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Statistics for Business and Economics 6th Edition Chapter 13 Multiple Regression Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-1 Chapter Goals After completing this chapter, you should be able to: Apply multiple regression analysis to business decisionmaking situations Analyze and interpret the computer output for a multiple regression model Perform a hypothesis test for all regression coefficients or for a subset of coefficients Fit and interpret nonlinear regression models Incorporate qualitative variables into the regression model by using dummy variables Discuss model specification and analyze residuals Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-2 The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi) Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error Y β0 β1X1 β2 X2 βk Xk ε Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-3 Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Multiple regression equation with k independent variables: Estimated (or predicted) value of y Estimated intercept Estimated slope coefficients yˆ i b0 b1x1i b2 x 2i bk xki In this chapter we will always use a computer to obtain the regression slope coefficients and other regression summary measures. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-4 Multiple Regression Equation (continued) Two variable model y yˆ b0 b1x1 b2 x 2 x2 x1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-5 Standard Multiple Regression Assumptions The values xi and the error terms εi are independent The error terms are random variables with mean 0 and a constant variance, 2. E[ε i ] 0 and E[ε i2 ] σ 2 for (i 1, ,n) (The constant variance property is called homoscedasticity) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-6 Standard Multiple Regression Assumptions (continued) The random error terms, εi , are not correlated with one another, so that E[ε iε j ] 0 for all i j It is not possible to find a set of numbers, c0, c1, . . . , ck, such that c 0 c1x1i c 2 x 2i cK xKi 0 (This is the property of no linear relation for the Xj’s) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-7 Example: 2 Independent Variables 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 ($100’s) Data are collected for 15 weeks Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-8 Pie Sales Example Week Pie Sales Price ($) Advertising ($100s) 1 350 5.50 3.3 2 460 7.50 3.3 3 350 8.00 3.0 4 430 8.00 4.5 5 350 6.80 3.0 6 380 7.50 4.0 7 430 4.50 3.0 8 470 6.40 3.7 9 450 7.00 3.5 10 490 5.00 4.0 11 340 7.20 3.5 12 300 7.90 3.2 13 440 5.90 4.0 14 450 5.00 3.5 15 300 7.00 2.7 Multiple regression equation: Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Sales = b0 + b1 (Price) + b2 (Advertising) Chap 13-9 Estimating a Multiple Linear Regression Equation Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression Excel: Tools / Data Analysis... / Regression PHStat: PHStat / Regression / Multiple Regression… Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-10 Multiple Regression Output Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations ANOVA Regression Sales 306.526 - 24.975(Pri ce) 74.131(Advertising) 15 df SS MS F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-11 The Multiple Regression Equation Sales 306.526 - 24.975(Pri ce) 74.131(Advertising) where Sales is in number of pies per week Price is in $ Advertising is in $100’s. b1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. b2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to price Chap 13-12 Coefficient of Determination, R2 Reports the proportion of total variation in y explained by all x variables taken together SSR regression sum of squares R SST total sum of squares 2 This is the ratio of the explained variability to total sample variability Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-13 Coefficient of Determination, R2 (continued) Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error Regression 52.1% of the variation in pie sales is explained by the variation in price and advertising 47.46341 Observations ANOVA SSR 29460.0 R .52148 SST 56493.3 2 15 df SS MS F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-14 Estimation of Error Variance Consider the population regression model Yi β0 β1x1i β2 x 2i βK xKi ε i The unbiased estimate of the variance of the errors is n s2e 2 e i SSE n K 1 n K 1 i1 where ei y i yˆ i The square root of the variance, se , is called the standard error of the estimate Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-15 Standard Error, se Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error Regression The magnitude of this value can be compared to the average y value 47.46341 Observations ANOVA se 47.463 15 df SS MS F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-16 Adjusted Coefficient of Determination, R 2 R2 never decreases when a new X variable is added to the model, even if the new variable is not an important predictor variable 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 explanatory power to offset the loss of one degree of freedom? Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-17 Adjusted Coefficient of Determination, R 2 (continued) Used to correct for the fact that adding non-relevant independent variables will still reduce the error sum of squares SSE / (n K 1) 2 R 1 SST / (n 1) (where n = sample size, K = number of independent variables) Adjusted R2 provides a better comparison between multiple regression models with different numbers of independent variables Penalize excessive use of unimportant independent variables Smaller than R2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-18 R Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations ANOVA Regression 15 df 2 R .44172 2 44.2% 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 SS MS F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-19 Coefficient of Multiple Correlation The coefficient of multiple correlation is the correlation between the predicted value and the observed value of the dependent variable R r(yˆ , y) R2 Is the square root of the multiple coefficient of determination Used as another measure of the strength of the linear relationship between the dependent variable and the independent variables Comparable to the correlation between Y and X in simple regression Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-20 Evaluating Individual Regression Coefficients Use t-tests for individual coefficients Shows if a specific independent variable is conditionally important Hypotheses: H0: βj = 0 (no linear relationship) H1: βj ≠ 0 (linear relationship does exist between xj and y) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-21 Evaluating Individual Regression Coefficients (continued) H0: βj = 0 (no linear relationship) H1: βj ≠ 0 (linear relationship does exist between xi and y) Test Statistic: t bj 0 (df = n – k – 1) Sb j Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-22 Evaluating Individual Regression Coefficients (continued) Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations ANOVA Regression 15 df t-value for Price is t = -2.306, with p-value .0398 t-value for Advertising is t = 2.855, with p-value .0145 SS MS F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-23 Example: Evaluating Individual Regression Coefficients From Excel output: H0: βj = 0 H1: βj 0 Price Advertising d.f. = 15-2-1 = 12 = .05 Coefficients Standard Error t Stat P-value -24.97509 10.83213 -2.30565 0.03979 74.13096 25.96732 2.85478 0.01449 The test statistic for each variable falls in the rejection region (p-values < .05) t12, .025 = 2.1788 Decision: /2=.025 /2=.025 Reject H0 for each variable Conclusion: Reject H0 Do not reject H0 -tα/2 -2.1788 0 Reject H0 tα/2 2.1788 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. There is evidence that both Price and Advertising affect pie sales at = .05 Chap 13-24 Confidence Interval Estimate for the Slope Confidence interval limits for the population slope βj b j tnK 1,α/2Sb j Coefficients Standard Error Intercept 306.52619 114.25389 Price -24.97509 10.83213 74.13096 25.96732 Advertising where t has (n – K – 1) d.f. Here, t has (15 – 2 – 1) = 12 d.f. Example: Form a 95% confidence interval for the effect of changes in price (x1) on pie sales: -24.975 ± (2.1788)(10.832) So the interval is Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. -48.576 < β1 < -1.374 Chap 13-25 Confidence Interval Estimate for the Slope (continued) Confidence interval for the population slope βi Coefficients Standard Error … Intercept 306.52619 114.25389 … 57.58835 555.46404 Price -24.97509 10.83213 … -48.57626 -1.37392 74.13096 25.96732 … 17.55303 130.70888 Advertising Lower 95% Upper 95% Example: Excel output also reports these interval endpoints: Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-26 Test on All Coefficients 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: H0: β1 = β2 = … = βk = 0 (no linear relationship) H1: at least one βi ≠ 0 (at least one independent variable affects Y) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-27 F-Test for Overall Significance Test statistic: MSR SSR/K F 2 se SSE/(n K 1) where F has k (numerator) and (n – K – 1) (denominator) degrees of freedom The decision rule is Reject H0 if F Fk,nK 1,α Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-28 F-Test for Overall Significance (continued) Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations ANOVA Regression 15 df MSR 14730.0 F 6.5386 MSE 2252.8 With 2 and 12 degrees of freedom SS MS P-value for the F-Test F 2 29460.027 14730.013 Residual 12 27033.306 2252.776 Total 14 56493.333 Coefficients Standard Error Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404 Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888 Advertising Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. t Stat 6.53861 Significance F P-value 0.01201 Lower 95% Upper 95% Chap 13-29 F-Test for Overall Significance (continued) H0: β1 = β2 = 0 H1: β1 and β2 not both zero = .05 df1= 2 df2 = 12 Critical Value: = .05 Do not reject H0 Reject H0 MSR F 6.5386 MSE Decision: Since F test statistic is in the rejection region (pvalue < .05), reject H0 F = 3.885 0 Test Statistic: Conclusion: F F.05 = 3.885 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. There is evidence that at least one independent variable affects Y Chap 13-30 Tests on a Subset of Regression Coefficients Consider a multiple regression model involving variables xj and zj , and the null hypothesis that the z variable coefficients are all zero: yi β0 β1x1i βK xKi α1z1i αr zri εi H0 : α1 α2 αr 0 H1 : at least one of α j 0 (j 1,..., r) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-31 Tests on a Subset of Regression Coefficients (continued) Goal: compare the error sum of squares for the complete model with the error sum of squares for the restricted model First run a regression for the complete model and obtain SSE Next run a restricted regression that excludes the z variables (the number of variables excluded is r) and obtain the restricted error sum of squares SSE(r) Compute the F statistic and apply the decision rule for a significance level ( SSE(r) SSE ) / r Reject H0 if F Fr,nK r 1,α 2 se Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-32 Prediction Given a population regression model yi β0 β1x1i β2 x 2i βK xKi εi (i 1,2,,n) then given a new observation of a data point (x1,n+1, x 2,n+1, . . . , x K,n+1) the best linear unbiased forecast of y^n+1 is yˆ n1 b0 b1x1,n1 b2 x 2,n1 bK xK,n1 It is risky to forecast for new X values outside the range of the data used to estimate the model coefficients, because we do not have data to support that the linear model extends beyond the observed range. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-33 Using The Equation to Make Predictions Predict sales for a week in which the selling price is $5.50 and advertising is $350: Sales 306.526 - 24.975(Pri ce) 74.131(Adv ertising) 306.526 - 24.975 (5.50) 74.131 (3.5) 428.62 Predicted sales is 428.62 pies Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Note that Advertising is in $100’s, so $350 means that X2 = 3.5 Chap 13-34 Predictions in PHStat PHStat | regression | multiple regression … Check the “confidence and prediction interval estimates” box Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-35 Predictions in PHStat (continued) Input values < Predicted y value < Confidence interval for the mean y value, given these x’s < Prediction interval for an individual y value, given these x’s Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-36 Residuals in Multiple Regression Two variable model y yˆ b0 b1x1 b2 x 2 < Residual = ei = (yi – yi) yi Sample observation < yi x2i x2 x1i x1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-37 Nonlinear Regression Models The relationship between the dependent variable and an independent variable may not be linear Can review the scatter diagram to check for non-linear relationships Example: Quadratic model Y β0 β1X1 β2 X12 ε The second independent variable is the square of the first variable Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-38 Quadratic Regression Model Model form: Yi β0 β1X1i β 2 X ε i 2 1i where: β0 = Y intercept β1 = regression coefficient for linear effect of X on Y β2 = regression coefficient for quadratic effect on Y εi = random error in Y for observation i Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-39 Linear vs. Nonlinear Fit Y Y X X Linear fit does not give random residuals Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. residuals residuals X X Nonlinear fit gives random residuals Chap 13-40 Quadratic Regression Model Yi β0 β1X1i β 2 X ε i 2 1i Quadratic models may be considered when the scatter diagram takes on one of the following shapes: Y Y β1 < 0 β2 > 0 X1 Y β1 > 0 β2 > 0 X1 Y β1 < 0 β2 < 0 X1 β1 > 0 β2 < 0 X1 β1 = the coefficient of the linear term β2 = the coefficient of the squared term Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-41 Testing for Significance: Quadratic Effect Testing the Quadratic Effect Compare the linear regression estimate yˆ b0 b1x1 with quadratic regression estimate yˆ b0 b1x1 b2 x12 Hypotheses H0: β2 = 0 (The quadratic term does not improve the model) H1: β2 0 (The quadratic term improves the model) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-42 Testing for Significance: Quadratic Effect (continued) Testing the Quadratic Effect Hypotheses H0: β2 = 0 (The quadratic term does not improve the model) H1: β2 0 (The quadratic term improves the model) The test statistic is b2 β2 t sb 2 d.f. n 3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. where: b2 = squared term slope coefficient β2 = hypothesized slope (zero) Sb = standard error of the slope 2 Chap 13-43 Testing for Significance: Quadratic Effect (continued) Testing the Quadratic Effect Compare R2 from simple regression to R2 from the quadratic model If R2 from the quadratic model is larger than R2 from the simple model, then the quadratic model is a better model Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-44 Example: Quadratic Model 3 1 7 2 8 3 15 5 22 7 33 8 40 10 54 12 67 13 70 14 78 15 85 15 87 16 99 17 Purity increases as filter time increases: Purity vs. Time 100 80 Purity Purity Filter Time 60 40 20 0 0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 5 10 15 20 Time Chap 13-45 Example: Quadratic Model (continued) Simple regression results: y^ = -11.283 + 5.985 Time Coefficients Standard Error -11.28267 3.46805 -3.25332 0.00691 5.98520 0.30966 19.32819 2.078E-10 Intercept Time t Stat t statistic, F statistic, and R2 are all high, but the residuals are not random: P-value Regression Statistics 0.96888 Adjusted R Square 0.96628 Standard Error 6.15997 373.57904 Time Residual Plot Significance F 10 2.0778E-10 Residuals R Square F 5 0 -5 0 5 10 15 20 -10 Time Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-46 Example: Quadratic Model (continued) Quadratic regression results: y^ = 1.539 + 1.565 Time + 0.245 (Time)2 Coefficients Standard Error Intercept 1.53870 2.24465 0.68550 0.50722 Time 1.56496 0.60179 2.60052 0.02467 Time-squared 0.24516 0.03258 7.52406 1.165E-05 Time Residual Plot P-value 10 Residuals t Stat 5 0 -5 R Square 0.99494 Adjusted R Square 0.99402 Standard Error 2.59513 F 1080.7330 5 2.368E-13 The quadratic term is significant and improves the model: R2 is higher and se is lower, residuals are now random Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. 10 15 20 Time Significance F Time-squared Residual Plot 10 Residuals Regression Statistics 0 5 0 -5 0 100 200 300 400 Time-squared Chap 13-47 The Log Transformation The Multiplicative Model: Original multiplicative model Y β0 X1β1 Xβ22 ε Transformed multiplicative model log(Y) log(β0 ) β1log(X1 ) β2log(X 2 ) log(ε) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-48 Interpretation of coefficients For the multiplicative model: log Yi log β0 β1 log X1i log ε i When both dependent and independent variables are logged: The coefficient of the independent variable Xk can be interpreted as a 1 percent change in Xk leads to an estimated bk percentage change in the average value of Y bk is the elasticity of Y with respect to a change in Xk Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-49 Dummy Variables A dummy variable is a categorical independent variable with two levels: yes or no, on or off, male or female recorded as 0 or 1 Regression intercepts are different if the variable is significant Assumes equal slopes for other variables If more than two levels, the number of dummy variables needed is (number of levels - 1) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-50 Dummy Variable Example yˆ b0 b1x1 b2 x 2 Let: y = Pie Sales x1 = Price x2 = Holiday (X2 = 1 if a holiday occurred during the week) (X2 = 0 if there was no holiday that week) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-51 Dummy Variable Example (continued) yˆ b0 b1x1 b 2 (1) (b 0 b 2 ) b1x1 Holiday yˆ b0 b1x1 b 2 (0) No Holiday y (sales) b0 + b2 b0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. b1 x 1 b0 Different intercept Same slope If H0: β2 = 0 is rejected, then “Holiday” has a significant effect on pie sales x1 (Price) Chap 13-52 Interpreting the Dummy Variable Coefficient Example: Sales 300 - 30(Price) 15(Holiday) Sales: number of pies sold per week Price: pie price in $ 1 If a holiday occurred during the week Holiday: 0 If no holiday occurred b2 = 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 Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-53 Interaction Between Explanatory Variables Hypothesizes interaction between pairs of x variables Response to one x variable may vary at different levels of another x variable Contains two-way cross product terms yˆ b0 b1x1 b 2 x 2 b3 x 3 b0 b1x1 b 2 x 2 b3 (x1x 2 ) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-54 Effect of Interaction Given: Y β β X (β β X )X 0 2 2 1 3 2 1 β0 β1X1 β2 X 2 β3 X1X 2 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 changes Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-55 Interaction Example Suppose x2 is a dummy variable and the estimated regression equation is yˆ 1 2x1 3x2 4x1x2 y 12 x2 = 1: ^y = 1 + 2x + 3(1) + 4x (1) = 4 + 6x 1 1 1 8 4 x2 = 0: ^y = 1 + 2x + 3(0) + 4x (0) = 1 + 2x 1 1 1 0 0 0.5 1 1.5 x1 Slopes are different if the effect of x1 on y depends on x2 value Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-56 Significance of Interaction Term The coefficient b3 is an estimate of the difference in the coefficient of x1 when x2 = 1 compared to when x2 = 0 The t statistic for b3 can be used to test the hypothesis H0 : β3 0 | β1 0, β 2 0 H1 : β3 0 | β1 0, β 2 0 If we reject the null hypothesis we conclude that there is a difference in the slope coefficient for the two subgroups Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-57 Multiple Regression Assumptions Errors (residuals) from the regression model: < ei = (yi – yi) Assumptions: The errors are normally distributed Errors have a constant variance The model errors are independent Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-58 Analysis of Residuals in Multiple Regression These residual plots are used in multiple regression: < Residuals vs. yi Residuals vs. x1i Residuals vs. x2i Residuals vs. time (if time series data) Use the residual plots to check for violations of regression assumptions Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-59 Chapter Summary Developed the multiple regression model Tested the significance of the multiple regression model Discussed adjusted R2 ( R2 ) Tested individual regression coefficients Tested portions of the regression model Used quadratic terms and log transformations in regression models Used dummy variables Evaluated interaction effects Discussed using residual plots to check model assumptions Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 13-60