Business statistics, 6e, 2005, groebner CH14

Business Statistics: A Decision-Making Approach 6th Edition Chapter 14 Multiple Regression Analysis and Model Building Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, Inc Chap 14-1 Chapter Goals After completing this chapter, you should be able to:  understand model building using multiple regression analysis  apply multiple regression analysis to business decision-making situations  analyze and interpret the computer output for a multiple regression model test the significance Business Statistics: A Decision- of the independent variables a multiple model Makingin Approach, 6e ©regression 2005 Prentice-Hall, Inc Chap 14-2  Chapter Goals (continued) After completing this chapter, you should be able to:  use variable transformations to model nonlinear relationships  recognize potential problems in multiple regression analysis and take the steps to correct the problems incorporate qualitative variables into the regression by using dummy variables Business Statistics: Amodel Decision Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-3 The Multiple Regression Model Idea: Examine the linear relationship between dependent (y) & or more independent variables (xi) Population model: Y-intercept Population slopes Random Error y =â +â1x1 +â x +K +â k x k +å Estimated multiple regression model: Estimated (or predicted) value of y Estimated intercept Business Statistics: A DecisionMaking Approach, 6e © 2005 1 Prentice-Hall, Inc Estimated slope coefficients yˆ =b +b x +b x +K + b x k k Chap 14-4 Multiple Regression Model Two variable model y pe o Sl fo i ar v r le ab yˆ =b0 +b1x1 +b x x1 x2 iable x r a v r o f Slope Business Statistics: A DecisionMaking Approach, 6e © 2005 x1 Prentice-Hall, Inc Chap 14-5 Multiple Regression Model Two variable model y yi Sample observation yˆ =b0 +b1x1 +b x < < yi e = (y – y) x2i < x1i Business Statistics: A DecisionMaking Approach, 6e © 2005 x1 Prentice-Hall, Inc x2 The best fit equation, y , is found by minimizing the sum of squared errors, e2 Chap 14-6 Multiple Regression Assumptions Errors (residuals) from the regression model: < e = (y – y)     The errors are normally distributed The mean of the errors is zero Errors have a constant variance The model errors are independent Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-7 Model Specification  Decide what you want to and select the dependent variable  Determine the potential independent variables for your model  Gather sample data (observations) for all variables Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-8 The Correlation Matrix  Correlation between the dependent variable and selected independent variables can be found using Excel:   Tools / Data Analysis… / Correlation Can check for statistical significance of correlation with a t test Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-9 Example  A distributor of frozen desert pies wants to evaluate factors thought to influence demand  Dependent variable: Pie sales (units per week)  Independent variables: Price (in $) Advertising ($100’s)  Data is collected for 15 weeks Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-10 Testing for Significance: Quadratic Model  Test for Overall Relationship   F test statistic = MSR MSE Testing the Quadratic Effect  Compare quadratic model y =â +â1x j +â x 2j +å with the linear model y =â +â1x j +å  Hypotheses (No 2nd order polynomial term) H : β = 0 Statistics: A Decision Business  Making Approach, © 2005 (2nd order polynomial term is needed) H6e A: β2  Prentice-Hall, Inc Chap 14-54 Higher Order Models y x If p = the model is a cubic form: j j y =â +â1x j +â x +â x +å Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-55 Interaction Effects   Hypothesizes interaction between pairs of x variables  Response to one x variable varies at different levels of another x variable Contains two-way cross product terms y =â +â1x1 +â x12 +â x +â x1x +â x12 x Basic Terms Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Interactive Terms Chap 14-56 Effect of Interaction  Given: y =â +â1x1 +â x +â x1x +å  Without interaction term, effect of x1 on y is measured by β1  With interaction term, effect of x1 on y is measured by β1 + β3 x2  Effect changes as x2 increases Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-57 Interaction Example y = + 2x1 + 3x2 + 4x1x2 y 12 where x2 = or (dummy variable) x2 = y = + 2x1 + 3(1) + 4x1(1) = + 6x1 x2 = y = + 2x1 + 3(0) + 4x1(0) = + 2x1 x1 Statistics: 0.5 A Decision1 1.5 Business MakingEffect Approach, 6e ©of2005 (slope) x1 on y does depend on x2 value Prentice-Hall, Inc Chap 14-58 Interaction Regression Model Worksheet Case, i yi x1i x2i x1i x2i : : : : 40 30 : multiply x1 by x2 to get x1x2, then Business Statistics: A Decisionrun regression with y, x1, x2 , x1x2 Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-59 Evaluating Presence of Interaction  Hypothesize interaction between pairs of independent variables y =â +â1x1 +â x +â x1x +å  Hypotheses:  H : β = (no interaction between x and x )  HA: β3 ≠ (x1 interacts with x2) Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-60 Model Building  Goal is to develop a model with the best set of independent variables    Stepwise regression procedure   Easier to interpret if unimportant variables are removed Lower probability of collinearity Provide evaluation of alternative models as variables are added Best-subset approach Try all combinations Business Statistics: A Decision- and select the best using the and lowest s highest adjusted R ε Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-61  Stepwise Regression  Idea: develop the least squares regression equation in steps, either through forward selection, backward elimination, or through standard stepwise regression  The coefficient of partial determination is the measure of the marginal contribution of each independent variable, given that other independent variables are in the model Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-62 Best Subsets Regression  Idea: estimate all possible regression equations using all possible combinations of independent variables  Choose the best fit by looking for the highest adjusted R2 and lowest standard error sε Stepwise regression and best subsets regression can be performed using PHStat, Business Statistics: A DecisionMinitab, or other statistical software packages Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-63 Aptness of the Model  Diagnostic checks on the model include verifying the assumptions of multiple regression:  Each x is linearly related to y i    Errors have constant variance Errors are independent Error are normally distributed Business Statistics: A DecisionErrors (or Residuals) are given by Making Approach, 6e © 2005 Prentice-Hall, Inc ei =( y - yˆ ) Chap 14-64 residuals residuals Residual Analysis x Constant variance x residuals residuals Non-constant variance Business Statistics: A DecisionNot Independent Making Approach, 6e © 2005 Prentice-Hall, Inc x x  Independent Chap 14-65 The Normality Assumption  Errors are assumed to be normally distributed  Standardized residuals can be calculated by computer  Examine a histogram or a normal probability plot of the standardized residuals to check for normality Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-66 Chapter Summary       Developed the multiple regression model Tested the significance of the multiple regression model Developed adjusted R2 Tested individual regression coefficients Used dummy variables Examined interaction in a multiple regression model Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-67 Chapter Summary (continued)    Described nonlinear regression models Described multicollinearity Discussed model building    Stepwise regression Best subsets regression Examined residual plots to check model assumptions Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-68 ... multiple regression analysis to business decision-making situations  analyze and interpret the computer output for a multiple regression model test the significance Business Statistics: A Decision-... correct the problems incorporate qualitative variables into the regression by using dummy variables Business Statistics: Amodel Decision Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 14-3 The... +å Estimated multiple regression model: Estimated (or predicted) value of y Estimated intercept Business Statistics: A DecisionMaking Approach, 6e © 2005 1 Prentice-Hall, Inc Estimated slope coefficients