Examining Relationships in Quantitative Research Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin.

Examining Relationships in Quantitative Research

Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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Learning Objectives_1

Understand and evaluate the types of relationships between variables

Explain the concepts of association and covariation

Discuss the differences between Pearson correlation and Spearman correlation

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Learning Objectives_2

Explain the concept of statistical significance versus practical significance

Understand when and how to use regression analysis

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Proctor & Gamble

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Describing Relationships Between Variables

Presence Direction

Strengthof association

Type

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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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Exhibit 12.1 Scatter Diagram Illustrates No Relationship

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Exhibit 12.2 Positive Relationship between X and Y

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Exhibit 12.3 Negative Relationship between X and Y

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Exhibit 12.4 Curvilinear Relationship between X and Y

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Correlation Analysis

Pearson Correlation Coefficient–statistical measure of the strength of a linear relationship between two metric variables Varies between – 1.00 and +1.00 The higher the correlation coefficient–the

stronger the level of association Correlation coefficient can be either

positive or negative

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Exhibit 12.5 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

Variables to be analyzed come from a bivariate normally distributed population

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Exhibit 12.6 SPSS Pearson Correlation Example

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Substantive Significance

Coefficient of Determination (r2) is a number measuring the proportion of variation in one variable accounted for by another The r2 measure can be thought of as a

percentage and varies from 0.0 to 1.00 The larger the size of the coefficient of

determination, the stronger the linear relationship between the two variables under study

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How to Measure the Relationship between Variables Measured

with Ordinal or Nominal 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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Exhibit 12.7 SPSS Example Spearman Rank Order Correlation

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Exhibit 12.8 SPSS Median Example for Restaurant Selection Factors

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What is Regression Analysis?

A method for arriving at more detailed answers (predictions) than can be provided by the correlation coefficient

Assumptions Variables are measured on interval or ratio

scales Variables come fro a normal population Error terms are normally and independently

distributed

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Exhibit 12.9 Straight Line Relationship in Regression

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Formula for a Straight Line

y = a + bX + ei

y = the dependent variable a = the intercept b = the slope X = the independent variable used to

predict y ei = the error for the prediction

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Exhibit 12.10 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 which produce the lowest sum of squared differences between the actual and predicted values of the dependent variable

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Exhibit 12.11 SPSS Results for Bivariate Regression

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Key Terms in Regression Analysis

Adjusted R-square Explained variance Unexplained variance Regression coefficient

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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? How much influence does the relationship

hold?

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Multiple Regression Analysis

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 for a straight line

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Beta Coefficient

A 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 probability

Evaluate the obtained r2 to see how large it is Examine the individual regression coefficient

and their t-test statistic to see which are statistically significant

Look at the beta coefficient to assess relative influence

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Exhibit 12.12 SPSS Example Multiple Regression

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Multicollinearity

Multicollinearity is a situation in which several independent variables are highly correlated with each other and can cause difficulty in estimating separate or independent regression coefficients for the correlated variables

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Marketing Research in Action: QualKote Manufacturing

How might the results of the regression model be useful to the QualKote plant manager?

Which independent variables are helpful in predicting customer satisfaction?

How would the manager interpret the mean values for the variables reported in Exhibit 12.12?

What other regression models might be examine with the questions from this survey?