Marketing Research

1. Marketing Research Aaker, Kumar, Day Seventh Edition Instructor’s Presentation Slides

2. Chapter Nineteen Correlation Analysis and Regression Analysis

3. Correlation Analysis • Measures the strength of the relationship between two or more variables • Correlation • Measures the degree to which there is an association between two intervally scaled variables Aaker, Kumar, Day

4. Correlation Analysis (Contd.) Positive Correlation • Tendency for a high value of one variable to be associated with a high value in the second Population Correlation () • Database includes entire population Aaker, Kumar, Day

5. Correlation Analysis (Contd.) Sample Correlation (r) • Measure is based on a sample • Reflects tendency for points to cluster systematically about a straight line or falling from left to right on a scatter diagram • R lies between -1 < r < + 1 • R = o ---> absence of linear association Aaker, Kumar, Day

6. Simple Correlation Coefficient • Plot points on scatter diagram cov (x,y) =σ(xi-x) * (yi - y) • Covariance between two variables gives measure of association • Divide the association expression by the sample size To ensure that the measure does not increase simply by increasing the sample size Aaker, Kumar, Day

7. Simple Correlation Coefficient (Contd.) • Divide the measure by the sample standard deviation of x and y • Sample coefficient of correlation, R is independent of sample size and units of measurement • It lies between -1 and +1 • It does not imply any causal relationship between the variables Aaker, Kumar, Day

8. Testing the Significance of the Correlation Coefficient • Null hypothesis: Ho equal to 0 • Alternative hypothesis: Ha not equal to 0 • Test statistic t = r  (n - 2) / (1 - r2) Aaker, Kumar, Day

9. Partial Correlation Coefficient • Provides a measure of association between two variables after controlling for the effects of one or more additional variables Aaker, Kumar, Day

10. Regression Analysis • Used to understand the nature of the relationship between two or more variables • A dependent or response variable (Y) is related to one or more independent or predictor variables (Xs) • Object is to build a regression model relating dependent variable to one or more independent variables • Model can be used to describe, predict, and control variable of interest on the basis of independent variables Aaker, Kumar, Day

11. Simple Linear Regression Yi = βo + β1 xi + εi Where Y • Dependent variable X • Independent variable βo • Model parameter • Mean value of dependent variable (Y) when the independent variable (X) is zero Aaker, Kumar, Day

12. Simple Linear Regression (Contd.) β1 • Model parameter • 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 Aaker, Kumar, Day

13. Assumption of the Regression Model • Error term is normally distributed (normality assumption) • Mean of error term is zero (E{Ei} = 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 is fixed (non-stochastic X) Aaker, Kumar, Day

14. 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 Yi = bo + b1 xi Aaker, Kumar, Day

15. Values of Least Squares Estimates bo and b1 b1 = n xiyi - (xi)(yi) n xi2 - (xi)2 bo = y - bi x Where y = yi ; x = xi n n Aaker, Kumar, Day

16. 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 sums of squares (SSE) SSE = ei2 = ((yi - yi)2 = Σ [yi-(bo + b1xi)]2 Aaker, Kumar, Day

17. Testing the Significance of the Independent Variables Null Hypothesis • There is no linear relationship between the independent & dependent variables Alternative Hypothesis • There is a linear relationship between the independent & dependent variables Aaker, Kumar, Day

18. Testing the Significance of the Independent Variables (Contd.) • Test Statistic t = b1 - β1 sb1 • Degrees of Freedom Vx = n - 2 • Testing for a Type II Error Ho: β1 equal to 0 Ha: β1 not equal to 0 • Decision Rule: Reject ho: β1 = 0 if α > p value Aaker, Kumar, Day

19. Predicting the Dependent Variable yi = bo + bixi • Error of prediction is yi - yi (yi - y)2 = σ(yi - y)2 + σ(yi - yi)2 • Total variation (SST) = Explained variation (SSM) + unexplained variation (SSE) Aaker, Kumar, Day

20. Predicting the Dependent Variable (Contd.) 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 Aaker, Kumar, Day

21. Coefficient of Determination (R2) • Measure of regression model's ability to predict R2 = SST - SSE SST = SSM SST = Explained Variation Total Variation Aaker, Kumar, Day

22. Multiple Linear Regression • A linear combination of predictor factors is used to predict the outcome or response factors • Involves computation of a multiple linear regression equation • More than one independent variable is included in a single linear regression model Aaker, Kumar, Day

23. Evaluating the Importance of Independent Variables • Which of the independent variables has the greatest influence on the dependent variable? • Consider t-value for βi's Ho : βi = 0 • If null hypothesis is true, bi (a non-zero estimate) was simply a sampling phenomenon Aaker, Kumar, Day

24. Examine the Size of the Regression Coefficients • Use beta coefficients when independent variables are in different units of measurement Standardized βi = bi (Standard deviation of xi) (Standard deviation of Y) • Compare β coefficients with the largest value representing the variable with the strongest impact on the dependent variable Aaker, Kumar, Day

25. Multicollinearity • Correlations among predictor variables • Discovered by examining the correlates among the X variables Selecting Predictor Variables • Include only those variables that account for most of the variation in the dependent variable Aaker, Kumar, Day

26. 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 Aaker, Kumar, Day

27. 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 prediction that no longer meet specified criteria at each stop Aaker, Kumar, Day