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LINEAR REGRESSION: What it Is and How it Works

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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  1. LINEAR REGRESSION:What it Is and How it Works

  2. Overview • What is BivariateLinear Regression? • The Regression Equation • How It’s Based on r • Assumptions

  3. 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

  4. Why is it Bivariate? • Two variables: X and Y • X - independent variable/predictor variable • Y - dependent/outcome/criterion variable

  5. 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

  6. 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)

  7. Regression Lines

  8. Review Question! Suppose we are doing a regression to predict alertness from amount of caffeine assumed. SPSS output shows that the slope is 1.50 and the intercept is -72.00. Write out the regression equation.

  9. Least-Squares Solution • Minimize squared error in prediction. • Error (residual) = difference between predicted y and actual y

  10. Residuals

  11. 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

  12. Assumptions for Bivariate Linear Regression • Quantitative data (or dichotomous) • Independent observations • Predict for same population that was sampled

  13. Assumptions for Bivariate Linear Regression • Linear relationship • Examine scatterplot • Homoscedasticity – equal spread of residuals at different values of predictor • Examine ZRESID vs ZPRED plot

  14. Checking for Homoscedasticity

  15. Assumptions for Bivariate Linear Regression • Independent errors • Durbin Watson should be close to 2 • (<1 or >3 = bad) • Normality of errors • Examine frequency distribution of residuals

  16. Checking for Normality of Errors

  17. Influential Cases • Influential cases have greater impact on the slope and y-intercept • Select casewise diagnostics and look for cases with large residuals

  18. Review Question! When predicting z-scores on Y from z-scores on X, what does the slope equal?

  19. 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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