1. The greatest blessing in life is
in giving and not taking.

2. Statistical Package Usage
Topic: Simple Linear Regression
By Prof Kelly Fan, Cal State Univ, East Bay

3. Overview Correlation analysis Linear regression model
Goodness of fit of the model
Model assumption checking
How to handle outliers

4. Example: Weight vs. Height
Gender
Height (inches)
Weight (pounds)
Age M
68.5
155 23 F
61.2 99 20
63.0
115 21
70.0
205 45
68.6
170 --
65.1
125 30
72.4
220 48

5. Correlation between X and Y
X and Y might be related to each other in many ways: linear or curved.
Correlation analysis provided here is only for the linear association.

6. (Pearson) Correlation Coefficient of X and Y
A measurement of the strength of the “LINEAR” association between X and Y
Sx: the standard deviation of the data values in X, Sy: the standard deviation of the data values in Y;
the correlation coefficient of X and Y is:

7. Correlation Coefficient of X and Y
The magnitude of r measures the strength of the linear association of X and Y
The sign of r indicate the direction of the association: “-”  negative association
“+”  positive association

8. Examples of Different Levels of Correlation
Median Linearity
r=.98
Strong Linearity

9. Examples of Different Levels of Correlation
Nearly Uncorrelated
r=.00
Nearly Curved

10. Example: Weight vs. Height

11. Graphical Summary of Two Quantitative Variable
Scatterplot of response variable against explanatory variable
What is the overall (average) pattern?
What is the direction of the pattern?
How much do data points vary from the overall (average) pattern?
Any potential outliers?

12. Summary for Height and Weight
Some Simple Conclusions
Scatterplot (Weight vs. Height)
Weight is somewhat Linearly related with Height
Weight is Increasing as Height increases.
Data points are more or less around the line.
No potential outlier.

13. Regression Equation
The regression line models the relationship between X and Y on average.
The math equation of a regression line is called regression equation.

14. The Usage of Regression Equation
Predict the value of Y for a given X value
Eg. What is the weight for a student with 60” high?

15. Predicted Value (Fitted Value)
is called “predicted Y,” pronounced as “y hat,” which estimates the average Y value for a specified X value.
Eg.
The predicted weight for a given height

16. The Limitation of the Regression Equation
The regression equation cannot be used to predict Y value for the X values which are (far) beyond the range in which data are observed.
Eg. The predicted WT of a given HT:
Given HT of 50”, the regression equation will give us WT of x50 = pounds!!

17. The Unpredicted Part
The value is the part the regression equation (model) cannot predict, and it is called “residual.”

18. } residual

19. Goodness of Fit
R^2 is the proportion of Y variance explained/accounted by the model we use to fit the data
When there is only one X (simple linear regression) R^2 = r^2.

20. SAS Output

21. Confidence Interval of Mean Y

22. Prediction Interval

23. Model Checking Scatter plot shows linear pattern
Normality: The residuals must follow a normal distribution
Equal Variances: The variance of residuals must be constant over the different values of X

24. Checking the Normality

25. Tests for Normality

26. Checking the Equal Variances

27. Identify Outliers using Residual Plots
Use “studentized” residuals!!
The cases with studentized residuals of size 3 or more are outliers

28. Transformation
If data show non-normal or unequal variances, then transform Y.
If data show a non-linear pattern, then add X^2 into the model

29. The Influence of Outliers
Type I Outlier
The slope becomes bigger
(toward outliers)
The r value becomes smaller (less linear)

30. The Influence of Outliers
Type II Outlier
The slope becomes clear (toward outliers)
The | r | value becomes larger (more linear: 0.1590.935)

31. How to Handle Outliers
Use Spearman correlation coef (the r of rank of X and rank of Y)
For Type I outliers:
Try to find the special factor behind the outliers.
Report two models with and without outliers.
For Type II outliers:
Report the model without outliers and mention the existence of outliers.
Whenever possible, collect more data in the range of X where outliers appear.

32. SAS Output