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

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

4. 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

5. 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

6. Bivariate Relationships
Plotting a Line

7. 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

8. Plot showing positive covariance

9. 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

10. 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?

11. 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

12. X and Y
Y X
2 1 2
4 3
3 4
6 5
5 6

13. X and Y
Y X
2 1 2
4 3
3 4
6 5
5 6

14. Theory
Living in an urban area allows better access to prenatal care.

15. Output Source SS df MS Number of obs = 41 F( 1, 39) = 9.09
Model Prob > F =
Residual R-squared =
Adj R-squared =
Total Root MSE =
prenatalcarepct Coef. Std. Err t P>t [95% Conf. Interval]
urbanpctoftotal
_cons

16. Linear Equation

17. 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