Augmented designs: when you have many candidates and few seeds

Federer (1956) introduced augmented designs for a problem every plant breeder eventually faces: hundreds of new candidate lines, each with enough seed for one plot only. There is no way to replicate every test entry. The trick is to replicate a small number of standard checks across blocks, use those checks to estimate block effects, and then adjust the test entry means.

When you reach for it

Early-generation testing of large breeding populations. Germplasm screening trials. Initial yield trials in any crop. Whenever the number of test entries is too large to allow replication and you still need a defensible analysis to rank candidates.

Worked example: 15 test entries plus 3 checks in 2 blocks

15 new sorghum test entries (T01 to T15), 3 checks (C1, C2, C3), arranged in 2 blocks. Each block contains all 3 checks (replicated across blocks) plus a non-overlapping subset of test entries.

block,entry,type,yield_t_ha
B1,C1,check,3.21
B1,C2,check,3.85
B1,C3,check,4.02
B1,T01,test,4.41
B1,T02,test,3.62
B1,T03,test,4.18
B1,T04,test,3.95
B1,T05,test,4.55
B1,T06,test,3.71
B1,T07,test,4.32
B2,C1,check,3.45
B2,C2,check,4.12
B2,C3,check,4.31
B2,T08,test,4.71
B2,T09,test,4.21
B2,T10,test,3.92
B2,T11,test,4.65
B2,T12,test,3.85
B2,T13,test,4.41
B2,T14,test,4.05
B2,T15,test,4.55

What the analysis does

Block effect is estimated from the checks (which appear in every block). Each test entry mean is then adjusted by subtracting the estimated block effect of the block it sat in. The result is an adjusted mean that is comparable across blocks, even though no test entry was itself replicated.

Standard errors of differences come in three flavours: check vs check (smallest, since checks are replicated), check vs test (intermediate), test vs test (largest, since test entries are unreplicated). StatVeda reports CD at 5 percent and 1 percent for all three.

What to read off the output

Rank the test entries by adjusted mean. Compare each test entry to the best check using the check-versus-test CD. Anything significantly better than the best check is a candidate to advance. Anything close to the worst check can usually be dropped.

Common mistakes

Skipping the block effect adjustment and ranking on raw means. Using too few checks (fewer than 3 makes the block effect estimate noisy). Putting all checks in one block (defeats the design). Reporting only one CD value and using it to compare any pair of entries (CD differs by entry-pair type).

Practical note on layout

Each block should contain every check at least once. Test entries are partitioned across blocks so that no test entry is repeated. Block size is chosen so that each block fits a reasonably homogeneous patch of land. The number of blocks is the number of partitions of the test entries you can comfortably make.

How many checks is enough

Three checks per block is the practical minimum. With two checks the block effect estimate has only one df per block, and one bad plot can swing it. With four or more, the precision improves but you are burning seed and land on entries you already know. The textbook recommendation is at least three checks of differing performance levels (one good, one average, one poor reference) so that the ranking of test entries against checks is meaningful at every level of the yield range.

Augmented CRD vs Augmented RBD

Augmented CRD treats blocks as exchangeable (no expected gradient between them). Augmented RBD treats blocks as fixed sources of variation (expected gradient). If the blocks correspond to physical strips across a field gradient or to replications spaced in time, Augmented RBD is the right call. If the blocks are arbitrary groupings of plots in a uniform area, Augmented CRD will do.