Regression - Regression Regression Statistical technique...

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Regression Regression: Statistical technique for determining the best-fitting straight line for a set of data The best-fitting straight line is called the regression line . The regression line is a mathematical model of the relationship between the variables, and it can be used to predict the value of Y (DV) from a known value of X (IV). 1. Equation of the regression line Ŷ = bX + a Ŷ= predicted value of Y; predicted score on DV b = slope of regression line; the change in Y for each unit of increase in X X = known value of X; known score on IV a = y-intercept; where the regression line crosses the Y axis; the value of Ŷ when X = 0 2. How is the best-fitting line determined? The equation for slope and y-intercept are derived from a calculus-based procedure called the least squares method . In this method, the goal is to minimize the distance between the line and the actual data points. (The squares part of this term comes from squaring the distance.) Y – Ŷ = error or residual (the difference between actual and predicted value of Y) ( 29 = - 2 ˆ Y Y SS error = sum of squared error The best-fitting regression line is defined as the line that has the smallest possible SS error 3. Equations for slope and y-intercept You are NOT required to memorize the following equations. I’m providing these equations here so you know that the values for b and a don’t just come out of thin air. b = ( 29 ( 29 ( 29 - - - 2 X Y X M X M Y M X a = X Y bM M - 4. Using the Regression Line to Make Predictions Example 1: We can use the data set I provided at the beginning of the unit on correlation to predict depression score from optimism score. I calculated the slope and y-intercept, and they are b = -.44 and a = 16.79 We can plug those values into the regression equation to predict depression Ŷ = -.44X + 16.79 Notation note : Ŷ is called Y hat. Notation note : M X is the mean of the X values, and M Y is the mean of the Y values.
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Page 2 of 6 Ŷ = predicted depression score X = known optimism score a) What if someone has an optimism score of 14? What do you predict for depression? Ŷ = -.44X + 16.79 = -.44(14) + 16.79 = 10.63
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Regression - Regression Regression Statistical technique...

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