The slope, explained variance, residuals Bivariate regression The slope, explained variance, residuals

What is the formula for a slope? A. e = mc2 B. Yi = a + bxi + ei C. ŷ = a + bx D. y ≥ x ≥ a

What information does the slope provide? A. whether the relationship is statistically significant B. whether a case is a severe outlier, like Buchanan’s share of the vote C. on average, what is the predicted value of y, given various values of x D. which baseball batter is likely to hit best in the next game

What is a? 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

Plot showing positive covariance

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.

Output Source SS df MS Number of obs = 41 F( 1, 39) = 9.09 Model 860.523694 1 860.523694 Prob > F = 0.0045 Residual 3693.55683 39 94.7065855 R-squared = 0.1890 Adj R-squared = 0.1682 Total 4554.08053 40 113.852013 Root MSE = 9.7317 prenatalcarepct Coef. Std. Err. t P>t [95% Conf. Interval] urbanpctoftotal .2517241 .083509 3.01 0.005 .0828111 .4206371 _cons 76.35186 4.367962 17.48 0.000 67.51682 85.18689

Linear Equation

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