Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-1 Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 14 Introduction to Multiple Regression

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-2 Learning Objectives In this chapter, you learn:  How to develop a multiple regression model  How to interpret the regression coefficients  How to determine which independent variables to include in the regression model  How to determine which independent variables are most important in predicting a dependent variable  How to use categorical variables in a regression model

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-3 The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (X i ). Multiple Regression Model with k Independent Variables: Y-intercept Population slopesRandom Error

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-4 Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Estimated (or predicted) value of Y Estimated slope coefficients Multiple regression equation with k independent variables: Estimated intercept In this chapter we will always use Excel to obtain the regression slope coefficients and other regression summary measures.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-5 Multiple Regression Equation Example with two independent variables Y X1X1 X2X2 Slope for variable X 1 Slope for variable X 2

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-6 Multiple Regression Equation 2 Variable Example  A distributor of frozen dessert 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

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-7 Multiple Regression Equation 2 Variable Example  Sales = b 0 + b 1 (Price) + b 2 (Advertising)  Sales = b 0 +b 1 X 1 + b 2 X 2 Where X 1 = Price X 2 = Advertising WeekPie Sales Price ($) Advertising ($100s) Multiple regression equation:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-8 Multiple Regression Equation 2 Variable Example, Excel Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 14-9 Multiple Regression Equation 2 Variable Example b 1 = : sales will decrease, on average, by pies per week for each $1 increase in selling price, net of the effects of changes due to advertising b 2 = : sales will increase, on average, by pies per week for each $100 increase in advertising, net of the effects of changes due to price where Sales is in number of pies per week Price is in $ Advertising is in $100’s.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Multiple Regression Equation 2 Variable Example Predict sales for a week in which the selling price is $5.50 and advertising is $350: Predicted sales is pies Note that Advertising is in $100’s, so $350 means that X 2 = 3.5

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Coefficient of Multiple Determination  Reports the proportion of total variation in Y explained by all X variables taken together

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Coefficient of Multiple Determination (Excel) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising % of the variation in pie sales is explained by the variation in price and advertising

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Adjusted r 2  r 2 never decreases when a new X variable is added to the model  This can be a disadvantage when comparing models  What is the net effect of adding a new variable?  We lose a degree of freedom when a new X variable is added  Did the new X variable add enough independent power to offset the loss of one degree of freedom?

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Adjusted r 2  Shows the proportion of variation in Y explained by all X variables adjusted for the number of X variables used (where n = sample size, k = number of independent variables)  Penalizes excessive use of unimportant independent variables  Smaller than r 2  Useful in comparing models

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Adjusted r 2 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising % of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap F-Test for Overall Significance  F-Test for Overall Significance of the Model  Shows if there is a linear relationship between all of the X variables considered together and Y  Use F test statistic  Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H 1 : at least one β i ≠ 0 (at least one independent variable affects Y)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap F-Test for Overall Significance  Test statistic:  where F has (numerator) = k and (denominator) = (n - k - 1) degrees of freedom

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap F-Test for Overall Significance Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising P-value for the F-Test

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap F-Test for Overall Significance  H 0 : β 1 = β 2 = 0  H 1 : β 1 and β 2 not both zero   =.05  df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Since F test statistic is in the rejection region (p-value <.05), reject H 0 There is evidence that at least one independent variable affects Y 0  =.05 F.05 = 3.89 Reject H 0 Do not reject H 0 Critical Value: F  = F

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Residuals in Multiple Regression Two variable model Y X1X1 X2X2 YiYi Y i < x 2i x 1i The best fitting linear regression equation, Y, is found by minimizing the sum of squared errors,  e 2 < Sample observation Residual = e i = (Y i – Y i ) <

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Multiple Regression Assumptions Assumptions:  The errors are independent  The errors are normally distributed  Errors have an equal variance e i = (Y i – Y i ) Errors ( residuals ) from the regression model: <

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Multiple Regression Assumptions  These residual plots are used in multiple regression:  Residuals vs. Y i  Residuals vs. X 1i  Residuals vs. X 2i  Residuals vs. time (if time series data) Use the residual plots to check for violations of regression assumptions <

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Individual Variables Tests of Hypothesis  Use t-tests of individual variable slopes  Shows if there is a linear relationship between the variable X i and Y  Hypotheses: H 0 : β i = 0 (no linear relationship) H 1 : β i ≠ 0 (linear relationship does exist between X i and Y)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Individual Variables Tests of Hypothesis H 0 : β j = 0 (no linear relationship) H 1 : β j ≠ 0 (linear relationship does exist between X i and Y)  Test Statistic: (df = n – k – 1)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Individual Variables Tests of Hypothesis Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations15 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Price Advertising t-value for Price is t = , with p- value.0398 t-value for Advertising is t = 2.855, with p-value.0145

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Individual Variables Tests of Hypothesis d.f. = = 12  =.05 t  /2 = H 0 : β j = 0 H 1 : β j ≠ 0 The test statistic for each variable falls in the rejection region (p-values <.05) There is evidence that both Price and Advertising affect pie sales at  =.05 Reject H 0 for each variable CoefficientsStandard Errort StatP-value Price Advertising Decision: Conclusion: Reject H 0  /2=.025 -t α/2 Do not reject H 0 0 t α/2  /2=.025

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Confidence Interval Estimate for the Slope Confidence interval for the population slope β i Example: Form a 95% confidence interval for the effect of changes in price (X 1 ) on pie sales, holding constant the effects of advertising: ± (2.1788)(10.832): So the interval is ( , ) CoefficientsStandard Error Intercept Price Advertising where t has (n – k – 1) d.f. Here, t has (15 – 2 – 1) = 12 d.f.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Confidence Interval Estimate for the Slope Confidence interval for the population slope β i Example: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between 1.37 to pies for each increase of $1 in the selling price, holding constant the effects of advertising. CoefficientsStandard Error … Lower 95%Upper 95% Intercept … Price … Advertising …

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Testing Portions of the Multiple Regression Model  Contribution of a Single Independent Variable X j SSR(X j | all variables except X j ) = SSR (all variables) – SSR(all variables except X j )  Measures the contribution of X j in explaining the total variation in Y (SST)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Testing Portions of the Multiple Regression Model Measures the contribution of X 1 in explaining SST From ANOVA section of regression for Contribution of a Single Independent Variable X j, assuming all other variables are already included (consider here a 3-variable model): SSR(X 1 | X 2 and X 3 ) = SSR (all variables) – SSR(X 2 and X 3 )

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap The Partial F-Test Statistic Consider the hypothesis test: H 0 : variable X j does not significantly improve the model after all other variables are included H 1 : variable X j significantly improves the model after all other variables are included Test using the F-test statistic: (with 1 and n-k-1 d.f.)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Testing Portions of Model: Example Test at the α =.05 level to determine whether the price variable significantly improves the model given that advertising is included Example: Frozen dessert pies

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Testing Portions of Model: Example H 0 : X 1 (price) does not improve the model with X 2 (advertising) included H 1 : X 1 does improve model α =.05, df = 1 and 12 F critical Value = 4.75 (For X 1 and X 2 )(For X 2 only) ANOVA dfSSMS Regression Residual Total ANOVA dfSS Regression Residual Total

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Testing Portions of Model: Example Conclusion: Reject H 0 ; adding X 1 does improve model (For X 1 and X 2 )(For X 2 only) ANOVA dfSSMS Regression Residual Total ANOVA dfSS Regression Residual Total

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Relationship Between Test Statistics  The partial F test statistic developed in this section and the t test statistic are both used to determine the contribution of an independent variable to a multiple regression model.  The hypothesis tests associated with these two statistics always result in the same decision (that is, the p-values are identical). Where a = degrees of freedom

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Coefficient of Partial Determination for k Variable Model Measures the proportion of variation in the dependent variable that is explained by X j while controlling for (holding constant) the other independent variables

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Using Dummy Variables  A dummy variable is a categorical independent variable with two levels:  yes or no, on or off, male or female  coded as 0 or 1  Assumes equal slopes for other variables  If more than two levels, the number of dummy variables needed is (number of levels - 1)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Example Let: Y = pie sales X 1 = price X 2 = holiday (X 2 = 1 if a holiday occurred during the week) (X 2 = 0 if there was no holiday that week)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Example Same slope X 1 (Price) Y (sales) b 0 + b 2 b0b0 Holiday No Holiday Different intercept Holiday (X 2 = 1) No Holiday (X 2 = 0) If H 0 : β 2 = 0 is rejected, then “Holiday” has a significant effect on pie sales

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Example Sales: number of pies sold per week Price: pie price in $ Holiday: 1 If a holiday occurred during the week 0 If no holiday occurred b 2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same price

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Model More Than Two Levels  The number of dummy variables is one less than the number of levels  Example: Y = house price ; X 1 = square feet  If style of the house is also thought to matter Style = ranch, split level, colonial  Three levels, so two dummy variables are needed

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Model More Than Two Levels Example: Let “colonial” be the default category, and let X 2 and X 3 be used for the other two categories: Y = house price X 1 = square feet X 2 = 1 if ranch, 0 otherwise X 3 = 1 if split level, 0 otherwise The multiple regression equation is:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Dummy Variable Model More Than Two Levels With the same square feet, a ranch will have an estimated average price of thousand dollars more than a colonial. With the same square feet, a split-level will have an estimated average price of thousand dollars more than a colonial. Consider the regression equation: For a colonial: X 2 = X 3 = 0 For a ranch: X 2 = 1; X 3 = 0 For a split level: X 2 = 0; X 3 = 1

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Interaction Between Independent Variables  Hypothesizes interaction between pairs of X variables  Response to one X variable may vary at different levels of another X variable  Contains a two-way cross product term 

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Effect of Interaction  Given:  Without interaction term, effect of X 1 on Y is measured by β 1  With interaction term, effect of X 1 on Y is measured by β 1 + β 3 X 2  Effect changes as X 2 changes

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Interaction Example X 2 = 1: Y = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 X 2 = 0: Y = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1 Slopes are different if the effect of X 1 on Y depends on X 2 value X1X Y = 1 + 2X 1 + 3X 2 + 4X 1 X 2 Suppose X 2 is a dummy variable and the estimated regression equation is

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Significance of Interaction Term  Can perform a partial F-test for the contribution of a variable to see if the addition of an interaction term improves the model  Multiple interaction terms can be included  Use a partial F-test for the simultaneous contribution of multiple variables to the model

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Simultaneous Contribution of Independent Variables  Use partial F-test for the simultaneous contribution of multiple variables to the model  Let m variables be an additional set of variables added simultaneously  To test the hypothesis that the set of m variables improves the model: (where F has m and n-k-1 d.f.)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Chapter Summary  Developed the multiple regression model  Tested the significance of the multiple regression model  Discussed adjusted r 2  Discussed using residual plots to check model assumptions In this chapter, we have

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap Chapter Summary  Tested individual regression coefficients  Tested portions of the regression model  Used dummy variables  Evaluated interaction effects In this chapter, we have