# Marketing Research Essentials

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```							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

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
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
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
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

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