Atcually it's not very difficult. Just include regressions and classification effects! That's part of the beauty (sic) of using a 'unbalanced' matrix approach to linear models. By the time that we have seen multiple regression, and 2-way ANOVA, then including more than one effect in a model is trivial; and exactly the same formulae can be used for analysis of covariance.
What is much more important is understanding whether covariances should or can be included in the model. We should only use a covariate if it is something that is measured on the experimental unit BEFORE the start of the treatment. For example, with dairy cattle on feeding trials, it is quite common to measure the milk yield for a period before the start of the actual comparative trial, and to use this pre-treatment milk yield as a covariate to try and reduce the residual variation, se2. Similarly, with humans on nutritional, dietetic trials, we might be measuring the cholersterol level in individuals receiving different diets. Suppose that we weigh each individual at the start of the experiment and aat the end. It would be permisible to include initial body weight as a covariate in the analysis (Yij = µ + trti + bwtij + eij). It would be much more dangerous to include the weight at the end of the trial as a covariate. Why? Because it is possible that the treatments may influence the final bodyweight, and hence including final bodyweight as a covariate will be 'adjusting out' part of the very effects that we are trying to look at! Not a good idea.
