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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

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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

### Learning Objectives_2

• Explain the concept of statistical significance versus practical significance

• Understand when and how to use regression analysis

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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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