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LINEAR REGRESSION:
What it Is and How it Works
Overview
• What is Bivariate Linear Regression?
• The Regression Equation
• How It’s Based on r
What is Bivariate
Linear Regression?
• Predict future scores on Y based on
measured scores on X
• Predictions are based on a correlation from
a sample where both X and Y were
measured
Why is it Bivariate?
• Two variables: X and Y
• X - independent variable/predictor variable
• Y - dependent/outcome/criterion variable
Why is it Linear?
• Based on the linear relationship
(correlation) between X and Y
• The relationship can be described by the
equation for a straight line
The Regression Equation
y = b0+ b1xi + ei
y = predicted score on criterion variable
b0 = intercept
xi = measured score on predictor variable
b1 = slope
ei = residual (error score)
Regression Lines
Least-Squares Solution
• Minimize squared error in prediction.
• Error (residual) = difference between
predicted y and actual y
Residuals
How It’s Based on r
Replace x and y with zX and zY:
zY = bo + b1zX
and the y-intercept becomes 0:
zY = b1zX
and the slope becomes r:
zY = rzX
Take-Home Point
• Linear regression is a way of using
information about a correlation to make
predictions