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# LINEAR REGRESSION: What it Is and How it Works PowerPoint PPT Presentation

LINEAR REGRESSION: What it Is and How it Works. Overview. What is Bivariate Linear Regression ? The Regression Equation How It’s Based on r Assumptions. What is Bivariate Linear Regression ?. Predict future scores on Y based on measured scores on X

LINEAR REGRESSION: What it Is and How it Works

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## LINEAR REGRESSION:What it Is and How it Works

### Overview

• What is BivariateLinear Regression?

• The Regression Equation

• How It’s Based on r

• Assumptions

### What is BivariateLinear 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 = b1xi+ b0 + ei

y = predicted score on criterion variable

b0 = intercept

xi = measured score on predictor variable

b1 = slope

ei = residual (error score)

### Least-Squares Solution

• Minimize squared error in prediction.

• Error (residual) = difference between predicted y and actual y

### How It’s Based on r

Replace x and y with zX and zY:

zY = b1zX + bo

and the y-intercept becomes 0:

zY = b1zX

and the slope becomes r:

zY = rzX

### Assumptions for Bivariate Linear Regression

• Quantitative data (or dichotomous)

• Independent observations

• Predict for same population that was sampled

### Assumptions for Bivariate Linear Regression

• Linear relationship

• Examine scatterplot

• Homoscedasticity – equal spread of residuals at different values of predictor

• Examine ZRESID vs ZPRED plot

### Assumptions for Bivariate Linear Regression

• Independent errors

• Durbin Watson should be close to 2

• Normality of errors

• Examine frequency distribution of residuals

### Influential Cases

• Influential cases have greater impact on the slope and y-intercept

• Select casewise diagnostics and look for cases with large residuals

### Choosing Stats

Participants are asked to pretend that they are jurors and, after watching a videotape of a defendant being questioned, indicate whether they think the defendant is guilty or not. The defendants are either African American or Caucasian. The researcher hypothesizes that participants will be more likely to think the African American defendants are guilty as compared to Caucasians.