Examining relationships in quantitative research
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Examining Relationships in Quantitative Research. 12. 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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Examining relationships in quantitative research

Examining Relationships in Quantitative Research

12


Learning objectives 1

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

Learning Objectives_2

  • Explain the concept of statistical significance versus practical significance

  • Understand when and how to use regression analysis


Proctor gamble

Proctor & Gamble


Describing relationships between variables

Describing Relationships Between Variables

Presence

Direction

Strength

of association

Type


Relationships between variables

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 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 12.1 Scatter Diagram Illustrates No Relationship


Exhibit 12 2 positive relationship between x and y

Exhibit 12.2 Positive Relationship between X and Y


Exhibit 12 3 negative relationship between x and y

Exhibit 12.3 Negative Relationship between X and Y


Exhibit 12 4 curvilinear relationship between x and y

Exhibit 12.4 Curvilinear Relationship between X and Y


Correlation analysis

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

Exhibit 12.5 Strength of Correlation Coefficients


Assumptions for pearson s correlation coefficient

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

Exhibit 12.6 SPSS Pearson Correlation Example


Substantive significance

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

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 12.7 SPSS Example Spearman Rank Order Correlation


Exhibit 12 8 spss median example for restaurant selection factors

Exhibit 12.8 SPSS Median Example for Restaurant Selection Factors


What is regression analysis

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

Exhibit 12.9 Straight Line Relationship in Regression


Formula for a straight line

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

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


Ordinary least squares ols

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

Exhibit 12.11 SPSS Results for Bivariate Regression


Key terms in regression analysis

Key Terms in Regression Analysis

  • Adjusted R-square

  • Explained variance

  • Unexplained variance

  • Regression coefficient


Significance of regression coefficients

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

  • 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

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

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

Exhibit 12.12 SPSS Example Multiple Regression


Multicollinearity

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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