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 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

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LINEAR REGRESSION: What it Is and How it Works

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Linear regression what it is and how it works

LINEAR REGRESSION:What it Is and How it Works


Overview

Overview

  • What is BivariateLinear Regression?

  • The Regression Equation

  • How It’s Based on r

  • Assumptions


What is bivariate linear regression

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

Why is it Bivariate?

  • Two variables: X and Y

  • X - independent variable/predictor variable

  • Y - dependent/outcome/criterion variable


Why is it linear

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

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)


Regression lines

Regression Lines


Least squares solution

Least-Squares Solution

  • Minimize squared error in prediction.

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


Residuals

Residuals


How it s based on r

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

Assumptions for Bivariate Linear Regression

  • Quantitative data (or dichotomous)

  • Independent observations

  • Predict for same population that was sampled


Assumptions for bivariate linear regression1

Assumptions for Bivariate Linear Regression

  • Linear relationship

    • Examine scatterplot

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

    • Examine ZRESID vs ZPRED plot


Checking for homoscedasticity

Checking for Homoscedasticity


Assumptions for bivariate linear regression2

Assumptions for Bivariate Linear Regression

  • Independent errors

    • Durbin Watson should be close to 2

  • Normality of errors

    • Examine frequency distribution of residuals


Checking for normality of errors

Checking for Normality of Errors


Influential cases

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

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.


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