Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-1 Chapter 14 Multiple Regression Model Building Statistics for Managers Using Microsoft ® Excel 4 th Edition

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-2 Chapter Goals After completing this chapter, you should be able to:  use quadratic terms in a regression model  measure the correlation among the independent variables  build a regression model using stepwise or best- subsets approach  understand the pitfalls involved in developing a multiple regression model

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-3  The relationship between the dependent variable and an independent variable may not be linear  Review the scatter diagram to check for non- linear relationships  Non-linear relationships might be modeled with a quadratic relationship Nonlinear Relationships

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-4 Quadratic Regression Model Quadratic models may be considered when the scatter diagram 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-5 Residual Analysis A non-linear relationship Multiple Regression with X 1 and X 2 X 1 Residual plot was randomly distributed

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-6 Quadratic Model Worksheet Multiply X 1 by X 2 to get X 1 *X 2. Run regression with Y, X 1, X 2, X 1 X 2 Item, iYiYi X 1i X 2i X 2 2i …………… Square X 2i to get X 2 2i Run regression with Y i, X 1i, X 2 2i

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-7 Testing the Overall Model  Test for Overall Relationship H 0 : β 1 = β 2 = 0 (no overall relationship between X and Y) H 1 : Not all β’s = 0 (there is a relationship between X and Y)  F test statistic =  Use Excel to obtain the quadratic model:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-8 Testing the Quadratic Term  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 where: b 2 = squared term slope coefficient β 2 = hypothesized slope (zero) S b2 = standard error of the slope

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 14-9 Testing the Quadratic Term  A second test of the quadratic effect Compare the adjusted r 2 from the simple regression to the adjusted r 2 from the quadratic model  If the adjusted r 2 of the quadratic model is greater than the adjusted r 2 of the simple linear regression then the quadratic model has explained a larger percentage of the variability in Y (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Example: Quadratic Model  Purity increases as filter time increases: Purity Filter Time

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Example: Quadratic Model (continued) Regression Statistics R Square Adjusted R Square Standard Error  Simple regression results: Y = Time Coefficients Standard Errort StatP-value Intercept Time E-10 FSignificance F E-10 ^ T-statistic, F-statistic, and r 2 are all high, but the residuals are not random:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Coefficients Standard Errort StatP-value Intercept Time Time-squared E-05 Regression Statistics R Square Adjusted R Square Standard Error FSignificance F E-13  Quadratic regression results: Y = Time (Time) 2 ^ Example: Quadratic Model (continued) The quadratic term is significant and improves the model: adj. r 2 is higher and S YX is lower, residuals are now random

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Collinearity  Collinearity: High correlation exists between two independent variables  This means the two variables contribute redundant information to the multiple regression model

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Collinearity  Including two highly correlated explanatory 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)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 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 insignificant 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Detecting Collinearity (Variance Inflationary Factor) VIF j is used to measure collinearity: If VIF j > 10, X j is correlated with the other explanatory variables where R 2 j is the coefficient of determination of variable X j with all other X variables

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Example: Pie Sales Model Sales = b 0 + b 1 (Price) + b 2 (Advertising) Week Pie Sales Price ($) Advertising ($100s) Recall the multiple regression model of chapter 13:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Detect Collinearity in PHStat Output for the pie sales example:  Since there are only two explanatory variables, only one VIF is reported  VIF is < 10  There is no evidence of collinearity between Price and Advertising Regression Analysis Price and all other X Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 VIF PHStat / regression / multiple regression … Check the “variance inflationary factor (VIF)” box

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 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  Best-subset approach  Try all combinations and select the best using the highest adjusted r 2 and lowest standard error

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  Idea: develop the least squares regression equation in steps, adding one explanatory 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 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

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 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)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Steps in Model Building  1. Choose explanatory variables to include in the model  2. Estimate full model and check VIFs  3. Check if any VIFs > 10  4.  If no VIF > 10, go to step 5  If one VIF > 10, remove this variable  If more than one, eliminate the variable with the highest VIF and go back to step 2  5. Perform best subsets regression with remaining variables …

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Steps in Model Building  6. List all models with C p close to or less than (k + 1)  7. Choose the best model  Consider parsimony  Do extra variable make a significant contribution?  8. Perform complete analysis with chosen model, including residual analysis  9. Transform the model if necessary to deal with violations of linearity or other model assumptions  10. Use the model for prediction (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Model Building Flowchart Choose X 1,X 2,…X k Run regression to find VIFs Remove variable with highest VIF Any VIF>10? Run subsets regression to obtain “best” models in terms of C p Do complete analysis Add quadratic term and/or transform variables as indicated Perform predictions No More than one? Remove this X Yes No Yes

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Pitfalls  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 :

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Additional Pitfalls  Obtain VIFs for each independent variable before determining which variables should be included in the model  Examine several alternative models using stepwise and best-subsets models  Use other methods when the assumptions necessary for least-squares regression have been seriously violated (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Chapter Summary  Developed the quadratic regression model  Described collinearity and the VIF model  Discussed model building  Stepwise regression  Best subsets  Addressed pitfalls in multiple regression