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Understanding Linear Regression and Variance Explained

This text provides a comprehensive overview of linear regression analysis, focusing on the correlation between variables X and Y. It highlights how, in cases of perfect correlation, one variable can perfectly predict the other. The discussion extends to imperfect correlations and introduces key concepts such as explained and unexplained variance, represented mathematically in terms of total variance and the correlation coefficient r². Additionally, practical examples are included to clarify these statistical concepts.

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Understanding Linear Regression and Variance Explained

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  1. Monday, October 17 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…

  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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