Efficiency factors in block designs

As a more complete, if a little pedestrian, example of a function, consider finding the efficiency factors for a block design. (Some aspects of this problem have already been discussed in §.)

A block design is defined by two factors, say blocks (b levels) and varieties, (v levels). If $R^{v\times v}$ and $K^{b\times b}$ are the replications and block size matrices, and $N^{b\times v}$ is the incidence matrix, then the efficiency factors are defined as the eigenvalues of the matrix $$E = I_v - R^{-1/2}N'K^{-1}NR^{-1/2} = I_v - A'A$$where A = K^-1/2NR^-1/2 . One way to write the function is as in Figure .

 

Figure:   A function for block design efficiencies

It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines.

The result of the function is a list giving not only the efficiency factors as the first component, but also the block and variety canonical contrasts, since sometimes these give additional useful qualitative information.