Many factor levels and small blocks
Incomplete designs arise in experiments with more treatments than experimental units or, if the experimental units are grouped into blocks, with more treatments than the block size. We have already studied one solution for multi-factor experiments:
Fractional factorial experiments leave each main factor effects orthogonal to other factors and the blocks — they are only confounded with high-order interactions.
A different situation arises in experiments with a single factor that has more levels than the block size. This often arises in variety trials in agriculture where 50 or more varieties may be assessed but the experimental units (often plots of land or containers in greenhouses) are grouped into relatively small blocks (fields or shelves in the greenhouse).
An incomplete design is again required if there are more treatments than the block size but, since there is only a single factor, fractional factorial designs cannot be used.
Which treatments should be allocated to each experimental unit?
This section describes some experimental designs for this situation.
Principles
Two main principles are usually followed in the design of experiments with more factor levels than the block size:
Neither of these principles is essential — we can analyse data from unbalanced experiments and ones with factor levels that have been used different numbers of times.
Experiments with 8 factor levels
The diagram below shows experimental designs with different block sizes for comparing 8 factor levels (A, B, ..., H). For each design,
Results are initially shown for an experiment in which there are 8 experimental units in each block. Since all factor levels can be used in each block, an orthogonal design can be used, so each treatment occurs with every other treatment in all blocks — the design is balanced — and all pairs of treatments can be compared with equal accuracy.
Select Block size 7 (balanced) from the pop-up menu. In this design, blocks and the factor are not orthogonal, but the design is balanced — all pairs of treatments occur together in 6 of the 8 blocks — so again all treatments can be compared with equal accuracy.
Balance cannot be achieved with smaller block sizes. Select Block size 6 (nearly balanced) from the pop-up menu. In this design, all pairs of treatments occur together in either 4 or 5 blocks. Observe that when pairs of treatments occur together 5 times, the difference is estimated more accurately (lower standard error), but because the design is almost balanced, the standard errors for all differences are similar.
Now select Block size 2 (unbalanced) from the pop-up menu. Each treatment only occurs in the same block as two others so the design is extremely unbalanced — some pairs of treatments can be compared much more accurately than others.
Finally use the pop-up menu to compare the two designs with Block size 3 and note how the design affects the accuracy of comparing factor levels. Picking a design that is nearly balanced is not always easy.