Marketing Research Essentials

							Essentials of Marketing
       Research
       Kumar, Aaker, Day
 Instructor’s Presentation Slides
  Essentials of Marketing Research   Kumar, Aaker, Day
      Chapter Seventeen
 Correlation Analysis and
   Regression Analysis

Essentials of Marketing Research   Kumar, Aaker, Day
                   Correlation Analysis
X and Y are random variables that are jointly normally distributed and, in addition,
   that the obtained data consists of a random sample of n independent pairs of
   observations (X1, Y1), (X2, Y2), . . . . (Xn, Yn) from an underlying bi-variate
   normal population.

                                              Y = f(X)

                                          any relationships?
Relationships – 3 goals                   if any, how strong?
                                          nature or form

Two of the most powerful and versatile approaches for investigating variable
  relationships are correlation analysis and regression analysis.




                Essentials of Marketing Research           Kumar, Aaker, Day
Product moment correlation coefficient - - Pearson Rho, Y - - computation
Interpretation of Y
Not designed to measure relationships other than linear symetric
Assumptions underlying Y
Continuous, distributions are of the same shape range restriction problem
Significance evaluation - -          statistical
                           --        substantive (Y2)
Contingency correlation, point bi-serial, Spearman partial correlation coefficients.




                Essentials of Marketing Research      Kumar, Aaker, Day
          Correlation Analysis
 Measures  the strength of the relationship
  between two or more variables
 Correlation
      Measuresthe degree to which there is an association
      between two internally scaled variables




       Essentials of Marketing Research   Kumar, Aaker, Day
  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


       Essentials of Marketing Research     Kumar, Aaker, Day
    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


          Essentials of Marketing Research   Kumar, Aaker, Day
         Simple Correlation Coefficient

  Cov ( x, y )          (X                i    X ) * (Yi  Y )

          1        X i  X (Yi  Y )
rxy            *        *
        (n  1)       Sx      Sy

                                    Cov xy
             rxy 
                                  Sx * S y

         Essentials of Marketing Research            Kumar, Aaker, Day
Computation of Correlation Coefficient




    Essentials of Marketing Research   Kumar, Aaker, Day
  Correlation Analysis (contd)
• This is called Pearson product - moment
  correlation coefficient and lies between
  -1 and +1.

• Correlation coefficient is NOT an
  indicator of causal relationship
  between variables

      Essentials of Marketing Research   Kumar, Aaker, Day
    Correlation Analysis (contd)

  Partial correlation coefficient - measure
  of association between two variables
  after controlling for the effects of one or
  more additional variables

                             rXY  rXZ * rYZ
rXY ,Z 
                    (1  r ) * (1  r )
                                      2
                                      XZ
                                                                2
                                                               YZ

       Essentials of Marketing Research    Kumar, Aaker, Day
Ranking for Cereal in Two Countries




   Essentials of Marketing Research   Kumar, Aaker, Day
Computation of Spearman Correlation
            Coefficient




      Essentials of Marketing Research   Kumar, Aaker, Day
        Correlation Analysis (contd)

Spearman Rank Correlation Coefficient - Index of
correlation between two rank-order variables.

                           2
                     6 Di 
                                                     
                                              6 Di2 
                                      rs  1   i 2 
                                               
                                               n n 1


            rs  1    i    
                      n n 1
                          2
                                                             
Where Di is the difference between ranks associated
with a brand and n is the number of brands evaluated.
           Essentials of Marketing Research               Kumar, Aaker, Day
  Testing the Significance of the
     Correlation Coefficient
 Null hypothesis:

                Ho : p equal to 0
 Alternative         hypothesis:
                Ha : p not equal to 0
 Test   statistic
                t = r  (n - 2) / (1 - r2)

         Essentials of Marketing Research   Kumar, Aaker, Day
                 Consider the Store example


In our example, n = 6 and r = .70. Hence,




If the test is done at  = .05 with n-2 = 4 degrees of freedom, then the critical
    value of t can be obtained from the tables to be 2.78. Since 1.96<2.78, we
    fail to reject the null hypothesis.



                Essentials of Marketing Research       Kumar, Aaker, Day
            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
         Essentials of Marketing Research   Kumar, Aaker, Day
        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

          Essentials of Marketing Research   Kumar, Aaker, Day
       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



         Essentials of Marketing Research   Kumar, Aaker, Day
 Assumptions of the Regression
           Model
 Errorterm 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)
          Essentials of Marketing Research   Kumar, Aaker, Day
Estimating the Model Parameters
 Calculatepoint 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
       Essentials of Marketing Research    Kumar, Aaker, Day
   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
    Essentials of Marketing Research          Kumar, Aaker, Day
                    Problem for Regression
 Y          X
Sales   Advertising
 3           7
 8          13
 17         13
 4          11
 15         16
 7           6




            Essentials of Marketing Research   Kumar, Aaker, Day
Therefore the regression model would be

                                      Ŷ = -2.55 + 1.05 Xi
                                      Y2 = (0.74)2 = 0.54

(Variance in sales (Y) explained by ad (X))
Assume that the Sbo = 0.51 and
Sb1 = 0.26 at  = 0.5, df = 4, CVt =

Is bo significant? Is b1 significant?




                 Essentials of Marketing Research           Kumar, Aaker, Day
                     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


          Essentials of Marketing Research       Kumar, Aaker, Day
  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




       Essentials of Marketing Research   Kumar, Aaker, Day
         Testing the Significance of the
        Independent Variables (Contd.)
   Test Statistic
                                    t = b 1 - β1
                                               sb1
   Degrees of Freedom
                                   V=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
           Essentials of Marketing Research             Kumar, Aaker, Day
     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)


          Essentials of Marketing Research     Kumar, Aaker, Day
    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
        Essentials of Marketing Research   Kumar, Aaker, Day
Coefficient of Determination (R2)
   Measure of regression model's ability to predict
                          R2          = SST - SSE
                                              SST
                                      = SSM
                                             SST
                                      = Explained Variation
                                              Total Variation



          Essentials of Marketing Research          Kumar, Aaker, Day
     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


         Essentials of Marketing Research   Kumar, Aaker, Day
   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
       Essentials of Marketing Research     Kumar, Aaker, Day
     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




          Essentials of Marketing Research       Kumar, Aaker, Day
                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

       Essentials of Marketing Research   Kumar, Aaker, Day
           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
        Essentials of Marketing Research   Kumar, Aaker, Day
  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
       Essentials of Marketing Research   Kumar, Aaker, Day