1. Chapter 11 Analysis of Covariance
BAE 6520 Applied Environmental Statistics
Biosystems and Agricultural Engineering Department
Division of Agricultural Sciences and Natural Resources
Oklahoma State University
Source
Dr. Dennis R. Helsel & Dr. Edward J. Gilroy
2006 Applied Environmental Statistics Workshop
and
Statistical Methods in Water Resources

2. Analysis of Covariance Application
Are there more than one regression line?
Are they parallel?
Do they have the same intercept?

3. Additional Examples

4. Analysis of Covariance (ANCOVA)
Technique that is between ANOVA and regression.
Similar to regression with dummy variables.
Covariate
Uncontrolled variable that influences the response but does not interact with any of the other factors being tested.
Removes variation due to covariate, which provides a more precise analysis
If not used, will be reflected in the error term.

5. Two Types of Explanatory Variables
Regression Analysis
Using Dummy Variables
Two Types of Explanatory Variables
Continuous Variable X
Y = b0 + b1 X
Models the relationship with Y
Binary (indicator) variable GROUP
GROUP = 0 or 1
Indicates a group assignment
0 = Winter
1 = Summer

6. Testing for a Shift in Intercept (Assuming Two Parallel Lines)
Y = b0 + b1 X + b2 GROUP
Winter: Group = 0 b2*GROUP = 0
Summer: Group = 1 b2*GROUP = b2
Null Hypothesis
b2 = 0
Alternative Hypothesis
b2 ≠ 0
Y = bo + b1X when GROUP=0 (winter)
Y = bo + b1X +b2 when GROUP=1 (summer)
Or Y = (bo + b2) + b1X
b2 is the shift in the intercept

7. MINITAB Output

8. Testing for Different Slopes (Assuming Equal Intercepts)
Y = b0 + b1 X + b3 GROUP X
Null Hypothesis
b3 = 0
Alternative Hypothesis
b3 ≠ 0
Y = bo + b1X when GROUP=0 (winter)
Y = bo + b1X + b3X when GROUP=1 (summer)
Or Y = bo + (b1 + b3) X

9. Testing for Different Slopes and Intercepts
Y = b0 + b1 X + b2 GROUP + b3 GROUP X
Null Hypothesis
b2 = 0 and/or b3 = 0
Alternative Hypothesis
b2 ≠ 0 and/or b3 ≠ 0
Y = bo + b1X GROUP=0 (winter)
Y = bo + b1X + b2 + b3X GROUP=1 (summer)
Or Y = (bo + b2) + (b1 + b3) X

10. Paired Watershed Study
ANCOVA Application
Paired Watershed Study
Paired Watershed Study - tool used to evaluate how land use change effects water quality in a treatment watershed while accounting for climatic variation with a calibration watershed.
A calibration period is used to establish a relationship between the watersheds and is then compared to the treatment period relationship.
If a significant difference can be found between the two relationships we can calculate the difference in slopes and intercepts.