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## Bivariate regression

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**Bivariate regression**The slope, explained variance, residuals**What is ŷ?**• A. The error of the estimate • B. the expected value of x • C. the expected value of y • D. the formula for a line**What is a residual?**• A. all the numbers above the slope • B. all the numbers below the slope • C. the distance between actual y and predicted y • D. the difference between actual x and predicted x**What is a in the formula ŷ = a +bx?**• A. the y intercept • B. the value of y when x = 0 • C. where the slope crosses the y a axis • D. all of the above**Bivariate Relationships**Plotting a Line**Review: Covariance**• When it tends to be the case that x is greater than the mean when y is greater than the mean AND x is lower than the mean when y is lower than the mean, then there is a positive covariation**Expected value**• But we may want to know more specific knowledge than that – we may want to know the expected value of y for each increased value of x • I may know the mean of everyone’s height in class • But if I know gender, then I can generate two expected values • If you remember, we are always trying to do better than the mean**Substantive effect**• For every 10K dollars given in humanitarian aid, there is an increase in 3K spent on weapons • For every 10K dollars given in humanitarian aid, there is a .5K increase spent on weapons • For every 10K dollars given in humanitarian aid, there is a 8K increase spent on weapons • Unit of analysis?**Regression equation**• y = a + bx + e • ŷ = a + bx • ŷ is also known as yhat • y is the dependent variable value • yhat is the predicted value • a is the intercept**X and Y**• Y X • 2 1 • 2 • 4 3 • 3 4 • 6 5 • 5 6**X and Y**• Y X • 2 1 • 2 • 4 3 • 3 4 • 6 5 • 5 6**Theory**Living in an urban area allows better access to prenatal care.**ŷ= a + bx**• b is slope – rise over run • a is the y intercept; constant • Standard error is the average error from the actual points to the slope • T is the ratio of the slope divided by the standard error • Beta = Pearson r in bivariate analysis**Policy Liberalism and Public Opinion (Erikson, Wright &**McIver, 1987) Q: Where does the diagonal line come from?**Extending Interval Correlations**• Regression and correlation are intimately related concepts. • You’ve probably all tried to map lines in Cartesian spaces before…**Policy Liberalism and Public Opinion (Erikson, Wright &**McIver, 1987) Q: Where does the diagonal line come from?**Estimating the magnitude of the effect from the slope**• Find the range of x • (x may vary from 0-4, 1-5, 0-100000) • Multiply the range by the slope to find the effect of change in y, going from the lowest value of x to the highest value of x • If x ranges from 0-4, the range is 4 • If the slope is .25, the magnitude of the effect is 1**An example from the literature**• http://lilt.ilstu.edu/gmklass/pos138/articles/womensrep.pdf