Alpha Lattice vs Balanced Lattice: incomplete-block designs for large variety trials

When a variety trial has 60, 80 or 150 entries, a complete block becomes too large to stay uniform. Incomplete-block designs split each replication into small blocks so the within-block variation stays small. The two textbook families are the balanced lattice (Yates 1936) and the alpha lattice (Patterson and Williams 1976). The choice is set by your entry count and how many replications you can run.

The decision in one sentence

Use a balanced lattice when the number of entries is a perfect square and you can afford k + 1 replications. Use an alpha lattice when the entry count is not a clean square, or when you can only run two or three replications, because the alpha design works for almost any number of entries and any practical replication count.

What each design demands

Design            Entries (v)        Replications     Block size
Balanced lattice  perfect square     k + 1            k = sqrt(v)
                  (e.g. 25, 49, 81)  (rigid)
Alpha lattice     almost any v       2, 3, or more    user-chosen,
                  (e.g. 60, 80, 96)  (flexible)       v divisible by k

A balanced lattice for 81 entries needs block size 9 and 10 replications. That is often impossible on seed and land. The alpha lattice for 80 entries can use block size 8 with 2 replications, and still recover information between blocks.

Worked example: 9 entries, blocks of 3

A small balanced lattice makes the mechanics visible. Nine entries (T1 to T9), block size 3, in three replications. Each replication is partitioned into 3 incomplete blocks, and every pair of entries appears together in a block the same number of times across the full design.

rep,block,entry,yield_t_ha
1,1,T1,4.21
1,1,T2,4.55
1,1,T3,3.95
1,2,T4,5.12
1,2,T5,4.78
1,2,T6,5.31
1,3,T7,4.02
1,3,T8,4.62
1,3,T9,4.35
2,1,T1,4.31
2,1,T4,5.21
2,1,T7,4.12
2,2,T2,4.62
2,2,T5,4.85
2,2,T8,4.71
2,3,T3,4.05
2,3,T6,5.38
2,3,T9,4.41

What the analysis does

The incomplete-block analysis estimates a block effect within each replication, then adjusts each entry mean for the blocks it appeared in. The payoff is the recovery of inter-block information: an adjusted mean that is more precise than the raw mean when the blocks really did absorb field variation. The output reports the efficiency of the lattice relative to a complete-block RBD on the same plots. If that efficiency is near 100 percent, the blocking did not help and a plain RBD would have been fine. If it is well above 100 percent, the incomplete blocks bought real precision.

How to read the output

Rank entries by adjusted mean, not raw mean. Use the reported CD for the adjusted comparisons. Check the relative efficiency: it tells you whether the design earned its extra complexity on this field.

How to pick before you plant

Count the entries. If it is a perfect square and you can run k + 1 replications, the balanced lattice is the most efficient option. If it is not a clean square, or seed and land limit you to two or three replications, use the alpha lattice. In modern multi-location breeding trials with awkward entry counts, the alpha lattice is the default for exactly this reason.

Common mistakes

Forcing entries into a balanced lattice by adding filler varieties to reach a perfect square, which wastes plots. Analysing a lattice as a plain RBD, which discards the block adjustment and the efficiency gain. Reporting raw means when the design produced adjusted means. Choosing block size with no relation to a uniform patch of land, which defeats the purpose of incomplete blocking.

When the efficiency comes back near 100 percent

That is not a failure, it is information. It means the field was uniform enough that incomplete blocking did not separate real variation, so the RBD analysis on the same data is essentially equivalent. You spent some df on blocks and learned the field was even. The cost was small. The reverse mistake, running an RBD on a heterogeneous field and never blocking, is the expensive one.