Regression Analysis in Marketing Research

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Regression Analysis in Marketing Research

• Dr. John T. Drea
• Professor of Marketing
• Western Illinois University

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Prediction

• Two approaches to prediction
• Extrapolation using past events to predict
  future events (like steering a canoe by looking
  behind you)
• Predictive modeling using one or more other
  variables to predict another variable
• Ex Could you predict success in a course if you
  knew the hours spent studying, number of other
  semester hours taken, and hours spent working?
• The difference between a predicted value and an
  actual value is called a residual.

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Variables

• The dependent variable is what we are trying to
  predict - it is typically represented by Y.
• The independent variable is a variable used to
  predict the dependent variable - it is typically
  represented by x.
• Note that independent variable predicts the
  dependent variable - it cannot be stated that the
  independent variable (x) causes changes in the
  dependent variable (Y).
• Regression typically uses interval/ratio scales
  variables as the independent and dependent
  variable. You can also use dummy coding (1, 0)
  for nominally scaled measures (a 1 if a
  characteristic is present, a 0 if that
  characteristic is absent.

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Bivariate Regression

• Bivariate linear regression (simple regression)
  investigates a straight line relationship of the
  type
• Regression basically fits the data to a straight
  line, where a is the intercept point and b is the
  slope of the line.

Y a bx e
a
where Y is the dependent variable, x is the
independent variable, and a and b are two
constants to be estimated.
SPSS fits the line to minimize vertical distances
between points and the regression line. This is
called the least squares criterion.
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Bivariate Regression in SPSS
Step 1
This is Y
Step 2
This is x
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Bivariate Regression in SPSS Results
19.0 of the variation in BIAmtrak can be
accounted for by AAmtrak., meaning 81 of the
variation is unaccounted for.
The equation is significantly better than chance,
as evidenced by the F-value
The significant t-value suggests that Aamtrak
belongs in the equation. The significant
constant indicates there is considerable
variation unexplained.
The unstandardized equation would be Y 3.132
.507(Aamtrak) Thus, if a subject had an Aamtrak
score of 2, the equation would predict Y 3.132
.507(2) 4.146
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Multiple Regression

• Multiple regression allows for the simultaneous
  investigation of two or more independent
  variables and a single dependent variable.
• Multiple regression is quite useful - it is
  likely that several variables are related to an
  independent variable.
• Regression is useful when we want to explain,
  predict, or control a dependent variable.
• The use of the unstandardized coefficients allows
  you to use the equation in a very practical way.

The form for an unstandardized equation is Y a
b1x1 b2x2 bixi
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Multiple Regression

• Each coefficient in multiple regression is also
  known as a coefficient of partial regression - it
  assesses the relationship between itself (Xi) and
  the dependent variable (Y) not accounted for by
  other variables in the model.
• Each variable introduced into the equation needs
  to account for variation in Y that has not be
  accounted for by any of the X variables already
  entered.
• We typically assume that the X variables are
  uncorrelated with one another. If they are not
  uncorrelated, we have a problem of
  multicollinearity.

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Multiple regression

• Multicollinearity is a problem in regression - it
  occurs when the independent variables are highly
  correlated with one another.
• Multicollinearity does not affect the models
  overall ability to predict, but it can impact the
  interpretation of individual coefficients.
• Multicollinearity can be assessed through the use
  of a statistic, the variance inflation factor
  (VIF)
• If VIF lt 10, multicollinearity is not a problem.
• If VIF gt 10, remove the variable from the
  independent variables and run the analysis again.

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Interpreting Regression Results

• R2
• It is a coefficient of determination - it
  indicates the percentage of of variation in Y
  explained by the variation in the independent
  variables (Xi). It determines the goodness of
  fit for your model (regression equation). It
  ranges from 0-1.0.
• Std. error of the estimate
• It measures the accuracy of predictions using the
  regression equation.
• The smaller the std. error of the estimate, the
  smaller the confidence interval (the more precise
  the prediction)

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Interpreting Regression Results

• F-values
• The F-value determines whether the equation is
  better than chance. A p-value of .05 or lower
  indicates we would reject the null hypothesis
  that the independent variables are not related to
  the dependent variable.
• The F-value does not measure whether your model
  does a good job of predicting - only that it is
  better than chance.
• T-tests
• Examine the t-values to determine whether to
  include additional variables into the model.
  T-values should be statistically significant to
  be included in your analysis.

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Interpreting Regression Results

• Unstandardized coefficients (abbreviated as B)
• These are written in the metric of the measure,
  which makes them useful for prediction.
• Standardized coefficients (beta)
• These are written in a standardized form, ranging
  from 0 to 1.
• The higher the value of the standardized
  coefficient, the more important the predictor is
  to the model. (i.e., the more unique variation in
  Y than can be accounted for by that variable)
• Introducing more variables into an equation
  typically explains more variation (increases R2),
  but each variable must be a significant
  contributor of otherwise unexplained variation to
  include in the model (see T-test results to
  determine this.)

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Multiple Regression in SPSS
Step 1
Step 2
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Multiple Regression in SPSS Results

• Note that the circled t-values for two of the
  variables are not significant these do not
  supply any unique variation to the prediction of
  the dependent variable, so they should be removed
  from analysis.
• Note the standardized coefficients (beta) the
  greater the beta, the more important a variable
  is to the prediction of the dependent variable.
• Finally, not the size of the t-value for the
  constant this suggests the model still has
  considerable unexplained variation.

Y 3.219 .235(Aamtrak, Good/Bad)
.245(Aamtrak, like/dislike) - .0638(Aauto,
goob/bad)
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Multiple Regression in SPSS Results
The model indicates that the five predictors
account for 21.5 of the variation in Aamtrak.
The F-value suggests that the equation is
significantly better than chance.
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Multiple Regression

• Example Toy Manufacturer Sales Hypothesis How
  are weekly toy sales affected by
• changes in levels of advertising,
• the use of sales reps vs. agents for calling on
  retailers, and
• local school enrollments?

Toy Sales Advertising(X1) sales rep/agent(X2)
school enrollment(X3) e To do this, we need to
dummy code sales rep 1 or agent 0. This
produces the following equation Y 102.18
3.87X1 115.2X2 6.73X3 R2 0.845 So what
does this mean?
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Multiple Regression

• So what do those coefficients mean?

Y 102.18 3.87X1 115.2X2 6.73X3 e If the
other variables are held constant, you could
state X1 1 spent in advertising yields 3.87
in sales. X2 The use of a salesperson instead of
an agent contributes 115.20 in additional
sales. X3 Each additional school enrollment
yields 6.73 in toy sales. These three variables
explain 84.5 of the variation in toy sales. If
we spent 1000 in advertising, used a sales rep.,
and there are 500 children in the local schools,
what would sales be? Y 102.18 3.87(1000)
115.2(1) 6.73(500) Y 7350.2