1. Marketing Research Aaker, Kumar, Day and Leone Tenth Edition
Instructor’s Presentation Slides

2. Correlation Analysis and Regression Analysis
Chapter Nineteen
Correlation Analysis and Regression Analysis

3. Definitions Correlation analysis Correlation coefficient
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)

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

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

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

7. Scatter Plots

8. Scatter Plots (Contd.)

9. Correlation Coefficient
Simple Correlation Coefficient
Pearson Product-moment Correlation Coefficient

10. Determining Sample Correlation Coefficient

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.

12. Partial Correlation Coefficient
Measure of association between two variables after controlling for the effects of one or more additional variables

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

14. Simple Linear Regression Model

15. Simple Linear Regression Model – A Graphical Illustration

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)

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
Where
Predicted value of Yi ,

18. Residual Value ei = yi - yi = yi - (bo + b1 xi)
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

19. Standard Error Mean Square Error Standard Error of b1

20. Testing the Significance of 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
H0: β1 = 0
Ha: β1 ≠ 0

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: β1 = 0
Ha: β1 ≠ 0
Decision Rule
Reject ho: β1 = 0 if α > p value

22. Sum of Squares obtained if we do not use x to predict y
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

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

24. Multiple Regression ε is the error or residual.
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.

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
b , bk are the partial regression coefficients.

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)

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

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

29. Stepwise Regression (Contd.)
Backward Elimination
Start with full regression equation
i.e. y = βo + β1x1 + β2 x 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

30. Residual Plots Heteroskedasticity Autocorrelation
Random distribution of residuals
Nonlinear pattern of residuals
Heteroskedasticity
Autocorrelation

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 cross-validation. The roles of the estimation and validation halves are then reversed, and the cross-validation is repeated

32. Regression with Dummy Variables
Yi = a + b1D1 + b2D2 + b3D3 + error
For rational buyer, Ŷi = a
For brand-loyal consumers, Ŷi = a + b1