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.

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

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

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

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

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

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

Exhibit 12.1 - No Relationship between X and Y

Exhibit 12.2 - Positive Relationship between X and Y

Exhibit 12.3 - Negative Relationship between X and Y

Exhibit 12.4 - Curvilinear Relationship between X and Y

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

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

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

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

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

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

Exhibit 12.7 - SPSS Spearman Rank Order Correlation

Exhibit 12.8 - Median Example for Restaurant Selection Factors

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

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

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

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

Exhibit 12.9 - The Straight Line Relationship in Regression

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

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

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

Exhibit 12.11 - SPSS Results for Bivariate Regression

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

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

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

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

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

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

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

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

Exhibit 12.12 - Example of Heteroskedasticity

Exhibit 12.13 - Example of a Normal Curve

Exhibit 12.14 - SPSS Results for Multiple Regression

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

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?