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Understanding Regression Analysis: Rules and Examples

Learn the logic behind regression analysis, from interval-scale data to the strength of relationships. Explore the rules of the game, the importance of the regression line, and delve into practical examples like education and voter turnout. Master the interpretation and application of statistical measurements for dependent and independent variables. Enhance your analysis skills with visual inspection techniques and readings on democracy and ideology.

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Understanding Regression Analysis: Rules and Examples

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  1. THE LOGIC OF REGRESSION

  2. OUTLINE • The Rules of the Game: Interval-Scale Data and PRE (Strength) • Understanding the Regression Line (Form) • Example: Education and Voter Turnout • On the Importance of Visual Inspection (Scattergram)

  3. READINGS • Pollock, Essentials, review chs. 5-6, read ch. 7 (pp. 154-165) • Pollock, SPSS Companion, ch. 8 • Course Reader, Selections 3-4 (Ideology and Law, Correlates of Democracy)

  4. REGRESSION ANALYSIS: • THE BASIC GOALS • Taking full advantage of interval-scale data • Measuring form, strength, and significance of statistical • relationships • Specify associations between dependent and independent • variables

  5. THE RULES OF THE GAME • PRE = (E1 – E2)/E1 • Guessing Y without knowing X: mean value of Y • E1 = Σ(Yi –Y)2 • Guessing Y given knowledge of X: • Yi = a + bXi • Stipulations: a linear relationship, such that sum of squared deviations of observed values of Y from predicted values is minimal—thus, the line of “least squares”

  6. E1 = sum of squared deviations from the mean E2 = sum of squared deviations from the regression line PRE = (E1 – E2)/E1 Which measures the strength of the relationship The regression line—that is, the equation—measures the form of the relationship.

  7. Understanding the Regression Line • Path of the mean values of Y upon X • Estimated “average” values of Y for values of X • A line that cuts through the exact middle of • the scattergram • 4. A very precise statement of the form of a relationship

  8. Path of Mean Values of Y for Values of X

  9. Scattergram and Least-Squares Line

  10. Visualizing Line of Least Squares

  11. Variations in Relationships

  12. Elements of the Regression Equation

  13. Example: % High School Graduates (X) and % Turnout (Y)

  14. Regression Equation: High School Graduates and Turnout

  15. Estimated turnout = -26.27 + .87 (% graduates) When X is zero, predicted y = - .26.27 Question: Where is X when predicted value of Y = 0? Answer: Around 30.2 (compare to minimal value of X) Slope = +.87 (for every 1 percent increase in high-school graduates, an increase of .87 percent in turnout)

  16. What About Wyoming?

  17. On the Importance of the Scattergram 1. Visual confirmation of observed relationship 2. Identify patterns in deviations from the line—that is, in patterns among “residual values” 3. This is crucial since different arrays of data can produce identical regression lines (same form, that is, but different strength) 4. Identification of “outliers” (extreme cases)

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