1. Happiness comes not from material wealth but less desire.

2. Applied Statistics Using SAS and SPSS
Topic: Simple linear regression
By Prof Kelly Fan, Cal State Univ, East Bay

3. Example: Computer Repair
 A company markets and repairs small computers. How fast (Time) an electronic component (Computer Unit) can be repaired is very important to the efficiency of the company. The Variables in this example are:
Time and Units.

4. How long will it take me to repair this unit?
Humm…
How long will it take me to repair this unit?
Goal: to predict the length of repair Time for a given number of computer Units

5. Computer Repair Data Units Min’s 1 23 6 97 2 29 7 109 3 49 8 119 4 64
149 74
145 5 87 10
154 96
166

6. 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?

7. Summary for Computer Repair Data
Some Simple Conclusions
Scatterplot (Time vs Units)
Time is Linearly related with computer Units.
(The length of) Time is Increasing as (the number of) Units increases.
Data points are closed to the line.
No potential outlier.

8. Numerical Summary of Two Quantitative Variable
Regression equation
Correlation

9. Review: Math Equation for a Line
Y: the response variable
X: the explanatory variable
Y=b0+b1X Y
} b1 1
} b0 X

10. 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.

11. The Usage of Regression Equation
Predict the value of Y for a given X value
Eg. How long will it take to repair 3 computer units?

12. General Notation
is called “predicted Y,” pronounced as “y hat,” which estimates the average Y value for a specified X value.
Eg.
The predicted repair time of a given # of units

13. 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 40”, the regression equation will give us WT of x40 = -5 pounds!!

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

15. residual {

16. Correlation between X and Y
X and Y might be related to each other in many ways: linear or curved.

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

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

19. (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:

20. 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

21. 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.

22. SPSS Output
Analyze >> Regression >> Linear

23. Confidence Intervals

24. Check for Normality

25. Check for Equal Variances
SCATTERPLOT of zresid & zpred

26. The Influence of Outliers
The slope becomes smaller (toward outliers)
The r value becomes smaller (less linear)

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

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