Alpha lattice for a 64-entry wheat yield trial: adjusted means and efficiency over RBD
Sixty-four wheat entries that will not fit a homogeneous block. An 8 by 8 alpha lattice with incomplete blocks, block-adjusted entry means, and the efficiency gain over a notional RBD.
Priya is running a preliminary yield trial with 64 advanced wheat entries. The available field is long and runs across a known fertility gradient, so a single block of 64 plots would not be uniform. A randomised block design needs every entry in one homogeneous block, which is not realistic at this plot count. She uses an alpha lattice instead: an 8 by 8 design with incomplete blocks of 8 entries, replicated twice.
Why an incomplete-block design
With 64 entries an RBD block would be 64 plots wide and would straddle the fertility gradient, inflating the error. An alpha lattice breaks each replicate into eight small incomplete blocks of eight plots. Small blocks are far more uniform, so the design removes block-to-block soil variation that an RBD would have left in the error term.
Question
After adjusting entry means for incomplete-block effects, which entries are genuinely top yielding, and how much precision did the lattice buy compared with a notional RBD on the same plots?
Data, in StatVeda format
One incomplete block per line, each cell EntryName:value, replicates separated by a line containing only three dashes. Eight blocks per replicate, two replicates, 128 plots. The trial is large, so an excerpt is shown; the full paste continues in the same pattern.
# Replicate 1, blocks of 8 (grain yield t/ha) E1:4.82, E2:5.11, E3:4.65, E4:5.30, E5:4.95, E6:5.02, E7:4.71, E8:5.18 E9:4.55, E10:5.40, E11:4.88, E12:5.05, E13:4.62, E14:5.22, E15:4.79, E16:5.10 # ... blocks 3 to 8 of replicate 1, same EntryName:value pattern ... --- # Replicate 2, blocks of 8 (entries re-grouped into new incomplete blocks) E1:4.75, E9:4.60, E17:5.01, E25:4.88, E33:4.92, E41:4.70, E49:5.12, E57:4.83 E2:5.05, E10:5.34, E18:4.79, E26:5.20, E34:4.66, E42:5.15, E50:4.90, E58:5.08 # ... blocks 3 to 8 of replicate 2 ...
What she does in StatVeda
Open Experimental Designs. Pick Alpha Lattice (PBIB) Design. Paste the full block-structured data with the three-dash separator between the two replicates. Run.
ANOVA: Replications (1 df), Blocks within replications (adjusted), Entries (63 df, adjusted for blocks), and intra-block Error. The entry effect is significant after block adjustment.
Adjusted (block-corrected) entry means are reported alongside unadjusted means. Several entries shift rank: an entry that sat in two high-fertility incomplete blocks has its mean adjusted downward, and a strong entry that fell in poor blocks is adjusted upward. Example: E10 unadjusted 5.37, adjusted 5.21; E25 unadjusted 4.88, adjusted 5.06.
CD(5%) for comparing two adjusted entry means is reported from the effective error variance, not the raw error.
Efficiency of the alpha lattice relative to a randomised block design = 1.34. Reading: the lattice extracted enough block variation that an RBD on the same plots would have needed about 34 percent more replications to reach the same precision.
What it means
The block adjustment is not cosmetic. Because each incomplete block of eight plots sat at a slightly different point on the fertility gradient, the unadjusted means partly reflect which block an entry happened to land in. The adjusted means remove that, so the ranking reflects genotype, not soil. The relative efficiency of 1.34 is the concrete payoff: the incomplete-block control bought precision equivalent to a third more replication, at no extra field cost.
Select the next-stage entries on adjusted means, not unadjusted means. Entries like E25 that rose after adjustment are genuine performers held back by poor blocks; entries like E10 that fell were flattered by good blocks.
Advance the top 16 adjusted entries that exceed the trial mean by more than the reported CD into a multi-location trial.
Keep the alpha lattice for the next preliminary trial. A relative efficiency of 1.34 means it is clearly worth the small extra layout effort over an RBD at this entry count.
Priya reports both mean columns side by side so reviewers can see which entries moved and why, plus the relative-efficiency number that justifies the design choice.
The trap of reading unadjusted means
If she had ignored the incomplete-block structure and ranked on raw means, she would have promoted E10 over E25, the exact reverse of the correct call. With 64 entries spread across a fertility gradient, a few placements in good or bad blocks are enough to flip a shortlist. The adjusted means are the entire reason for using a lattice rather than just averaging.
What she will do next season
Carry the top 16 adjusted entries into a multi-location replicated trial, where GxE becomes the question and an AMMI or GGE biplot becomes the right tool. The alpha lattice did its job at the single-site screening stage: it filtered 64 entries down to a manageable 16 with the block noise removed.