The Multivariate Linear Model
Our analysis of the multivariate linear model will attempt to repeat all of the previous material for the univariate case in one fell swoop for the multivariate case. Since the foundation for many of the issues has already been laid, this attempt may not be as ambitious as it may appear. Often, it will just be necessary to discuss how the multivariate case is similar or different from the univariate. Themain themeof themultivariate linearmodel is the change froma single response
yi for individual i to many responses for a single individual. In general, the different responses will have different units, as we may be looking at height, weight, and so on from a plant in response to fertilizer, water, or soil factors. In other cases, we may have multiple responses in the same units, such as concentrations of lead in different tissues, or blood pressure at different time points, for the same individual. We will leap into Gauss-Markov estimation with the goal of repeating Chapters
2-4 for the multivariate case. Then we will look at maximum likelihood estimation and distribution theory in Section 9.3. Hypothesis testing in the multivariate case will be discussed in Section 9.4. Repeated measures problems, where the multiple responses are all the same type, will be treated in Section 9.5.