Randomization for a balanced superimposition of one Youden square on another

Some experimentation, particularly on trees in orchards, requires the superimposition of a new set of treatments on an existing experiment, on the assumption that the effects of the new treatments do not interact with those of the old. Different approaches to the randomization of such new treatments were considered by Preece, Bailey and Patterson, who investigated in detail the case when one Latin square is superimposed on another to produce a Graeco-Latin square. For non-orthogonal superimposition the theory is in general much more complicated. However, this paper shows that fairly simple randomization results are obtainable for certain balanced superimpositions of one Youden square on another (i) when each Youden square is of size (t-1) \times t and (ii) when each Youden square is of size k \times t, where t is a prime power of the form 4s-1 and k=2s-1 or k=2s. In each case, t is the number of treatments in each Youden square. The valid randomization procedure obtained for the second case is unlike any previously known randomization procedure and is unlikely to be acceptable in practice; it is, however, an excellent indication of the difficulties that arise when randomization is attempted in unusual circumstances.