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Understanding Correlation and Regression for Perfect Prediction

Explore the relationship between variables through correlation and linear regression with practical examples. Learn how to predict outcomes when variables are perfectly or imperfectly correlated. Understand variance explanation and prediction accuracy.

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Understanding Correlation and Regression for Perfect Prediction

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  1. Monday, October 8 Wednesday, October 10 Correlation and Linear Regression

  2. zy = zx When X and Y are perfectly correlated

  3. We can say that zx perfectly predicts zy zy’ = zx Or zy = zx ^

  4. When they are imperfectly correlated, i.e., rxy ≠ 1 or -1 zy’ = rxyzx

  5. Example from hands… • Let’s double-check our understanding of what a correlation coefficient is with respect to z-scores on X and Y variables.

  6. When we want to express the prediction in terms of raw units: zy’ = rxyzx Y’ = bYXX + aYX bYX = rYX (σy / σx) aYX = Y - bYXX _ _

  7. SStotal = SSexplained+SSunexplained N N N Explained and unexplained variance SStotal = SSexplained + SSunexplained

  8. σ2Y’ [ =unexplained] σ2Y [ =total] Explained and unexplained variance r2XY = 1 - σ2Y - σ2Y’ = σ2Y r2 is the proportion explained variance to the total variance.

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