Split-plot irrigation by variety in wheat: reading the right error term
Three irrigation regimes as whole plots, four varieties as sub-plots, four blocks. Main effects and the irrigation by variety interaction read against the correct split-plot error structure.
Rakesh is an agronomist running an irrigation trial on bread wheat. He has three irrigation regimes (I1 one irrigation at crown root, I2 two irrigations, I3 four irrigations) and four varieties (V1 to V4). Irrigation is hard to randomise on a small plot because water moves through the soil, so it has to sit on large blocks of land. Variety can be randomised freely within each watered strip. He lays the trial out in four blocks.
Why split-plot, not factorial RBD
Irrigation is the whole-plot factor: each irrigation regime is applied to one large plot per block, and there are only four replications of it. Variety is the sub-plot factor: each whole plot is split into four sub-plots, one per variety. The two factors are tested against two different error terms. Treating this as an ordinary factorial RBD would test irrigation against the wrong, too-small error and inflate its significance.
Question
Does irrigation raise yield, do varieties differ, and does the best variety change with the amount of water (the irrigation by variety interaction)?
Data, in StatVeda format
One row per whole-plot (irrigation) treatment within each block. Each row holds the four sub-plot (variety) values in order V1, V2, V3, V4. The four blocks stack in sequence. Grain yield in t/ha.
# Block 1: I1, I2, I3 (each row = V1,V2,V3,V4) 3.92, 4.15, 4.05, 3.78 4.61, 4.88, 5.02, 4.40 5.05, 5.31, 5.58, 4.92 # Block 2 3.85, 4.05, 3.98, 3.71 4.55, 4.79, 4.95, 4.34 5.12, 5.40, 5.65, 5.01 # Block 3 4.01, 4.22, 4.12, 3.90 4.70, 4.95, 5.10, 4.51 5.18, 5.45, 5.71, 5.08 # Block 4 3.88, 4.10, 4.00, 3.74 4.58, 4.82, 4.98, 4.38 5.09, 5.36, 5.61, 4.97
What he does in StatVeda
Open Experimental Designs. Pick Split-Plot Design. Set the whole-plot factor name to Irrigation with three levels, the sub-plot factor to Variety with four levels, four blocks. Paste the data block. Run.
Source, df, MS, F, p:
Block: 3 df
Irrigation (whole plot): 2 df, F tested against Error(a) (6 df), F = 312.4, p < 0.001
Error(a), whole-plot error: 6 df
Variety (sub plot): 3 df, F tested against Error(b) (27 df), F = 96.7, p < 0.001
Irrigation x Variety: 6 df, F against Error(b), F = 5.81, p = 0.0007
Error(b), sub-plot error: 27 df
Three separate CD(5%) values are reported: irrigation means (uses Error(a)) CD = 0.121; variety means (uses Error(b)) CD = 0.104; variety means within the same irrigation level CD = 0.180.
CV(a) = 2.9 percent on the whole-plot, CV(b) = 3.6 percent on the sub-plot.
What it means
All three terms are significant. Yield rises strongly with irrigation: the I3 mean (around 5.3) is far above I1 (around 3.9), a difference far larger than the irrigation CD of 0.121. Varieties differ, and the significant interaction (p = 0.0007) says the variety ranking is not the same at every water level. Reading the means, V3 is the best variety under I2 and I3 but under I1 (water stress) V2 edges ahead of V3. The interaction is real but modest: the responsive variety V3 simply gains more from extra water than the others.
Recommend V3 under assured irrigation (two or more irrigations): it has the highest yield wherever water is not limiting, and the gap exceeds the within-irrigation CD of 0.180.
Recommend V2 for the one-irrigation, water-scarce situation, where it is the top performer under I1.
Report irrigation against Error(a), not Error(b). Using the sub-plot error for irrigation would have shrunk the CD roughly fourfold and overstated precision on the factor that is actually the least replicated.
Rakesh attaches the StatVeda output to the AICRP agronomy report. Because the two error lines and the three distinct CD values are printed on one page, the reviewer can see at a glance that the split-plot structure was respected.
The mistake this design avoids
If he had run a two-factor factorial RBD, irrigation would have been tested against the pooled error with 33 df instead of Error(a) with 6 df. The F for irrigation would have looked even larger and the irrigation CD would have collapsed to about 0.06, implying a precision the four whole-plot replications cannot support. The split-plot keeps the error structure honest: the factor that is physically harder to replicate is tested against its own, larger error.
What he will do next season
Carry forward V3 and V2 only, add a fifth variety, and add a moisture-sensor covariate so the next analysis can be a split-plot with a soil-moisture covariate. The interaction found here, that the best variety depends on water, is exactly the kind of result that justifies a second confirmatory season before a regional recommendation.