1. Yesterday Correlation Regression -Definition
-Deviation Score Formula, Z score formula
-Hypothesis Test
Regression
Intercept and Slope
Unstandardized Regression Line
Standardized Regression Line
Hypothesis Tests

2. Summary Correlation: Pearson’s r Unstandardized Regression Line

3. Some issues with r Outliers have strong effects
Restriction of range can suppress or augment r
Correlation is not causation
No linear correlation does not mean no association

4. Outliers
Child 19 is lowering r
Child 18 is increasing r

5. The restricted range problem
The relationship you see between X and Y may depend on the range of X
For example, the size of a child’s vocabulary has a strong positive association with the child’s age
But if all of the children in your data set are in the same grade in school, you may not see much association

6. Common causes, confounds
Two variables might be associated because they share a common cause.
There is a positive correlation between ice cream sales and drownings.
Also, in many cases, there is the question of reverse causality

7. Non-linearity
Some variables are not linearly related, though a relationship
obviously exists
For monotonic relationships that are not linear we use Spearman’s r

8. Regression: Analyzing the “Fit”
How well does the regression line describe the data?
Assessing “fit” relies on analysis of residuals
Are the residuals randomly distributed? (If no, perhaps a linear model is inappropriate)
How large are the residuals? Too big? (low correlation means big residuals)

9. Assumptions of Regression
The residuals have mean of 0 and variance of sresid2
The residuals are uncorrelated with X
The residuals are homoscedastic (similarly sized across the range of x)

10. Residual Diagnostics I: Graphing

11. Residual Diagnostics I: Graphing
Residual Plot
resid
Problem: curvilinearity

12. Residual Diagnostics I: Graphing
Agreeableness Time 2

13. Residual Diagnostics I: Graphing
Residual Plot
Residuals
Problem: heteroscedasticity

14. Regression: Analyzing the “Fit”
How well does the regression line describe the data?
Assessing “fit” relies on analysis of residuals
Are the residuals randomly distributed? (If no, perhaps a linear model is inappropriate)
How large are the residuals? Too big? (low correlation means big residuals)
Residual plots
ANOVA

15. Regression ANOVA
SSY
SSmodel
SSresid Y Y’

16. Regression ANOVA Source SS df s2 Model Error Total F=t2
“the amount of variance in Y explained by our model”

17. Exercise X Y Fill in the ANOVA table 1 3 4 5 6 9 7 Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5

18. Exercise X Y Y’ (Y-Y’)2 1 3 4 5 6 9 7 SSresid = … SSmodel = …
Predicted value
(Y-Y’)2
Residual
(Unpredicted deviation)
(Predicted Deviation) 1 3 4 5 6 9 7
SSresid = …
SSmodel = …
Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5

19. Exercise X Y Y’ (Y-Y’)2 1 3 3.5 (-0.5)2 (-1.5)2 4 (0.5)2 5 (-1)2 (0)2
Predicted value
(Y-Y’)2
Residual
(Unpredicted deviation)
(Predicted Deviation) 1 3
3.5
(-0.5)2
(-1.5)2 4
(0.5)2 5
(-1)2
(0)2 6
(1)2 9
6.5
(1.5)2 7
SSresid = …
SSmodel = …
Mean: 5 5
Stdevp: r=
b= 0.375
a= 3.125
X Y Y'
5 4 5
5 6 5 3 9

20. Regression ANOVA
Source SS df s2 F
Model
Error
Total

21. Regression ANOVA
Source SS df s2 F
Model 9 1 12
Error 3 4
.75
Total 5