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Intro to Regression

Intro to Regression. POL 242. Summary. Regression is the process by which we fit a line to depict the relationship between two variables (usually both interval or close to it). This line allows us to predict a value of the dependent variable at each level of the independent variable.

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Intro to Regression

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  1. Intro to Regression POL 242

  2. Summary • Regression is the process by which we fit a line to depict the relationship between two variables (usually both interval or close to it). • This line allows us to predict a value of the dependent variable at each level of the independent variable. • And the predicted change in the DV for a one-unit change in the IV. • There is only one line that minimizes the distance between the actual, observed values of the independent value and the predicted values of the independent variable. • The equation of a line is written with a slope and a y-intercept. For regression, the slope is the [unstandardized] coefficient and the y-intercept is called the constant.

  3. Residuals • The vertical distance between each point and the line represents the magnitude of an error in prediction. Observed (or Real) Value = Y Predicted Value = Y' • The difference between the real and the predicted value is called the residual. The residual is the error in the prediction = Y - Y' • To draw the best line, we want to minimize this error (shorten the distance between Y and Y') for all observations.

  4. Residuals Squared • However, we cannot simply take the sum of the differences because some predicted Y' values will be greater than observed Y values (those points that fall below the line) and some Y' will be less than Y (illustrated with points above the line). • To keep these negative and positive numbers from canceling each other out, we square the difference between the observed value and the predicted value = (Y-Y')2 • There is only one line that will minimize Σ(Y-Y')2

  5. Equation of the regression line • The equation of the line that predicts Y for values of X is: Y' = by X + αy Y' = predicted or estimated value of Y by = slope of the line for minimizing errors in predicting Y. This is the unstandardized coefficent. It represents how much Y is predicted to change for every one unit change in X. αy  = Y axis intercept for minimizing errors in predicting Y. This is the constant. X = the [observed] value of X

  6. Slope • How to calculate the slope: • Where SSx = sum of the squares of X  =    • N = number of paired observations.

  7. The constant • How to calculate the Y-axis intercept: • A bar above the variable letter means that it is an average. For example:

  8. Three Examples • Study hard, get good grades? • http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html • http://standards.nctm.org/document/eexamples/chap7/7.4/

  9. Webstats Output Predicted change in Y for every one unit change in X (Slope) ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T AGE 4.668449 2.550202 .568988 1.831 .1098 (Constant) -107.526203 95.822840 -1.122 .2988 Y-intercept

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