1. Examining Relationships in Quantitative Research
Chapter 12
Examining Relationships in Quantitative Research
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Learning Objectives
Understand and evaluate the types of relationships between variables
Explain the concepts of association and co-variation
Discuss the differences between Pearson correlation and Spearman correlation

3. Learning Objectives
Explain the concept of statistical significance versus practical significance
Understand when and how to use regression analysis

4. Examining Relationships between Variables
Relationships between variables can be described through:
Presence
Direction

5. Examining Relationships between Variables
Strength of association:
No relationship
Weak relationship
Moderate relationship
Strong relationship

6. Examining Relationships between Variables
Type
Linear relationship: An association between two variables whereby the strength and nature of the relationship remains the same over the range of both variables
Curvilinear relationship: A relationship between two variables whereby the strength and/or direction of their relationship changes over the range of both variables

7. Covariation and Variable Relationships
Covariation: The amount of change in one variable that is consistently related to the change in another variable of interest
Scatter diagram: A graphic plot of the relative position of two variables using a horizontal and a vertical axis to represent the values of the respective variables
A way of visually describing the covariation between two variables

8. Exhibit 12.1 - No Relationship between X and Y

9. Exhibit 12.2 - Positive Relationship between X and Y

10. Exhibit 12.3 - Negative Relationship between X and Y

11. Exhibit 12.4 - Curvilinear Relationship between X and Y

12. 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
0 represents absolutely no association between two variables
– 1.00 or 1.00 represent a perfect link between two variables
Correlation coefficient can be either positive or negative

13. Exhibit 12.5 - Rules of Thumb about the Strength of Correlation Coefficients

14. Assumptions for Calculating Pearson’s Correlation Coefficient
The two variables have been measured using interval- or ratio-scaled measures
Nature of the relationship to be measured is linear
A straight line describes the relationship between the variables of interest
Variables to be analyzed need to be from a normally distributed population

15. Exhibit 12.6 - SPSS Pearson Correlation Example for Santa Fe Grill Customers

16. Substantive Significance of the Correlation Coefficient
Coefficient of determination (r2): A number measuring the proportion of variation in one variable accounted for by another
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 being examined

17. Influence of Measurement Scales on Correlation Analysis
Spearman rank order correlation coefficient: A statistical measure of the linear association between two variables where both have been measured using ordinal (rank order) scales

18. Exhibit 12.7 - SPSS Spearman Rank Order Correlation

19. Exhibit 12.8 - Median Example for Restaurant Selection Factors

20. What is Regression Analysis?
A method for arriving at more detailed answers (predictions) than can be provided by the correlation coefficient
A number of ways to make such predictions:
Extrapolation from past behavior of the variable
Simple guesses
Use of a regression equation that includes information about related variables to assist in the prediction

21. What is Regression Analysis?
Bivariate regression analysis: A statistical technique that analyzes the linear relationship between two variables by estimating coefficients for an equation for a straight line
One variable is designated as a dependent variable
The other is called an independent or predictor variable

22. What is Regression Analysis?
Use of a simple regression model assumes:
Variables of interest are measured on interval or ratio scales
Variables come from a normal population
Error terms associated with making predictions are normally and independently distributed

23. Fundamentals of Regression Analysis
General formula for a straight line:
Where,
Y = The dependent variable
a = The intercept (point where the straight line intersects the Y-axis when X = 0)
b = The slope (the change in Y for every 1 unit change in X )
X = The independent variable used to predict Y
ei = The error of the prediction

24. Exhibit 12.9 - The Straight Line Relationship in Regression

25. Least Squares Procedure
A regression approach that determines the best-fitting line by minimizing the vertical distances of all the points from the line
Unexplained Variance
The amount of variation in the dependent variable that cannot be accounted for by the combination of independent variables

26. Exhibit 12.10 - Fitting the Regression Line Using the “Least Squares” Procedure

27. Ordinary Least Squares
A statistical procedure that estimates regression equation coefficients that produce the lowest sum of squared differences between the actual and predicted values of the dependent variable
Regression Coefficient
An indicator of the importance of an independent variable in predicting a dependent variable
Large coefficients are good predictors and small coefficients are weak predictors

28. Exhibit 12.11 - SPSS Results for Bivariate Regression

29. Significance of Regression Coefficients
Is there a relationship between the dependent and independent variable?
How strong is the relationship?

30. Multiple Regression Analysis
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

31. Beta Coefficient
An estimated regression coefficient that has been recalculated to have a mean of 0 and a standard deviation of 1
Such a change enables independent variables with different units of measurement to be directly compared on their association with the dependent variable

32. Examining the Statistical Significance of Each Coefficient
Each regression coefficient is divided by its standard error to produce a t statistic
Which is compared against the critical value to determine whether the null hypothesis can be rejected

33. Examining the Statistical Significance of Each Coefficient
Model F statistic: Compares the amount of variation in the dependent measure “explained” or associated with the independent variables to the “unexplained” or error variance
A larger F statistic indicates that the regression model has more explained variance than error variance

34. Substantive Significance
The multiple r2 describes the strength of the relationship between all the independent variables and the dependent variable
The larger the r2 measure, the more of the behavior of the dependent measure is associated with the independent measures we are using to predict it

35. Multiple Regression Assumptions
Linear relationship
Homoskedasticity: The pattern of the co-variation is constant (the same) around the regression line, whether the values are small, medium, or large
Heteroskedasticity: The pattern of covariation around the regression line is not constant around the regression line, and varies in some way when the values change from small to medium and large

36. Multiple Regression Assumptions
Normal distribution
Normal curve: A curve that indicates the shape of the distribution of a variable is equal both above and below the mean

37. Exhibit 12.12 - Example of Heteroskedasticity

38. Exhibit 12.13 - Example of a Normal Curve

39. Exhibit 12.14 - SPSS Results for Multiple Regression

40. Multicollinearity
A situation in which several independent variables are highly correlated with each other
Can result in difficulty in estimating independent regression coefficients for the correlated variables

41. Marketing Research in Action: The Role of Employees in Developing a Customer Satisfaction Program
Will the results of this regression model be useful to the QualKote plant manager?
If yes, how?
Which independent variables are helpful in predicting A36–Customer Satisfaction?
How would the manager interpret the mean values for the variables reported in Exhibit 12.16?
What other regression models might be examined with the questions from this survey?