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1 Marketing Research Aaker, Kumar, Day and Leone Tenth Edition Instructor’s Presentation Slides 2 Chapter Nineteen Correlation Analysis and Regression Analysis http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 3 Definitions • Correlation analysis ▫ Measures strength of the relationship between two variables • Correlation coefficient ▫ Provides a measure of the degree to which there is an association between two variables (X and Y) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 4 Regression Analysis • Statistical technique that is used to relate two or more variables • Objective is to build a regression model or a prediction equation relating the dependent variable to one or more independent variables • The model can then be used to describe, predict, and control the variable of interest on the basis of the independent variables • Multiple regression analysis - Regression analysis that involves more than one independent variable http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 5 Correlation Analysis • Pearson correlation coefficient ▫ Measures the degree to which there is a linear association between two interval-scaled variables ▫ A positive correlation reflects a tendency for a high value in one variable to be associated with a high value in the second ▫ A negative correlation reflects an association between a high value in one variable and a low value in the second variable http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 6 Correlation Analysis (Contd.) • Population correlation (p) - If the database includes an entire population • Sample correlation (r) - If measure is based on a sample R lies between -1 < r < + 1 R = 0 ---> absence of linear association http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 7 Scatter Plots http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 8 Scatter Plots (Contd.) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 9 Correlation Coefficient Simple Correlation Coefficient Cov( x, y ) ( X i X ) * (Yi Y ) Pearson Product-moment Correlation Coefficient Xi X (Yi Y ) 1 rxy * * ( n 1) Sx Sy rxy http://www.drvkumar.com/mr10/ Cov xy Sx * S y Marketing Research 10th Edition 10 Determining Sample Correlation Coefficient http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 11 Testing the Significance of the Correlation Coefficient • Null hypothesis: Ho : p = 0 • Alternative hypothesis: Ha : p ≠ 0 • Test statistic Example: n = 6 and r = .70 At = .05 , n-2 = 4 degrees of freedom, Critical value of t = 2.78 Since 1.96<2.78, we fail to reject the null hypothesis. http://www.drvkumar.com/mr10/ t .70 62 1 0.70 2 1.96 Marketing Research 10th Edition 12 Partial Correlation Coefficient Measure of association between two variables after controlling for the effects of one or more additional variables rXY , Z http://www.drvkumar.com/mr10/ rXY rXZ * rYZ (1 r ) * (1 r ) 2 XZ 2 YZ Marketing Research 10th Edition 13 Regression Analysis Simple Linear Regression Model Yi = βo + β1xi + εi Where ▫ Y = Dependent variable ▫ X =Independent variable ▫ β o = Model parameter that represents mean value of dependent variable (Y) when the independent variable (X) is zero ▫ β1 = Model parameter that represents the slope that measures change in mean value of dependent variable associated with a one-unit increase in the independent variable ▫ εi = Error term that describes the effects on Yi of all factors other than value of Xi http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 14 Simple Linear Regression Model http://www.drvkumar.com/mr10/ Marketing Research 10th Edition Simple Linear Regression Model – A Graphical Illustration http://www.drvkumar.com/mr10/ 15 Marketing Research 10th Edition 16 Assumptions of the Simple Linear Regression Model • Error term is normally distributed (normality assumption) • Mean of error term is zero [E(εi) = 0) • Variance of error term is a constant and is independent of the values of X (constant variance assumption) • Error terms are independent of each other (independent assumption) • Values of the independent variable X are fixed (non-stochastic X) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 17 Estimating the Model Parameters • Calculate point estimate bo and b1 of unknown parameter βo and β1 • Obtain random sample and use this information from sample to estimate βo and β1 • Obtain a line of best "fit" for sample data points - least squares line Predicted value of Yi , Where http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 18 Residual Value • Difference between the actual and predicted values • Estimate of the error in the population ei = yi - yi = yi - (bo + b1 xi) • bo and b1 minimize the residual or error sum of squares (SSE) SSE = ei2 = ((yi - yi)2 = Σ [yi-(bo + b1xi)]2 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 19 Standard Error • Mean Square Error • Standard Error of b1 • Standard Error of b0 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 20 Testing the Significance of Independent Variables • Null Hypothesis ▫ There is no linear relationship between the independent & dependent variables H0: β1 = 0 • Alternative Hypothesis ▫ There is a linear relationship between the independent & dependent variables H a : β1 ≠ 0 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 21 Testing the Significance of Independent Variables (Contd.) • Test Statistic t = b1 - β1 sb1 • Degrees of Freedom V = n – 2 • Testing for a Type II Error Ho: Ha: β1 = 0 β1 ≠ 0 • Decision Rule Reject ho: β1 = 0 if α > p value http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 22 Sum of Squares SST Sum of squared prediction error that would be obtained if we do not use x to predict y SSE Sum of squared prediction error that is obtained when we use x to predict y SSM Reduction in sum of squared prediction error that has been accomplished using x in predicting y http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 23 Predicting the Dependent Variable • Dependent variable, yi = bo + bixi • Error of prediction is yi – y • Total variation (SST) = Explained variation (SSM) + Unexplained variation (SSE) (Yi - Y)2 = (Yi - Y)2 + (Yi – Yi)2 Coefficient of Determination (r2) • Measure of regression model's ability to predict r2 = (SST - SSE) / SST = SSM / SST = Explained Variation / Total Variation http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 24 Multiple Regression • A linear combination of predictor factors is used to predict the outcome or response factors • The general form of the multiple regression model is explained as: where β1 , β2, . . . , βk are regression coefficients associated with the independent variables X1, X2, . . . , Xk and ε is the error or residual. http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 25 Multiple Regression (Contd.) • The prediction equation in multiple regression analysis is Ŷ = α + b1X1 + b2X2 + …….+bkXk where Ŷ is the predicted Y score and b1 . . . , bk are the partial regression coefficients. http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 26 Partial Regression Coefficients Y = α + b1X1 + b2X2 + error • b 1 is the expected change in Y when X1 is changed by one unit, keeping X 2 constant or controlling for its effects. • b 2 is the expected change in Y for a unit change in X2, when X1 is held constant. • If X1 and X2 are each changed by one unit, the expected change in Y will be (b1 / b2) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 27 Evaluating the Importance of Independent Variables • Consider t-value for βi's • Use beta coefficients when independent variables are in different units of measurement Standardized βi = bi Standard deviation of xi Standard deviation of Y • Check for multicollinearity http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 28 Stepwise Regression • Predictor variables enter or are removed from the regression equation one at a time • Forward Addition ▫ Start with no predictor variables in regression equation i.e. y = βo + ε ▫ Add variables if they meet certain criteria in terms of F-ratio http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 29 Stepwise Regression (Contd.) • Backward Elimination ▫ Start with full regression equation i.e. y = βo + β1x1 + β2 x2 ...+ βr xr + ε ▫ Remove predictors based on F- ratio • Stepwise Method ▫ Forward addition method is combined with removal of predictors that no longer meet specified criteria at each step http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 30 Residual Plots Random distribution of residuals Heteroskedasticity http://www.drvkumar.com/mr10/ Nonlinear pattern of residuals Autocorrelation Marketing Research 10th Edition 31 Predictive Validity • Examines whether any model estimated with one set of data continues to hold good on comparable data not used in the estimation. • Estimation Methods 1. The data are split into the estimation sample (with more than half of the total sample) and the validation sample, and the coefficients from the two samples are compared. 2. The coefficients from the estimated model are applied to the data in the validation sample to predict the values of the dependent variable Yi in the validation sample, and then the model fit is assessed. 3. The sample is split into halves – estimation sample and validation sample for conducting cross-validation. The roles of the estimation and validation halves are then reversed, and the cross-validation is repeated http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 32 Regression with Dummy Variables Yi = a + b1D1 + b2D2 + b3D3 + error • For rational buyer, Ŷi = a • For brand-loyal consumers, Ŷi = a + b1 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition