1. Analysis of Covariance
Comparison of Head-Lengths for Trout from 2 Lakes
Source: C. McC. Mottley (1941). “The Covariance Method of Comparing the Head-Lengths of Trout From Different Environments,” Copeia, Vol.1941, #3, pp

2. Analysis of Covariance
Goal: To Compare 2 or More Group/Treatment Means for One Variable, after controlling for one or more concomitant variables/Covariates
Procedure: Fit a Regression Model Relating Response to Group (Using Dummy Variables) and Covariate(s)
Adjusted Means: Fitted Values from regression for each group evaluated at common value(s) of covariate(s), typically at the mean value(s)
Treatments can be combinations of factors in a Factorial Structure
Model:

3. Example – Fale Trout Heads from 2 Lakes (Environments)
Units: Female Fish Sampled from Lakes
Response Variable: Head Length (mm)
Groups: Environments (Kootenay Lake, Wilson Lake)
Covariate: Standard-Length (mm)
Samples: n1 = 50 from Kootenay, n2 = 10 from Wilson
Transformation: Base10 logarithms of Head-Length and Fish-Length Taken for Biological Reasons
Summary Statistics:

4. T-test – Ignoring Covariate

5. T-test – Ignoring Covariate

6. Adjusting for Body Length

7. Adjusted Means
Goal: Compare Means Head Lengths after adjusting for Body Lengths
Start with the Unadjusted Mean, and Subtract off the product of the slope from the regression and the difference between the group mean and the overall mean for the covariate. Effect of adjustment (assuming positive slope):
Adjust down for groups with high levels of covariate
Adjust up for groups with low levels of covariate

8. Trout Head-Length Data