《商务统计学概论》（英文版） CHAPTER 17 Model Building

CHAPTER 17 Model building to accompany Introduction to business statistics fourth edition, by Ronald m. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. stengel o The Wadsworth Group

CHAPTER 17 Model Building to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group

l Chapter 17-learning objectives Build polynomial regression models to describe curvilinear relationships Apply qualitative variables representing two or three categories Use logarithmic transforms in constructing exponential and multiplicative models Identify and compensate for multicollinearity apply stepwise regression Select the most suitable among competing models o 2002 The Wadsworth Group

Chapter 17 - Learning Objectives • Build polynomial regression models to describe curvilinear relationships • Apply qualitative variables representing two or three categories. • Use logarithmic transforms in constructing exponential and multiplicative models. • Identify and compensate for multicollinearity • Apply stepwise regression • Select the most suitable among competing models © 2002 The Wadsworth Group

l Polynomial models with One Quantitative predictor variable Simple linear regression equation =tbx Equation for second-order polynomial model y=b+6x+b,x Equation for third-order polynomial model b.+ tb.x Equation for general polynomial model: y=6+bx+6x+bx'+.+bxp o 2002 The Wadsworth Group

Polynomial Models with One Quantitative Predictor Variable • Simple linear regression equation: • Equation for second-order polynomial model: • Equation for third-order polynomial model: • Equation for general polynomial model: © 2002 The Wadsworth Group y b b x 0 1 ˆ = + 2 0 1 2 y ˆ = b + b x + b x 3 3 2 0 1 2 y ˆ = b + b x + b x + b x p p y ˆ = b + b x + b x + b x + ...+ b x 3 3 2 0 1 2

l Polynomial models with Two Quantitative predictor variables First-order model with no interaction 6+6,x+b,x2 First-order model with interaction y=6+b,,+ b,x,+b,, x, Second-order model with no interaction y=6 +bx+b,x,+bx,+ bx Second-order model with interaction: i=b+b,,+bx+6x f+6x+bx o 2002 The Wadsworth Group

Polynomial Models with Two Quantitative Predictor Variables • First-order model with no interaction: • First-order model with interaction: • Second-order model with no interaction: • Second-order model with interaction: © 2002 The Wadsworth Group 0 1 1 2 2 y ˆ = b + b x + b x 0 1 1 2 2 3 1 2 y ˆ = b + b x + b x + b x x 2 4 2 2 0 1 1 2 2 3 1 y ˆ = b + b x + b x + b x + b x 5 1 2 2 4 2 2 0 1 1 2 2 3 1 y ˆ = b +b x +b x +b x +b x +b x x

l Models with qualitative variables Equation for a model with a categorical independent variable with two possible states D=6+b,x where state 1 is shown=1 where state 2 is shown x=0 Equation for a model with a categorical independent variable with three possible states y=b+b,x,+b,x where state 1 is shown,=1x=0 where state 2 is shown x1=0, x2=1 Where state 3 is shown x,=0,x=0 o 2002 The Wadsworth Group

Models with Qualitative Variables • Equation for a model with a categorical independent variable with two possible states: – where state 1 is shown x = 1 – where state 2 is shown x = 0 • Equation for a model with a categorical independent variable with three possible states: – where state 1 is shown x1 = 1, x2 = 0 – where state 2 is shown x1 = 0, x2 = 1 – Where state 3 is shown x1 = 0, x2 = 0 © 2002 The Wadsworth Group y b b x 0 1 ˆ = + 0 1 1 2 2 y ˆ = b + b x + b x

l Models with data Transformations Exponential Model General equation for an exponential model y=风0·B1 Corresponding linear regression equation for an exponential model logy=logb+(ogb)·x Multiplicative Model: General equation for a multiplicative model: y=Bo A1,B2 Corresponding linear regression equation for a multiplicative model: log y=log bo+ 6, log x,+b2. log x o 2002 The Wadsworth Group

Models with Data Transformations Exponential Model: • General equation for an exponential model: • Corresponding linear regression equation for an exponential model: Multiplicative Model: • General equation for a multiplicative model: • Corresponding linear regression equation for a multiplicative model: © 2002 The Wadsworth Group x y 0 1 =  log y ˆ = log b + (log b ) x 0 1 1 2 0 1 2   y =   x  x 0 1 1 2 2 log y ˆ = log b + b log x + b log x

Example, problem 17. 8 International Data Corporation has reported the following costs per gigabyte of hard drive storage space for years 1995 through 2000. Using x =1 through 6 to represent years 1995 through 2000, fit a second-order polynomial model to the data and estimate the cost per gigabyte for the year 2008 Y ear x=Yr y=Cost 1995 s26184 1996 137. 94/ The regression equation 1997 69.68 will have the form 1998 29.30 y=b +bx+b,x 1999 123456 13.09 2000 6.46 o 2002 The Wadsworth Group

Example, Problem 17.8 • International Data Corporation has reported the following costs per gigabyte of hard drive storage space for years 1995 through 2000. Using x = 1 through 6 to represent years 1995 through 2000, fit a second-order polynomial model to the data and estimate the cost per gigabyte for the year 2008. The regression equation will have the form: Year x = Yr y = Cost 1995 1 $261.84 1996 2 137.94 1997 3 69.68 1998 4 29.30 1999 5 13.09 2000 6 6.46 © 2002 The Wadsworth Group2 0 1 2 y ˆ = b + b x + b x

l Example, Problem 17.8,cont Microsoft Excel Output SUMMARY OUTPUT Regression Statistics Multiple r 0.99655892 R Square 0.99312968 AdjRSquare.98854948 Standard Error 10.5650522 Observations o 2002 The Wadsworth Group

Example, Problem 17.8, cont. Microsoft Excel Output © 2002 The Wadsworth Group SUMMARY OUTPUT Regression Statistics Multiple R 0.99655892 R Square 0.99312968 Adj R Square 0.98854948 Standard Error 10.5650522 Observations 6

l Example, Problem 17.8, cont Microsoft Excel Output Standard Coefficients E rror t Stat value Intercept 38799318899339920.5294470.002527 x-14765675123644646-11942030.0012629 214.18839291729112558.205592400037879 The regression equation is y=38799-147.66x+1419x2 o 2002 The Wadsworth Group

Example, Problem 17.8, cont. Microsoft Excel Output The regression equation is: © 2002 The Wadsworth Group Coefficients Standard Error t Stat P-value Intercept 387.993 18.8993399 20.529447 0.0002527 x -147.65675 12.3644646 -11.94203 0.0012629 x^2 14.1883929 1.72911255 8.2055924 0.0037879 2 y ˆ = 387.99 −147.66x +14.19x

Example, problem 17.8, cont To estimate the cost per gigabyte for the year 2008, evaluate i when x= 14 j=38799-14766·14+14.19142 y=1101.99 So the cost per gigabyte in 2008 is estimated to be$110199 Does this make sense? Of course not Explanation: Although the polynomial equation provides a good fit for the data during the period 1995-2000, this form is not appropriate to extrapolate the data out to 2008 o 2002 The Wadsworth Group

Example, Problem 17.8, cont. • To estimate the cost per gigabyte for the year 2008, evaluate when x = 14. • So the cost per gigabyte in 2008 is estimated to be $1101.99. • Does this make sense? Of course not. • Explanation: Although the polynomial equation provides a good fit for the data during the period 1995-2000, this form is not appropriate to extrapolate the data out to 2008. © 2002 The Wadsworth Group y ˆ ˆ 1101.99 ˆ 387.99 147.66 14 14.19 142 = = −  +  y y