1. Correlation and Regression

2. Correlation Analysis
Pearson’s correlation coefficient (r, rho for population) measures the degree to which there is a linear association between two metric variables.
Measures the strength of the relationship between two or more variables.
Correlation coefficient lies between –1 and + 1
Correlation coefficient is NOT an indicator of causal relationship between variables

3. Positive Linear Correlation
General trend in the plotted points is from bottom left to top right.
Negative Linear Correlation
General trend in the plotted points is from top left to bottom right.
No Linear Correlation
No general trend in plotted points, or a non-linear trend.
The strength of the linear correlation can be judged by looking at how closely the points approximate a straight line.

4. Scatter diagram

7. Curious?

8. Variable x y xy x2 y2
3.545 30
106.35
900
2.6 32
83.2
6.76
1024
3.245
97.35
3.93 24
94.32
576
3.995 26
103.87
676
3.115
93.45
3.235 33
1089
3.225 27
87.075
729
2.44 37
90.28
5.9536
1369
3.24
103.68
2.29
84.73
5.2441
2.5 34 85
6.25
1156
4.02
104.52
Sums
41.38
398
12388

9. Testing the Significance of the Correlation Coefficient (why?)
Null hypothesis:
Ho : ρ equal to 0
Alternative hypothesis:
Ha : ρ not equal to 0

10. Output 1

11. Regression Analysis
Used to understand the nature of the relationship between two or more variables
A dependent or response variable (Y) is related to one or more independent or predictor variables (Xs)
Object is to build a regression model relating dependent variable to one or more independent variables (how is y changing with x?)
Model can be used to describe, predict, and control variable of interest on the basis of independent variables

12. Simple Linear Regression
Yi = βo + β1 xi + εi
Where Y
Dependent variable X
Independent variable βo
Intercept
Mean value of dependent variable (Y) when the independent variable (X) is zero

13. Simple Linear Regression (Contd.)β1
Model parameter
Slope that measures change in mean value of dependent variable associated with a one-unit increase in the independent variable εi
Error term that describes the effects on Yi of all factors other than value of Xi

14. Regression – Illustrative Example
Let us check whether x is related to y.
Calculate point estimate bo and b1 of unknown parameter βo and β1

15. Output linear regression
How do we write the regression equation?

16. Testing the Significance of the Independent Variables
Null Hypothesis
There is no linear relationship between the independent & dependent variables
Alternative Hypothesis
There is a linear relationship between the independent & dependent variables

17. Output linear regression
How do we interpret the results?

18. Coefficient of Determination (R2)
Measure of regression model's ability to predict
R2 = SST - SSE
SST
= SSM
= Explained Variation
Total Variation

19. Output linear regression

20. Multiple Linear Regression
A linear combination of predictor factors is used to predict the outcome or response factors
Involves computation of a multiple linear regression equation
More than one independent variable is included in a single linear regression model

21. Evaluating the Importance of Independent Variables
Which of the independent variables has the greatest influence on the dependent variable?
Consider the standardized estimate.
CAVEAT: Use standardized to compare independent variables within a sample ONLY. Do not use it for across-sample comparisons.