Fertiliser Trial with a Confounded Block
An agronomy institute tests four fertiliser formulations (A, B, C, D) on 16 plots arranged in four blocks of four. The blocks follow a fertility gradient across a slope. The statistician planned a randomised complete block design, but the field technician, working from a hand-drawn map, assigned treatments so that A always occupies the top plot of each block and D always the bottom. The recorded yields (tonnes per hectare) are: block 1: 4.1, 4.6, 5.0, 5.3; block 2: 3.9, 4.4, 4.8, 5.1; block 3: 4.3, 4.7, 5.2, 5.4; block 4: 4.0, 4.5, 4.9, 5.2, listed in plot order top to bottom. The institute wants to report whether formulations differ, and the funder wants a result within a month.
1. Diagnose the design problem and state what can and cannot be estimated from these data.
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The technician's assignment is not randomisation: treatment is perfectly aligned with plot position within block, and plot position is plausibly aligned with the fertility gradient. Treatment and position are therefore confounded. The block effect absorbs the average gradient across blocks, but the within-block gradient remains entangled with the treatment contrast.
What can still be estimated: the block totals and the overall treatment means, because each treatment appears once per block. A standard two-way analysis - blocks plus treatments - will run and produce an F test for treatments. What cannot be separated is the treatment effect from the within-block position effect. If position matters, the treatment F test is biased, and no amount of arithmetic recovers the missing information.
There is a diagnostic clue in the numbers. Yields rise monotonically with plot position in every block, from roughly 4.0 at the top to 5.25 at the bottom. That pattern is exactly what a position effect would produce, and it is consistent with A being worst and D best. But it is equally consistent with no treatment effect at all and a pure gradient. The data cannot distinguish these, which is the core finding to report.
2. The funder wants a conclusion. Propose what the institute should report and what it should do next.
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Report honestly: the planned randomised complete block analysis is invalid because randomisation was not carried out. Present the two-way ANOVA as descriptive only, explicitly flagging the confounding, and do not attach a p-value to the treatment test as if it were confirmatory.
A useful supporting analysis is a sensitivity check. Fit yield on plot position within block and on treatment; the position term will be estimable and, if it is large, it demonstrates that the apparent treatment ordering could be an artefact. If position explains most of the between-plot variation, the treatment contrast shrinks toward zero and the honest conclusion is 'no evidence of a treatment difference can be established'.
For the next step, the institute should repeat the trial with proper randomisation - a fresh random allocation per block, generated and recorded before planting - and with position recorded as a covariate so any residual gradient can be modelled. Sixteen plots is adequate for four treatments in four blocks if randomisation is done correctly; the design was never the problem, the execution was. The month-long deadline should be met with a candid interim report, not a fabricated significant result.