Business statistics, 7e, by groebner ch15

Business Statistics: A Decision-Making Approach 7th Edition Chapter 15 Multiple Regression Analysis and Model Building Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-1 Chapter Goals After completing this chapter, you should be able to:  explain 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, 7e ©regression 2008 Prentice-Hall, Inc Chap 15-2  Chapter Goals (continued) After completing this chapter, you should be able to:  recognize potential problems in multiple regression analysis and take steps to correct the problems  incorporate qualitative variables into the regression model by using dummy variables use variable transformations to model nonlinear Business Statistics: A Decisionrelationships  Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-3 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 β0  β1x1  β x    βk x k  ε Estimated multiple regression model: Estimated (or predicted) value of y Estimated intercept Business Statistics: A DecisionMaking Approach, 7e © 2008 1 Prentice-Hall, Inc Estimated slope coefficients yˆ b  b x  b x   Chap bk15-4 xk 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, 7e © 2008 x1 Prentice-Hall, Inc Chap 15-5 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, 7e © 2008 x1 Prentice-Hall, Inc x2 The best fit equation, y , is found by minimizing the sum of squared errors, e2 Chap 15-6 Multiple Regression Assumptions Errors (residuals) from the regression model: < e = (y – y)     The model errors are independent and random The errors are normally distributed The mean of the errors is zero Errors have a constant variance Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-7 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, 7e © 2008 Prentice-Hall, Inc Chap 15-8 The Correlation Matrix  Correlation between the dependent variable and selected independent variables can be found using Excel:   Formula Tab: Data Analysis / Correlation Can check for statistical significance of correlation with a t test Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-9 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 are collected for 15 weeks Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-10 Testing for Significance: Quadratic Model  Test for Overall Relationship   F test statistic = MSR MSE Testing the Quadratic Effect  Compare quadratic model y β0  β1x j  β x 2j  ε with the linear model y β0  β1x j  ε  Hypotheses (No 2nd order polynomial term) H : β = 0 Statistics: A Decision Business  Making Approach, © 2008 (2nd order polynomial term is needed) H7e A: β2  Prentice-Hall, Inc Chap 15-56 Higher Order Models y x If p = the model is a cubic form: j j y β0  β1x j  β x  β3 x  ε Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-57 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 β0  β1x1  β x12  β3 x  β x1x  β5 x12 x Basic Terms Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Interactive Terms Chap 15-58 Effect of Interaction  Given: y β0  β1x1  β2 x  β3 x1x  ε  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, 7e © 2008 Prentice-Hall, Inc Chap 15-59 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, 7e ©of2008 (slope) x1 on y does depend on x2 value Prentice-Hall, Inc Chap 15-60 Interaction Regression Model Worksheet multiply x1 by x2 to get x1x2, then Business Statistics: A Decisionrun regression with y, x1, x2 , x1x2 Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-61 Evaluating Presence of Interaction  Hypothesize interaction between pairs of independent variables y β0  β1x1  β x  β3 x1x  ε  Hypotheses:  H : β = (no interaction between x and x )  HA: β3 ≠ (x1 interacts with x2) Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 15-62 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, 7e © 2008 Prentice-Hall, Inc Chap 15-63  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, 7e © 2008 Prentice-Hall, Inc Chap 15-64 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, 7e © 2008 Prentice-Hall, Inc Chap 15-65 Aptness of the Model  Diagnostic checks on the model include verifying the assumptions of multiple regression:  Errors are independent and random  Error are normally distributed  Errors have constant variance  Each x is linearly related to y i Business Statistics: A DecisionErrors (or Residuals) are given by Making Approach, 7e © 2008 Prentice-Hall, Inc ei ( y  yˆ ) Chap 15-66 residuals residuals Residual Analysis x Constant variance x residuals residuals Non-constant variance Business Statistics: A DecisionNot Independent Making Approach, 7e © 2008 Prentice-Hall, Inc x x  Independent Chap 15-67 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, 7e © 2008 Prentice-Hall, Inc Chap 15-68 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, 7e © 2008 Prentice-Hall, Inc Chap 15-69 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, 7e © 2008 Prentice-Hall, Inc Chap 15-70 ... 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-... incorporate qualitative variables into the regression model by using dummy variables use variable transformations to model nonlinear Business Statistics: A Decisionrelationships  Making Approach,... b1x1  b x < < yi e = (y – y) x2i < x1i Business Statistics: A DecisionMaking Approach, 7e © 2008 x1 Prentice-Hall, Inc x2 The best fit equation, y , is found by minimizing the sum of squared errors,