2.2 Sixteen Treatments in a 4x4 Lattice


This is a problem for which there is a classical statistical design solution (lattice ) but it is included here to
illustrate the methods. The experiment, at Poza Rica, is to compare 16 varieties for drought tolerance. Four
replicates, each split into four blocks are to be used.

This time we have no structure to guide us so we make an arbitrary split in the first replicate.

Replicate 1
 Block 1 Block 2 Block 3 Block 4
 (1,2,3,4) (5,6,7,8) (9,10,11,12) (13,14,15,16)

 For the second replicate each block must include one variety from each of the first four blocks.

 Replicate 2
 Block 5 Block 6 Block 7 Block 8
 (1,5,9,13) (2,6,10,14) (3,7,11,15) (4,8,12,16)

 For the third replicate each block must include one variety from each of the first four blocks and one
 variety from each of the second four blocks. This requires slightly more thought than the second replicate
 but can be solved for the first block(9) and systematically thereafter.

 Replicate 3
 Block 9 Block 10 Block 11 Block 12
 (1,6,11,16) (2,5,12,15) (3,8,9,14) (4,7,10,13)

 That was probably the hardest stage, and for the fourth replicate the choices are reduced and the problem
 gets a little easier. If we had made a less fortunate choice in the third replicate then we could have found
 ourselves with no choice in the fourth replicate. We would then have had to try a different third replicate.

 Replicate 4
 Block 13 Block 14 Block 15 Block 16
 (1,7,12,14) (2,8,11,13) (3,5.10,16) (4,6,9,15)


This completes the required design. Note that if a fifth replicate were needed there is one more division
into four blocks which brings together all those pairs not previously linked. Note also that if we had only
needed three replicates we could have stopped after the third because of the sequential nature of the
construction.

Precision:
The range of standard errors for the estimated treatment differences is

0.791 a to 0.817 a with a mean of 0.796 o.

The minimum possible S.E. would be

 a0 q(2/4) = 0.707 o.