Correlation Analysis and Regression Analysis

Marketing ResearchAaker, Kumar, Leone and Day Twelfth EditionInstructor’s Presentation Slides1Chapter Nineteen2Correlation Analysis and Regression AnalysisMarketing Research 12th Edition /DefinitionsCorrelation analysis Measures strength of the relationship between two variablesCorrelation coefficient Provides a measure of the degree to which there is an association between two variables (X and Y)3Marketing Research 12th Edition /Regression AnalysisStatistical 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 variablesMultiple regression analysis - Regression analysis that involves more than one independent variable4Marketing Research 12th Edition /Correlation AnalysisPearson correlation coefficient Measures the degree to which there is a linear association between two interval-scaled variablesA positive correlation reflects a tendency for a high value in one variable to be associated with a high value in the secondA negative correlation reflects an association between a high value in one variable and a low value in the second variable 5Marketing Research 12th Edition /Correlation Analysis (Contd.)Population correlation (p) - If the database includes an entire populationSample correlation (r) - If measure is based on a sample6R lies between -1 absence of linear associationMarketing Research 12th Edition /Scatter Plots7Marketing Research 12th Edition /Scatter Plots (Contd.)8Marketing Research 12th Edition /Correlation Coefficient9Simple Correlation CoefficientPearson Product-moment Correlation CoefficientMarketing Research 12th Edition /Determining Sample Correlation Coefficient10Marketing Research 12th Edition /Testing the Significance of the Correlation CoefficientNull hypothesis: Ho : p = 0Alternative hypothesis: Ha : p ≠ 0Test statistic 11Example: n = 6 and r = .70At  = .05 , n-2 = 4 degrees of freedom, Critical value of t = 2.78Since 1.96 p valueMarketing Research 12th Edition /Sum of SquaresSST Sum of squared prediction error that would be obtained if we do not use x to predict ySSE Sum of squared prediction error that is obtained when we use x to predict ySSM Reduction in sum of squared prediction error that has been accomplished using x in predicting y22Marketing Research 12th Edition /Predicting the Dependent VariableDependent variable, yi = bo + bixi Error of prediction is yi – yTotal variation (SST) = Explained variation (SSM) + Unexplained variation (SSE) 23(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 VariationMarketing Research 12th Edition /Multiple RegressionA linear combination of predictor factors is used to predict the outcome or response factorsThe general form of the multiple regression model is explained as:24where 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 /Multiple Regression (Contd.)The prediction equation in multiple regression analysis is25Ŷ = α + b1X1 + b2X2 + .+bkXk where Ŷ is the predicted Y score and b1 . . . , bk are the partial regression coefficients.Marketing Research 12th Edition /Partial Regression Coefficientsb 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)26Y = α + b1X1 + b2X2 + errorMarketing Research 12th Edition /Evaluating the Importance of Independent VariablesConsider t-value for βi'sUse beta coefficients when independent variables are in different units of measurement Standardized βi = bi Standard deviation of xi Standard deviation of YCheck for multicollinearity27Marketing Research 12th Edition /Stepwise RegressionPredictor variables enter or are removed from the regression equation one at a timeForward AdditionStart with no predictor variables in regression equation i.e. y = βo + εAdd variables if they meet certain criteria in terms of F-ratio28Marketing Research 12th Edition /Stepwise Regression (Contd.)Backward EliminationStart with full regression equationi.e. y = βo + β1x1 + β2 x2 ...+ βr xr + εRemove predictors based on F- ratioStepwise MethodForward addition method is combined with removal of predictors that no longer meet specified criteria at each step29Marketing Research 12th Edition /Residual Plots30Random distribution of residualsNonlinear pattern of residualsHeteroskedasticityAutocorrelationMarketing Research 12th Edition /Predictive ValidityExamines whether any model estimated with one set of data continues to hold good on comparable data not used in the estimation.Estimation MethodsThe 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 cross-validation. The roles of the estimation and validation halves are then reversed, and the cross-validation is repeated31Marketing Research 12th Edition /Regression with Dummy Variables32Yi = a + b1D1 + b2D2 + b3D3 + error For rational buyer, Ŷi = a For brand-loyal consumers, Ŷi = a + b1Marketing Research 12th Edition /33End of Chapter NineteenMarketing Research 12th Edition /