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**Chapter 12 Multiple Regression**

Statistics for Business and Economics 7th Edition Chapter 12 Multiple Regression Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

Chapter Goals After completing this chapter, you should be able to: Apply multiple regression analysis to business decision-making 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**The Multiple Regression Model**

12.1 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 In this chapter we will always use a computer to obtain the regression slope coefficients and other regression summary measures. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Multiple Regression Equation**

(continued) Two variable model y Slope for variable x1 x2 Slope for variable x2 x1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Multiple Regression Model**

Two variable model y Sample observation yi < yi ei = (yi – yi) < x2i x2 x1i The best fit equation, y , is found by minimizing the sum of squared errors, e2 < x1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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. (The constant variance property is called homoscedasticity) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Standard Multiple Regression Assumptions**

(continued) The random error terms, εi , are not correlated with one another, so that It is not possible to find a set of numbers, c0, c1, , ck, such that (This is the property of no linear relation for the Xj’s) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

Pie Sales Example Week Pie Sales Price ($) Advertising ($100s) 1 350 5.50 3.3 2 460 7.50 3 8.00 3.0 4 430 4.5 5 6.80 6 380 4.0 7 4.50 8 470 6.40 3.7 9 450 7.00 3.5 10 490 5.00 11 340 7.20 12 300 7.90 3.2 13 440 5.90 14 15 2.7 Multiple regression equation: Sales = b0 + b1 (Price) + b2 (Advertising) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Estimating a Multiple Linear Regression Equation**

12.2 Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression Data / Data Analysis / Regression Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Multiple Regression Output**

Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 95% Upper 95% Intercept Price Advertising Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**The Multiple Regression Equation**

where Sales is in number of pies per week Price is in $ Advertising is in $100’s. b1 = : 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 b2 = : sales will increase, on average, by pies per week for each $100 increase in advertising, net of the effects of changes due to price Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Coefficient of Determination, R2**

12.3 Coefficient of Determination, R2 Reports the proportion of total variation in y explained by all x variables taken together This is the ratio of the explained variability to total sample variability Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Coefficient of Determination, R2**

(continued) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 95% Upper 95% Intercept Price Advertising 52.1% of the variation in pie sales is explained by the variation in price and advertising Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Estimation of Error Variance**

Consider the population regression model The unbiased estimate of the variance of the errors is where The square root of the variance, se , is called the standard error of the estimate Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Regression Statistics**

Standard Error, se Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 95% Upper 95% Intercept Price Advertising The magnitude of this value can be compared to the average y value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Adjusted Coefficient of Determination,**

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? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Adjusted Coefficient of Determination,**

(continued) Used to correct for the fact that adding non-relevant independent variables will still reduce the error sum of squares (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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Regression Statistics**

Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 95% Upper 95% Intercept Price Advertising 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Coefficient of Multiple Correlation**

The coefficient of multiple correlation is the correlation between the predicted value and the observed value of the dependent variable 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Evaluating Individual Regression Coefficients**

12.4 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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Evaluating Individual Regression Coefficients**

(continued) H0: βj = 0 (no linear relationship) H1: βj ≠ 0 (linear relationship does exist between xi and y) Test Statistic: (df = n – k – 1) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Evaluating Individual Regression Coefficients**

(continued) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Example: Evaluating Individual Regression Coefficients**

From Excel output: H0: βj = 0 H1: βj 0 Coefficients Standard Error t Stat P-value Price Advertising d.f. = = 12 = .05 t12, .025 = The test statistic for each variable falls in the rejection region (p-values < .05) Decision: Conclusion: Reject H0 for each variable a/2=.025 a/2=.025 There is evidence that both Price and Advertising affect pie sales at = .05 Reject H0 Do not reject H0 Reject H0 -tα/2 tα/2 2.1788 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Confidence Interval Estimate for the Slope**

Confidence interval limits for the population slope βj where t has (n – K – 1) d.f. Coefficients Standard Error Intercept Price Advertising 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: ± (2.1788)(10.832) So the interval is < β1 < Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Confidence Interval Estimate for the Slope**

(continued) Confidence interval for the population slope βi Coefficients Standard Error … Lower 95% Upper 95% Intercept Price Advertising 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Test on All Coefficients**

12.5 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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**F-Test for Overall Significance**

Test statistic: where F has k (numerator) and (n – K – 1) (denominator) degrees of freedom The decision rule is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**F-Test for Overall Significance**

(continued) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 ANOVA df SS MS F Significance F Regression 2 Residual 12 Total 14 Coefficients t Stat P-value Lower 95% Upper 95% Intercept Price Advertising With 2 and 12 degrees of freedom P-value for the F-Test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**F-Test for Overall Significance**

(continued) Test Statistic: Decision: Conclusion: H0: β1 = β2 = 0 H1: β1 and β2 not both zero = .05 df1= 2 df2 = 12 Critical Value: F = 3.885 Since F test statistic is in the rejection region (p-value < .05), reject H0 = .05 F There is evidence that at least one independent variable affects Y Do not reject H0 Reject H0 F.05 = 3.885 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

Prediction 12.6 Given a population regression model 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 yn+1 is 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. ^ Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Using The Equation to Make Predictions**

Predict sales for a week in which the selling price is $5.50 and advertising is $350: Note that Advertising is in $100’s, so $350 means that X2 = 3.5 Predicted sales is pies Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Nonlinear Regression Models**

12.7 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 The second independent variable is the square of the first variable Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Quadratic Regression Model**

Model form: 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Linear vs. Nonlinear Fit**

Y Y X X X X residuals residuals Linear fit does not give random residuals Nonlinear fit gives random residuals Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Quadratic Regression Model**

Quadratic models may be considered when the scatter diagram takes on one of the following shapes: Y Y Y Y X1 X1 X1 X1 β1 < 0 β1 > 0 β1 < 0 β1 > 0 β2 > 0 β2 > 0 β2 < 0 β2 < 0 β1 = the coefficient of the linear term β2 = the coefficient of the squared term Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Testing for Significance: Quadratic Effect**

Testing the Quadratic Effect Compare the linear regression estimate with quadratic regression estimate Hypotheses (The quadratic term does not improve the model) (The quadratic term improves the model) H0: β2 = 0 H1: β2 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Testing for Significance: Quadratic Effect**

(continued) Testing the Quadratic Effect Hypotheses (The quadratic term does not improve the model) (The quadratic term improves the model) The test statistic is H0: β2 = 0 H1: β2 0 where: b2 = squared term slope coefficient β2 = hypothesized slope (zero) Sb = standard error of the slope 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Example: Quadratic Model**

Purity increases as filter time increases: Purity Filter Time 3 1 7 2 8 15 5 22 33 40 10 54 12 67 13 70 14 78 85 87 16 99 17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Example: Quadratic Model**

(continued) Simple regression results: y = Time ^ Coefficients Standard Error t Stat P-value Intercept Time 2.078E-10 Regression Statistics R Square Adjusted R Square Standard Error F Significance F 2.0778E-10 t statistic, F statistic, and R2 are all high, but the residuals are not random: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Example: Quadratic Model**

(continued) Quadratic regression results: y = Time (Time)2 ^ Coefficients Standard Error t Stat P-value Intercept Time Time-squared 1.165E-05 Regression Statistics R Square Adjusted R Square Standard Error F Significance F 2.368E-13 The quadratic term is significant and improves the model: R2 is higher and se is lower, residuals are now random Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**The Log Transformation**

The Multiplicative Model: Original multiplicative model Transformed multiplicative model Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Interpretation of coefficients**

For the multiplicative model: 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

Dummy Variables 12.8 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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Dummy Variable Example**

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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Dummy Variable Example**

(continued) Holiday No Holiday Different intercept Same slope y (sales) If H0: β2 = 0 is rejected, then “Holiday” has a significant effect on pie sales b0 + b2 Holiday (x2 = 1) b0 No Holiday (x2 = 0) x1 (Price) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Interpreting the Dummy Variable Coefficient**

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 b2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same price Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

Effect of Interaction Given: 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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Interaction Example Suppose x2 is a dummy variable and the estimated regression equation is y 12 x2 = 1: y = 1 + 2x1 + 3(1) + 4x1(1) = 4 + 6x1 ^ 8 4 x2 = 0: y = 1 + 2x1 + 3(0) + 4x1(0) = 1 + 2x1 ^ x1 0.5 1 1.5 Slopes are different if the effect of x1 on y depends on x2 value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 If we reject the null hypothesis we conclude that there is a difference in the slope coefficient for the two subgroups Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Multiple Regression Assumptions**

12.9 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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**Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall**

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 Explained dummy variables Evaluated interaction effects Discussed using residual plots to check model assumptions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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