Chap 15-1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-1 Chapter 15 Multiple Regression Model Building Basic Business Statistics.

Chap 15-2 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-2 Learning Objectives In this chapter, you learn: To use quadratic terms in a regression model To use transformed variables in a regression model To measure the correlation among the independent variables To build a regression model using either the stepwise or best-subsets approach To avoid the pitfalls involved in developing a multiple regression model

Chap 15-3 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-3 The relationship between the dependent variable and an independent variable may not be linear Can review the scatter plot to check for nonlinear relationships Example: Quadratic model The second independent variable is the square of the first variable Nonlinear Relationships DCOVA

Chap 15-4 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-4 Quadratic Regression Model where: β 0 = Y intercept β 1 = regression coefficient for linear effect of X on Y β 2 = regression coefficient for quadratic effect on Y ε i = random error in Y for observation i Model form: DCOVA

Chap 15-5 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-5 Linear fit does not give random residuals Linear vs. Nonlinear Fit Nonlinear fit gives random residuals X residuals X Y X Y X DCOVA

Chap 15-6 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-6 Quadratic Regression Model Quadratic models may be considered when the scatter plot takes on one of the following shapes: X1X1 Y X1X1 X1X1 YYY β 1 < 0β 1 > 0β 1 < 0β 1 > 0 β 1 = the coefficient of the linear term β 2 = the coefficient of the squared term X1X1 β 2 > 0 β 2 < 0 DCOVA

Chap 15-7 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-7 Testing the Overall Quadratic Model Test for Overall Relationship H 0 : β 1 = β 2 = 0 (no overall relationship between X and Y) H 1 : β 1 and/or β 2 ≠ 0 (there is a relationship between X and Y) F STAT = Estimate the quadratic model to obtain the regression equation: DCOVA

Chap 15-8 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-8 Testing for Significance: Quadratic Effect Testing the Quadratic Effect Compare quadratic regression equation with the linear regression equation DCOVA

Chap 15-9 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-9 Testing for Significance: Quadratic Effect Testing the Quadratic Effect Consider the quadratic regression equation Hypotheses (The quadratic term does not improve the model) (The quadratic term improves the model) H 0 : β 2 = 0 H 1 : β 2  0 (continued) DCOVA

Chap 15-10 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-10 Testing for Significance: Quadratic Effect Testing the Quadratic Effect Hypotheses (The quadratic term does not improve the model) (The quadratic term improves the model) The test statistic is H 0 : β 2 = 0 H 1 : β 2  0 (continued) where: b 2 = squared term slope coefficient β 2 = hypothesized slope (zero) S b = standard error of the slope 2 DCOVA

Chap 15-11 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-11 Testing for Significance: Quadratic Effect Testing the Quadratic Effect Compare adjusted r 2 from simple regression model to adjusted r 2 from the quadratic model If adjusted r 2 from the quadratic model is larger than the adjusted r 2 from the simple model, then the quadratic model is likely a better model (continued) DCOVA

Chap 15-12 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-12 Example: Quadratic Model Purity increases as filter time increases: Purity Filter Time 3 1 7 2 8 3 15 5 22 7 33 8 4010 5412 6713 7014 7815 8515 8716 9917 DCOVA

Chap 15-13 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-13 Example: Quadratic Model (continued) Regression Statistics R Square0.96888 Adjusted R Square0.96628 Standard Error6.15997 Simple regression results: Y = -11.283 + 5.985 Time Coefficients Standard Errort StatP-value Intercept-11.282673.46805-3.253320.00691 Time5.985200.3096619.328192.078E-10 FSignificance F 373.579042.0778E-10 ^ t statistic, F statistic, and r 2 are all high, but the residuals are not random: DCOVA

Chap 15-14 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-14 Coefficients Standard Errort StatP-value Intercept1.538702.244650.685500.50722 Time1.564960.601792.600520.02467 Time-squared0.245160.032587.524061.165E-05 Quadratic regression results: Y = 1.539 + 1.565 Time + 0.245 (Time) 2 ^ Example: Quadratic Model in Excel & Minitab (continued) The quadratic term is statistically significant (p-value very small) DCOVA The regression equation is Purity = 1.54 + 1.56 Time + 0.245 Time Squared Predictor Coef SE Coef T P Constant 1.5390 2.24500 0.69 0.507 Time 1.5650 0.60180 2.60 0.025 Time Squared 0.24516 0.03258 7.52 0.000 S = 2.59513 R-Sq = 99.5% R-Sq(adj) = 99.4% ExcelMinitab

Chap 15-15 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-15 Regression Statistics R Square0.99494 Adjusted R Square0.99402 Standard Error2.59513 Quadratic regression results: Y = 1.539 + 1.565 Time + 0.245 (Time) 2 ^ Example: Quadratic Model in Excel & Minitab (continued) The adjusted r 2 of the quadratic model is higher than the adjusted r 2 of the simple regression model. The quadratic model explains 99.4% of the variation in Y. DCOVA The regression equation is Purity = 1.54 + 1.56 Time + 0.245 Time Squared Predictor Coef SE Coef T P Constant 1.5390 2.24500 0.69 0.507 Time 1.5650 0.60180 2.60 0.025 Time Squared 0.24516 0.03258 7.52 0.000 S = 2.59513 R-Sq = 99.5% R-Sq(adj) = 99.4%

Chap 15-16 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Example: Quadratic Model Residual Plots Quadratic regression results: Y = 1.539 + 1.565 Time + 0.245 (Time) 2 The residuals plotted versus both Time and Time-squared show a random pattern. (continued) DCOVA

Chap 15-17 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-17 Using Transformations in Regression Analysis Idea: non-linear models can often be transformed to a linear form Can be estimated by least squares if transformed transform X or Y or both to get a better fit or to deal with violations of regression assumptions Can be based on theory, logic or scatter plots DCOVA

Chap 15-18 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-18 The Square Root Transformation The square-root transformation Used to overcome violations of the constant variance assumption fit a non-linear relationship DCOVA

Chap 15-19 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-19 Shape of original relationship X The Square Root Transformation (continued) b 1 > 0 b 1 < 0 X Y Y Y Y Relationship when transformed DCOVA

Chap 15-20 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-20 Original multiplicative model The Log Transformation Transformed multiplicative model The Multiplicative Model: Original multiplicative model Transformed exponential model The Exponential Model: DCOVA

Chap 15-21 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-21 Interpretation of Coefficients For the multiplicative model: When both dependent and independent variables are lagged: The coefficient of the independent variable X k can be interpreted as : a 1 percent change in X k leads to an estimated b k percentage change in the average value of Y. Therefore b k is the elasticity of Y with respect to a change in X k. DCOVA

Chap 15-22 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-22 Collinearity Collinearity: High correlation exists among two or more independent variables This means the correlated variables contribute redundant information to the multiple regression model DCOVA

Chap 15-23 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-23 Collinearity Including two highly correlated independent variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and low t-values) Coefficient signs may not match prior expectations (continued) DCOVA

Chap 15-24 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-24 Some Indications of Strong Collinearity Incorrect signs on the coefficients Large change in the value of a previous coefficient when a new variable is added to the model A previously significant variable becomes non- significant when a new independent variable is added The estimate of the standard deviation of the model increases when a variable is added to the model DCOVA

Chap 15-25 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-25 Detecting Collinearity (Variance Inflationary Factor) VIF j is used to measure collinearity: If VIF j > 5, X j is highly correlated with the other independent variables where R 2 j is the coefficient of determination of variable X j with all other X variables DCOVA

Chap 15-26 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-26 Example: Pie Sales Sales = b 0 + b 1 (Price) + b 2 (Advertising) Week Pie Sales Price ($) Advertising ($100s) 13505.503.3 24607.503.3 33508.003.0 44308.004.5 53506.803.0 63807.504.0 74304.503.0 84706.403.7 94507.003.5 104905.004.0 113407.203.5 123007.903.2 134405.904.0 144505.003.5 153007.002.7 Recall the multiple regression equation of chapter 14: DCOVA

Chap 15-27 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-27 Detecting Collinearity in Excel using PHStat Output for the pie sales example: Since there are only two independent variables, only one VIF is reported VIF is < 5 There is no evidence of collinearity between Price and Advertising Regression Analysis Price and all other X Regression Statistics Multiple R0.030438 R Square0.000926 Adjusted R Square-0.075925 Standard Error1.21527 Observations15 VIF1.000927 PHStat / regression / multiple regression … Check the “variance inflationary factor (VIF)” box DCOVA

Chap 15-28 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-28 Detecting Collinearity in Minitab Predictor Coef SE Coef T P VIF Constant 306.50 114.3 2.68 0.020 Price - 24.98 10.83 -2.31 0.040 1.001 Advertising 74.13 25.97 2.85 0.014 1.001 Output for the pie sales example: Since there are only two independent variables, the VIF reported is the same for each variable VIF is < 5 There is no evidence of collinearity between Price and Advertising

Chap 15-29 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-29 Model Building Goal is to develop a model with the best set of independent variables Easier to interpret if unimportant variables are removed Lower probability of collinearity Stepwise regression procedure Provide evaluation of alternative models as variables are added and deleted Best-subset approach Try all combinations and select the best using the highest adjusted r 2 and lowest standard error DCOVA

Chap 15-30 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-30 Idea: develop the least squares regression equation in steps, adding one independent variable at a time and evaluating whether existing variables should remain or be removed 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 Stepwise Regression DCOVA

Chap 15-31 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-31 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 r 2 and lowest standard error Stepwise regression and best subsets regression can be performed using PHStat DCOVA

Chap 15-32 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-32 Alternative Best Subsets Criterion Calculate the value C p for each potential regression model Consider models with C p values close to or below k + 1 k is the number of independent variables in the model under consideration DCOVA

Chap 15-33 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-33 Alternative Best Subsets Criterion The C p Statistic Where k = number of independent variables included in a particular regression model T = total number of parameters to be estimated in the full regression model = coefficient of multiple determination for model with k independent variables = coefficient of multiple determination for full model with all T estimated parameters (continued) DCOVA

Chap 15-34 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-34 Steps in Model Building 1. Compile a listing of all independent variables under consideration 2. Estimate full model and check VIFs 3. Check if any VIFs > 5 If no VIF > 5, go to step 4 If one VIF > 5, remove this variable If more than one, eliminate the variable with the highest VIF and go back to step 2 4.Perform best subsets regression with remaining variables DCOVA

Chap 15-35 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-35 Steps in Model Building 5. List all models with C p close to or less than (k + 1) 6. Choose the best model Consider parsimony Do extra variables make a significant contribution? 7.Perform complete analysis with chosen model, including residual analysis 8.Transform the model if necessary to deal with violations of linearity or other model assumptions 9.Use the model for prediction and inference (continued) DCOVA

Chap 15-36 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-36 Model Building Flowchart Choose X 1,X 2,…X k Run regression to find VIFs Remove variable with highest VIF Any VIF>5? Run subsets regression to obtain “best” models in terms of C p Do complete analysis Add quadratic and/or interaction terms or transform variables Perform predictions No More than one? Remove this X Yes No Yes DCOVA

Chap 15-37 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-37 Pitfalls and Ethical Considerations Understand that interpretation of the estimated regression coefficients are performed holding all other independent variables constant Evaluate residual plots for each independent variable Evaluate interaction terms To avoid pitfalls and address ethical considerations:

Chap 15-38 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-38 Additional Pitfalls and Ethical Considerations Obtain VIFs for each independent variable before determining which variables should be included in the model Examine several alternative models using best- subsets regression Use other methods when the assumptions necessary for least-squares regression have been seriously violated To avoid pitfalls and address ethical considerations: (continued)

Chap 15-39 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-39 Chapter Summary Developed the quadratic regression model Discussed using transformations in regression models The multiplicative model The exponential model Described collinearity Discussed model building Stepwise regression Best subsets Addressed pitfalls in multiple regression and ethical considerations

Chap 15-41 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Learning Objective To learn how to measure the influence of individual observations using: The hat matrix elements h i The Studentized deleted residuals t i Cook’s distance statistic D i

Chap 15-42 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Influence Analysis Helps Identify Individual Observations Having A Disproportionate Impact On The Analysis There are three commonly used measures of influence: The diagonal values of the hat matrix h i The Studentized deleted residuals t i Cook’s distance statistic D i Each of these measure influence in slightly different ways. If potentially influential observations are present you may want to delete them and redo the analysis. DCOVA

Chap 15-44 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall If h i > 2(k + 1)/n Then X i Is An Influential Observation Here n = the number of observations & k = the number of independent variables For the OmniPower data n = 34 and k = 2 Thus you flag any h i > 2(2 + 1)/34 = 0.1765 From the Minitab output on the previous page none of the h i values are greater than this value Therefore none of the observations are candidates for removal from the analysis

Chap 15-45 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall What Is A Studentized Deleted Residual? A residual is the difference between an observed value of Y and the predicted value of Y from the model: A Studentized residual is a t statistic obtained by dividing e i by the standard error of The i th deleted residual is the difference between Y i and the predicted value of Y i from a regression model developed without using the i th observation The Studentized deleted residual is a t statistic obtained by dividing a deleted residual by the standard error of

Chap 15-46 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall If |t i | > t 0.05 With (n – (k + 1)) Then X i Is Highly Influential On The Regression Equation A highly influential observation is a candidate for removal from the analysis For the OmniPower data n = 34 and k = 2 so t 0.05 = 1.6973 Using this value since t 14 = -3.08402, t 15 = 2.20612, and t 20 = 2.27527 so observations 14, 15, & 20 are identified as highly influential observations These observations should be looked at further to determine whether or not they should be removed from the analysis

Chap 15-47 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Cook’s D i Statistic Is Based On Both h i and The Studentized Residual If D i > F 0.50 with numerator df = k = 1 and denominator df = n – k – 1, then the observation is highly influential on the regression equation and is a candidate for removal. For the OmniPower data F 0.50 = 0.807. Since no observations have a value that exceeds this value there are no observations flagged as highly influential by Cook’s D i

Chap 15-48 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Topic Summary Discussed how to measure the influence of individual observations using: The hat matrix elements h i The Studentized deleted residuals t i Cook’s distance statistic D i

Chap 15-51 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Data Mining Is A Sophisticated Quantitative Analysis That Attempts To Discover Useful Patterns Or Relationships In A Group Of Data Some examples of data mining in use today are: Mortgage acceptance & default -- predicting which applicants will qualify for a specific type of mortgage Retention -- who will remain a customer of the financial services company Response to promotions -- predicting who will respond to promotions Purchasing associations -- predicting what product a consumer will purchase given that they have purchased another product Product failure -- Predicting the type of product that will fail during a warranty period DCOVA

Chap 15-52 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Data Mining Typically Involves The Analysis Of Very Large Data Sets Large data sets have many observations and many variables Working with such data sets increases the importance of how data is collected so that it can be readily analyzed to arrive at useful conclusions

Chap 15-53 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Data Mining Can be Divided Into Two Types Of Tasks Exploratory Data Analysis Examining the basic features of a data set via Tables, Descriptive Statistics, & Graphic Displays Predictive Modeling Building a model to predict a variable based on the value of other variables via Simple Linear Regression, Multiple Regression Model Building, Stepwise Regression, & Logistic Regression

Chap 15-54 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Data Mining Also Uses Other Methods Not Discussed In This Book Classification & Regression Trees (CART) Chi-square Automatic Interaction Dectector (CHAID) Neural Nets Cluster Analysis Multidimensional Scaling

Chap 15-1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-1 Chapter 15 Multiple Regression Model Building Basic Business Statistics.

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Chap 15-1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 15-1 Chapter 15 Multiple Regression Model Building Basic Business Statistics.

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