Introduction to Linear Regression and Correlation Analysis
Introduction to Linear Regression and Correlation Analysis
Goals
After this, you should be able to:
• • • • •
Calculate and interpret the simple correlation between two variables
Determine whether the correlation is significant Calculate and interpret the simple linear regression equation for a set of data Understand the assumptions behind regression analysis Determine whether a regression model is significant
Goals
(continued)
After this, you should be able to:
• Calculate and interpret confidence intervals for the regression coefficients • Recognize regression analysis applications for purposes of prediction and description • Recognize some potential problems if regression analysis is used incorrectly • Recognize nonlinear relationships between two variables
Scatter Plots and Correlation
• A scatter plot (or scatter diagram) is used to show the relationship between two variables • Correlation analysis is used to measure strength of the association (linear relationship) between two variables – Only concerned with strength of the relationship – No causal effect is implied
Scatter Plot Examples
Linear relationships y y Curvilinear relationships
x y y
x
x
x
Scatter Plot Examples
(continued)
Strong relationships y y Weak relationships
x y y
x
x
x
Scatter Plot Examples
(continued)
No relationship y
x y
x
Correlation Coefficient
(continued)
• The population correlation coefficient ρ (rho) measures the strength of the association between the variables
• The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations
Features of ρ and r
• Unit free • Range between -1 and 1 • The closer to -1, the stronger the negative linear relationship • The closer to 1, the stronger the positive linear relationship • The closer to 0, the weaker the linear relationship
Examples of Approximate r Values y y y
x
Goals
After this, you should be able to:
• • • • •
Calculate and interpret the simple correlation between two variables
Determine whether the correlation is significant Calculate and interpret the simple linear regression equation for a set of data Understand the assumptions behind regression analysis Determine whether a regression model is significant
Goals
(continued)
After this, you should be able to:
• Calculate and interpret confidence intervals for the regression coefficients • Recognize regression analysis applications for purposes of prediction and description • Recognize some potential problems if regression analysis is used incorrectly • Recognize nonlinear relationships between two variables
Scatter Plots and Correlation
• A scatter plot (or scatter diagram) is used to show the relationship between two variables • Correlation analysis is used to measure strength of the association (linear relationship) between two variables – Only concerned with strength of the relationship – No causal effect is implied
Scatter Plot Examples
Linear relationships y y Curvilinear relationships
x y y
x
x
x
Scatter Plot Examples
(continued)
Strong relationships y y Weak relationships
x y y
x
x
x
Scatter Plot Examples
(continued)
No relationship y
x y
x
Correlation Coefficient
(continued)
• The population correlation coefficient ρ (rho) measures the strength of the association between the variables
• The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations
Features of ρ and r
• Unit free • Range between -1 and 1 • The closer to -1, the stronger the negative linear relationship • The closer to 1, the stronger the positive linear relationship • The closer to 0, the weaker the linear relationship
Examples of Approximate r Values y y y
x