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

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

Additional Examples

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.

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

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

MINITAB Output

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

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

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.