Transcript Hair_13 - Homestead

13: Examining Relationships in
Quantitative Research
Relationships between
Variables
Is there a relationship between the two
variables we are interested in?
How strong is the relationship?
How can that relationship be best
described?
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Covariation and Variable
Relationships
Covariation is amount of change in one
variable that is consistently related to
the change in another variable
A scatter diagram graphically plots the
relative position of two variables using a
horizontal and a vertical axis to
represent the variable values
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Positive Relationship
between X and Y
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Negative Relationship
between X and Y
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Curvilinear Relationship
between X and Y
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Scatter Diagram –
No Relationship between X and Y
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Correlation Analysis
 Pearson Correlation Coefficient (r) – measures the
strength of a linear relationship between two metric
variables
 Varies between – 1.00 and +1.00
 The higher the absolute value of the correlation
coefficient, the stronger the level of association
 Positive sign – direct relationship
 Negative sign – inverse relationship
 Check for significance – p < .05 good, but consider
“marginal” cases
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Strength of Correlation
Coefficients
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Assumptions for Pearson’s
Correlation Coefficient
The two variables are assumed to have
been measured using interval or ratio-scaled
measures
Nature of the relationship to be measured is
linear (NOT curvilinear or other)
Variables come from normally distributed
populations
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SPSS Pearson Correlation
Example
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Substantive Significance
Coefficient of Determination (R-squared)
measures the proportion of variation in
one variable accounted for by another
The r2 measure is a percentage varying from
0.0 to 1.00
The larger the r2, the stronger the linear
relationship between the two variables under
study
Explained and unexplained variation
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Relationships Between
Variables Measured with Rank
(Ordinal) Scales
Spearman Rank Order
Correlation Coefficient
is a statistical measure
of the linear association
between two variables where
both have been measured
using ordinal (rank order) scales
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SPSS Spearman Rank Order
Correlation
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SPSS Median Example for
Restaurant Selection Factors
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Regression Analysis
A method for arriving at more mathematically
detailed relationships (predictions) than those
provided by the correlation coefficient
Assumptions
Variables are measured on interval or ratio
scales
Variables come from a normal population
Error terms are normally and independently
distributed
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Straight Line Relationship in
Regression
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Formula for a Straight
Regression Line
y
a
b
X
ei
y
=
a + bX + ei
=
=
=
=
=
the dependent variable
the intercept
the slope
the independent variable used to predict y
the error for the prediction
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Fitting the Regression Line Using
the “Least Squares” Procedure
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Ordinary Least Squares
(OLS)
OLS is a statistical procedure
that estimates regression equation
Coefficients ( ‘a’ and ‘b’) which produce
the lowest sum of squared differences
between the actual and predicted
values of the dependent variable
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Key Terms in
Regression Analysis
Adjusted R-square
Explained vs. Unexplained variance
Regression coefficient
F-Test
Regression Significance
Coefficient Significance
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SPSS Results for Bivariate
Regression
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Significance of
Regression Coefficients
Answers these questions:
Is there a relationship between the
dependent and independent variable?
How strong is the relationship?
p- or t- values are indicators of
relationship strength and significance
Look for p < .05, but consider “marginal”
cases
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Multiple Regression
Multiple regression analysis is a
statistical technique which analyzes the
linear relationship between a dependent
variable and multiple independent
variables by estimating coefficients for
the equation of a straight line
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SPSS Multiple Regression
Example
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Beta Coefficient
A Standardized Beta Coefficient is an
estimated regression coefficient that
has been recalculated to have a mean
of 0 and a standard deviation of 1 in
order to enable independent variables
with different units of measurement to
be directly compared on their
association with the dependent variable
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Evaluating a
Regression Analysis
Assess the statistical significance of the
overall regression model using the F statistic
and its associated p-value
Evaluate the regression’s R-squared
Examine the individual regression
coefficients and their t-test statistics (or their
p-values) to see which are statistically
significant
Look at values of the beta coefficients to
assess relative influence of each predictor
(IV) on Dependent Variable (DV)
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Multicollinearity
Multicollinearity is when several
independent variables are highly
correlated with each other. This causes
difficulty in accurately estimating the
regression coefficients of the
correlated variables.
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Multicollinearity –
How to Avoid
Eliminate or replace highly correlated IVs
Perform a correlation matrix
Typically search for correlations higher than
.5 in absolute value
Factor Analysis Techniques (also called
“Principal Components Analysis”)
“Live with it” 
Hair/Wolfinbarger/Ortinau/Bush, Essentials of
Marketing Research 1e © McGraw-Hill/Irwin2008
Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved. 13-29