The subplots are assumed to be nested within the whole plots, so that a whole plot consists of a cluster of subplots and a level of A is applied to the entire cluster. Such an arrangement is often necessary by nature of the experiment, the classical example being the application of fertilizer to large plots of land and different crop varieties planted in subdivisions of the large plots. For this example, fertilizer is the whole plot factor A and variety is the subplot factor B . The first example is a split-plot design for which the whole plots are arranged in a randomized block design. Not only do you have as in the general linear model, but you have unknown parameters in , G , and R as well.
Least squares is no longer the best method.
Lacking such information, one approach is to use estimated GLS, in which you insert some reasonable estimate for V into the minimization problem. The goal thus becomes finding a reasonable estimate of G and R . Using calculus, it is possible to reduce this maximization problem to one over only the parameters in G and R . X ' V -1 y and p is the rank of X . One advantage of using the Newton-Raphson algorithm is that the second derivative matrix of the objective function evaluated at the optima is available upon completion.