Marketing Research

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Marketing Research

• Aaker, Kumar, Day and Leone
• Tenth Edition
• Instructors Presentation Slides

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Chapter Nineteen
Correlation Analysis and Regression Analysis
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Definitions

• Correlation analysis
• 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)

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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

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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

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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 lt r lt 1 R 0 ---gt absence
of linear association
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Scatter Plots
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Scatter Plots (Contd.)
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Correlation Coefficient
Simple Correlation Coefficient
Pearson Product-moment Correlation Coefficient
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Determining Sample Correlation Coefficient
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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 1.96lt2.78, we fail to reject the null
  hypothesis.

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Partial Correlation Coefficient

• Measure of association between two variables
  after controlling for the effects of one or more
  additional variables

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Regression Analysis

• Simple Linear Regression Model
• Yi ßo ß1xi ei
• Where
• Y Dependent variable
• X Independent variable
• ß o Model parameter that represents mean value
  of dependent variable (Y) when the independent
  variable (X) is zero
• ß1 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
• ei Error term that describes the effects on Yi
  of all factors other than value of Xi

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Simple Linear Regression Model
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Simple Linear Regression Model A Graphical
Illustration
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Assumptions of the Simple Linear 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 are fixed
  (non-stochastic X)

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Estimating the Model Parameters

• Calculate point estimate bo and b1 of unknown
  parameter ßo and ß1
• Obtain random sample and use this information
  from sample to estimate ßo and ß1
• Obtain a line of best "fit" for sample data
  points - least squares line

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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 sum of
  squares (SSE)
• SSE ?ei2 (?(yi - yi)2
• S yi-(bo b1xi)2

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Standard Error

• Mean Square Error
• Standard Error of b1
• Standard Error of b0

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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 ß1 0
Ha ß1 ? 0
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Testing the Significance of Independent Variables
(Contd.)

• Test Statistic t b1 - ß1
• sb1
• Degrees of Freedom V n 2
• Testing for a Type II Error
• Ho ß1 0
• Ha ß1 ? 0
• Decision Rule

Reject ho ß1 0 if a gt p value
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Sum of Squares

• 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

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Predicting the Dependent Variable

• Dependent variable, yi bo bixi
• Error of prediction is yi y
• Total variation (SST)
• Explained variation (SSM) Unexplained
  variation (SSE)

Coefficient of Determination (r2)

• Measure of regression model's ability to predict

r2 (SST - SSE) / SST SSM / SST
Explained Variation / Total Variation
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Multiple Regression

• 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 ß1 , ß2, . . . , ßk are regression
coefficients associated with the independent
variables X1, X2, . . . , Xk and e is the error
or residual.
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Multiple Regression (Contd.)

• The prediction equation in multiple regression
  analysis is

Y a b1X1 b2X2 .bkXk
where Y is the predicted Y score and b1 . . .
, bk are the partial regression coefficients.
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Partial Regression Coefficients
Y a 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)

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Evaluating the Importance of Independent Variables

• Consider t-value for ßi's
• Use beta coefficients when independent variables
  are in different units of measurement
• Standardized ßi bi Standard deviation of
  xi
• Standard deviation of Y
• Check for multicollinearity

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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 ßo e
• Add variables if they meet certain criteria in
  terms of F-ratio

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Stepwise Regression (Contd.)

• Backward Elimination
• Start with full regression equation
• i.e. y ßo ß1x1 ß2 x2 ... ßr xr e
• 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

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

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Regression with Dummy Variables
Yi a b1D1 b2D2 b3D3 error

• For rational buyer, Yi a
• For brand-loyal consumers, Yi a b1