Examining Relationships in Quantitative Research 12 McGraw-Hill/Irwin Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.

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

Learning Objectives_2 Explain the concept of statistical significance versus practical significance Understand when and how to use regression analysis

Proctor & Gamble

Describing Relationships Between Variables Presence Direction Strength of association Type

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?

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

Exhibit 12.1 Scatter Diagram Illustrates No Relationship

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 The higher the correlation coefficient–the stronger the level of association Correlation coefficient can be either positive or negative

Exhibit 12.5 Strength of Correlation Coefficients

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

Exhibit 12.6 SPSS Pearson Correlation Example

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

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

Exhibit 12.7 SPSS Example Spearman Rank Order Correlation

Exhibit 12.8 SPSS 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 Assumptions Variables are measured on interval or ratio scales Variables come fro a normal population Error terms are normally and independently distributed

Exhibit 12.9 Straight Line Relationship in Regression

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

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

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

Exhibit 12.11 SPSS Results for Bivariate Regression

Key Terms in Regression Analysis Adjusted R-square Explained variance Unexplained variance Regression coefficient

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?

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

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

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

Exhibit 12.12 SPSS Example Multiple Regression

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

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?