Transcript Execute the research

Chapter Fourteen
Data Analysis: Testing for
Association
Copyright  2004 McGraw-Hill Pty Ltd.
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Learning Objectives
 Understand and evaluate associations
between variables
 Explain the concept of covariation
 Discuss the differences in Chi-square
value, Pearson product moment
correlation and Spearman rank order
correlation
 Understand when and how to use
analysis
 Apply discriminant analysis and
conjoint analysis to examine marketing
research problems
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Introduction
PHASE III:
Execute the research
Marketing Research
Analyse the data
Step 8:
 The concept of relationship:
 A relationship is a consistent and
systematic link between two or
more variables
 Visit correlation and regression at:
<www.med.umkc.edu>
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Introduction
PHASE III:
Execute the research
Marketing Research
Analyse the data
Step 8:
 Data analysis also entails the
process of testing and assessing
associations between variables
 Associations between variables
include:




Presence of association
Direction of association
Strength of association
Type of association
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Using Covariation to Describe
Variable Relationships
 Covariation is the amount of
change in one variable that is
consistently related to the change
in another variable of interest
 A scatter diagram is a graphic plot
of the relative position of two
variables using a horizontal axis
and a vertical axis to represent the
values of the respective variables
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Using Covariation to Describe
Variable Relationships
 Nature of scatter diagrams
 Changes in the value of Y are often
systematically related to changes in
the value of X
 Relationships can be either positive
or negative
 If patterns are not linear, they may be
curvilinear
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Types of Association:
Linear—Positive and Negative
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Types of Association:
Nonexistent and Curvilinear
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The Chi-square Value
 Chi-square (χ2)
 Used to determine whether the presence of
one variable is systematically associated
with the presence of another
 Does not assess direction or strength
 A Chi-square test is often referred to as a
‘goodness-of-fit’ test
 Has categorical questions and allows a research
team to assess how closely what’s ‘observed’ fits
the pattern of what was ‘expected’ or anticipated
 For example, did the responses to a survey follow a
particular pattern?
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Chi-square Analysis
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Treatment and Measures of
Correlation Analysis
 The Pearson product moment correlation
coefficient is a statistical measure of the
strength and direction of a linear
relationship between two interval- and/or
ratio-scaled (metric) variables
 Ranges from –1.00 to 1.00. The higher the
correlation coefficient, the stronger the level of
association
 The correlation coefficient can be either + or –
depending on the relationship between 2 variables
 If the correlation coefficient is statistically
significant, the null hypothesis is rejected. If it is
not significant then it has no meaning
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Pearson Correlation Coefficient
Formula
 The Pearson correlation coefficient
formula makes several assumptions
about the nature of the data:
 Two variables are assumed to have been
measured using interval- and/or ratio-scaled
measures
 The nature of the relationship to be measured is
linear
 Variables to be analysed come from a bivariate
normally distributed population
 Null hypothesis states that there is no
association and that the correlation coefficient
is 0
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Correlation Analysis
 A correlation analysis is often
referred to as ‘plus or minus one’
by some practitioners of marketing
research
 Correlation allows a research team to
determine the sturdiness of a linear
relationship between two variables
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Rules of Thumb about the
Strength of Correlation
Coefficients
Range of Coefficient
Description of Strength
.81 to 1.00
Very strong
.61 to .80
Strong
.41 to .60
Moderate
.21 to .40
Weak
.00 to .20
None
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Coefficient of Determination
 When the Pearson product moment
correlation is strong and significant, then
the variables are associated in a linear
fashion
 The coefficient of determination (Rsquare) is the square of the Pearson
product moment correlation
 Between 0.0 and 1.0
 Shows the proportion of variation in one
variable accounted for by another

For example, R-square = 0.36 means that 36%
of the variation in one variable is associated
with the related variable
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Treatment and Measures of
Correlation Analysis
Statistical
significance
Correlation coefficient is
converted into a student’s t test
statistic.
Can only show association;
doesn’t allow researcher to make
causal statements.
Substantive
significance
Coefficient of determination is a
number measuring the proportion
of variation in one variable
accounted for by another.
The larger the number, the
stronger the linear relationship
between two variables under
study.
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SPSS Application: Pearson
Product Moment Correlation
Coefficient Example
Descriptive Statistics
Mean
Std.
Deviation
N
Recommend to
Friend
4.68
.98
50
Satisfaction Level
4.78
.95
50
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Correlations Example
Correlations
Recommend
to Friend
Recommend to Friend
Pearson
correlation
1.000
Sig. (2-tailed)
N
Satisfaction Level
Pearson
correlation
Satisfaction
Level
.601**
.000
50
.601**
Sig. (2-tailed)
.000
N
50
50
1.000
50
** Correlation is significant at the .01 level (2-tailed)
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The Spearman Rank Order
Correlation Coefficient
 The Spearman rank order correlation
coefficient is a statistical measure of
the strength and direction of a linear
relationship between two variables
where at least one is ordinal-scaled
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Spearman’s Rank Order
Correlation
Correlations
Food Quality Rank
Food Variety Rank
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
Spearman’s Rho
Food Quality
Food Variety
Rank
Rank
1.000
-.495**
.000
50
50
-.495**
1.000
.000
50
50
**Correlation is significant at the .01 level (2-tailed).
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Describing Relationships and
Making Predictions
 Beyond describing relationships,
researchers also may wish to
predict potential effects of X on Y
 A predictive model
 A mathematical model based on
relationships found among Y and X
variables to make a prediction
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Regression Analysis
 Regression equations include 2
basic approaches:
 Bivariate regression analysis—X1 and Y1
 Multivariate regression analysis—X1, X1, X2
… Xn and Y1
Assumptions
•Two variables are assumed to have
been measured using interval- and/or
ratio-scaled measures
•Nature of the relationship to be
measured is linear
•Variables to be analysed come from a
bivariate normally distributed
population
•Error terms associated with making
predictions are normally and
independently distributed
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Regression Analysis
 A regression analysis is often
referred to as the R-square test
 Regression analysis designates an
independent variable as a ‘predictor’,
even though few (if any) relationships
on the marketing landscape are truly
deterministic in nature.
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What is Regression Analysis?
 Marketing managers need to make
predictions about situations affecting
the company and they may use a
regression equation.
Bivariate
regression
analysis
A statistical technique which
analyses the linear relationship
between two variables by
estimating coefficients for an
equation for a straight line.
One variable is the dependent
variable and the other is the
independent variable.
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Regression Analysis
 Once a regression equation has been
developed to predict values of Y, the
interest is in trying to find out how
accurate that prediction is.
 Look at the actual value collected in the
sample
 Compare this with what has been
predicted and see what the difference is
 This is the common way to judge the
accuracy of a regression equation or
model
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Points About Comparing
Actual and Predicted Values
Point 1—Differences between actual and predicted Y
values are represented by the error term of the
regression equation—ei
Point 2—By squaring e for each observation, and
summing them, the total represents an aggregate or
overall measure of the accuracy of the regression
equation
Point 3—Error terms can also be used to diagnose
potential problems
The pattern of errors produced by comparing actual Y
values with predicted Y values can show as errors are
normally distributed and/or have equal variances
across the range of X values
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Patterns of Residuals—
Three Examples
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Patterns of Residuals—
Three Examples (continued)
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Developing and Estimating
The Regression Coefficients
 Statistical
significance
 Is there a
relationship?
 If so, we need to
know how strong
that relationship is.
So …
Copyright  2004 McGraw-Hill Pty Ltd.
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 Substantive
significance
 Coefficient of
determination describes
the percentage of the
total variation in the
dependent variable that
can be explained by
using the independent
variable.
 Look at size of
R-square and
regression
coefficient.
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Summary of Selected
Multivariate Methods
 Multiple regression enables the
marketing researcher to predict a single
dependent metric variable from two or
more metrically measured independent
variables.
 Discriminant analysis can predict a
single dependent non-metric variable
from two or more metrically measured
independent variables.
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Summary of Selected
Multivariate Methods
 Conjoint analysis is used to estimate
the value (utility) that respondents
associate with different product and/or
service features, so that the most
preferred combination of features can
be determined.
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Multiple Regression Analysis
 A statistical technique which analyses
the linear relationship between a
dependent variable and multiple
independent variables by estimating
coefficients for the equation for a
straight line
 Regression coefficient is calculated for each
variable (X) that describes its relationship
with the dependent variable (Y)
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Multiple Regression Analysis
 Multiple regression analysis is a useful
technique because most information
problems (and market opportunities)
involve several independent variables
 Decision makers benefit from multiple
regression, since management can explore
how a ‘batch’ of independent variables
influence a dependent variable from the
output
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Multiple Regression Analysis
 These coefficients allow the marketing
manager to examine the relative
influence of each independent
variable on the dependent variable
 This relationship is still linear
 However, with more than one
independent variable we have to think
in terms of multiple independent
dimensions
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Multiple Regression Analysis
 To investigate these relationships
 Examine the regression coefficients
 They describe the average amount of change
expected in Y given a unit change in the
value of X (whichever X you are considering)
 Each regression coefficient describes the
relationship of that variable to the dependent
variable
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Multiple Regression Analysis
 When multiple independent variables
have different scales, is it not possible
to make relative comparisons?
 Calculate the standardised regression
coefficient
 Beta coefficient—an estimated regression
coefficient that has been recalculated to have a
mean of 0 and a standard deviation of 1
 This allows for variables with different units to be
compared directly on their association with Y
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Multiple Regression Analysis
 What is the strength of association?
 R-square—Coefficient of determination
 A measure of the amount of variation in Y
associated with the variation in the independent
variables, considered together
 The larger the R-square, the more the behaviour
of Y is associated with the independent measures
used to predict it
 R-square = .78, means that 78% of the variation in
sales revenue in the Canon example is accounted
for, or explained by, the variation in sales force,
advertising budget and customer attitudes
towards the products
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Examining Residuals:
Standardised Residuals vs Normal
Distribution
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Multicollinearity
 Multicollinearity is a very important
problem marketing researchers need to
address
 When multicollinearity exists in a data
set, the research team is challenged
because it makes it difficult (sometimes
impossible) to estimate separate
regression coefficients for correlated
variables
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Multicollinearity and Multiple
Regression Analysis
 Multicollinearity is a situation in which
several independent variables are
highly correlated with each other
 The effect makes it difficult for the
regression equation to separate out the
independent contributions of the
independent variables
 This situation inflates the standard error of
the coefficient and lowers the t-statistic
associated with it
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Discriminant Analysis
 A research team uses discriminant
analysis to classify groups or objects
by a set of independent variables
 Discriminant analysis enables a marketing
researcher to determine linear
combinations of the independent variables
of interest to the client
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Discriminant Analysis
 Discriminant score—the basis for
predicting which group an individual
belongs to:
Zi = b1X1i + b2X2i + b3X3i + . . . +
bnXni
Z
= ith individual’s discriminant score
bn = Discriminant coefficient of the nth variable
Xni = Individual’s value on the nth X variable
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Discriminant Analysis
bn = Discriminant coefficient of the
nth variable
 Estimates of the discriminant power of a
particular X variable
 The size of the coefficient is determined by
the variance structure of the variables
 X’s with a large discriminant power will have
large weights
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Discriminant Analysis—An
Example to Consider
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Discriminant Analysis
 Critical discriminators of likelihood
 Is about the prediction of a categorical
variable—group membership, likelihood
to buy, recommend, etc.
 Is the direction of group differences
based on finding a linear combination of
X’s
 The discriminant function shows large
differences in group means
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Conjoint Analysis
 Conjoint analysis is used to estimate
the value (utility) that respondents
associate with different product
and/or service features, so that the
most preferred combination
of features can be determined
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Conjoint Analysis
 Estimates the relative importance
placed on the different attributes of a
product or service
 As well as the value that is attached to
the various levels of each attribute
 The assumption is that consumers
form preferences for products by
evaluating the overall utility or value
of the product
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Conjoint Analysis
 Marketing researchers draw upon the
power of conjoint analysis to estimate
the value (utility) respondents associate
with different product and/or service
features
 Conjoint analysis, then, allows a marketing
research team to communicate the most
preferred combination of features to a client
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Conjoint Analysis
 Collect data, then produce a partworth estimate for each level of each
attribute
 Estimate the total worth of the product
profiles, then compare it to
consumer’s choice ranking
 Attribute importance estimate—the
importance of an attribute of an object
as estimated by conjoint analysis
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Conjoint Analysis
 To quantify the value that people
associate with different levels of
product/service attributes
 However, there are limitations
 Suffers from artificiality:
 Respondents may be more deliberate than
in a real situation
 Respondents may have additional
information when compared with reality
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Conjoint Analysis—An
Example to Consider
Attribute
Restaurant Profile A
Restaurant Profile B
Price level
Inexpensive ($6–$10)
Moderate ($10–$20)
Atmosphere
Family style
Smart
Menu type
Sandwiches
Salad, main, dessert
Service level
Self-service
Table service
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Conjoint Analysis—An
Example to Consider
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