: B2 = 0-using the conditional Student'st or F statistic. If a quadratic model fits the data better than a linear model, then the coefficient of the quadratic variable-Z2 = x; -will be significantly differ- ent from 0. The same approach applies if we have variables such as Zx = X or 24 = x X2. 12.7 Transformations for Nonlinear Regression Models 495 Example 12.11 Production Costs (Quadratic Model Estimation) Arnold Sorenson, production manager of New Frontiers Instruments, Inc., was inter- ested in estimating the mathematical relationship between the number of electronic assemblies produced during an 8-hour shift and the average cost per assembly. This function would then be used to estimate cost for various production order bids and to determine the production level that would minimize average cost. Data are found in the data file Production Cost