Chapter 19 - Correlation analysis and regression analysis. In this chapter, the following content will be discussed: Definitions, regression analysis, correlation analysis, determining sample correlation coefficient, testing the significance of the correlation coefficient,...
1 Marketing Research Aaker, Kumar, Leone and Day Twelfth Edition Instructor’s Chapter Nineteen Correlation Analysis and Regression Analysis / Marketing Research 12th Edition 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) / Marketing Research 12th Edition 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 Marketing Research 12th Edition Correlation Analysis • Pearson correlation coefficient ▫ Measures the degree to which there is a linear association between two intervalscaled 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 / Marketing Research 12th Edition 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 p value / Marketing Research 12th 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 Marketing Research 12th 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) ( Ψι − Ψ) = + ( Ψι ( Ψι − Ψ) Ψι) Coefficient of Determination (r2) • Measure of regression model's ability to predict r2 = (SST SSE) / SST = SSM / SST = Explained Variation / Total Variation / Marketing Research 12th 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 / Marketing Research 12th 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 / Marketing Research 12th 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) / Marketing Research 12th Edition 27 Evaluating the Importance of Independent Variables • • Consider tvalue 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 Marketing Research 12th 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 Marketing Research 12th 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 / Marketing Research 12th Edition 30 Residual Plots Random distribution of residuals Heteroskedasticity / Nonlinear pattern of residuals Autocorrelation Marketing Research 12th 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 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 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 The sample is split into halves – estimation sample and validation sample for conducting crossvalidation. The roles of the estimation and validation halves are then reversed, and the crossvalidation is repeated Marketing Research 12th Edition 32 Regression with Dummy Variables Yi = a + b1D1 + b2D2 + b3D3 + error For rational buyer, Ŷi = a • • / For brandloyal consumers, Ŷi = a + b1 Marketing Research 12th Edition 33 End of Chapter Nineteen / Marketing Research 12th Edition ...2 Chapter Nineteen Correlation Analysis and Regression Analysis / Marketing Research 12th Edition Definitions • Correlation analysis ▫ Measures strength of the ... independent variables Multiple regression analysis Regression analysis that involves more than one independent variable Marketing Research 12th Edition Correlation Analysis • Pearson correlation coefficient ... εi = Error term that describes the effects on Yi of all factors other than value of Xi / Marketing Research 12th Edition 14 Simple Linear Regression Model / Marketing Research 12th Edition 15 Simple Linear Regression Model – A Graphical Illustration / Marketing Research