5.1 The design
Suppose \(t\) treatments are to be compared and the experimental units fall into \(b\) groups, the blocks, of \(t\) units each, with units in the same block more alike than units in different blocks. The blocks may be fields divided into plots, batches of raw material, days, machines, operators or litters of animals. In a randomised complete block design (RCBD) each treatment is applied once in every block, so each block is a complete replicate of the experiment, and the treatments are allocated to the units at random within each block, with a fresh randomisation in every block.
Comparisons between treatments are then made within blocks, where the units are alike, and the differences between blocks are removed from the experimental error. The paired comparison design of Section 2 is the RCBD with \(t = 2\).
Example 5.1.1. Four fertilisers \(A\), \(B\), \(C\), \(D\) were compared on five fields. Each field was divided into four plots of equal size and the fertilisers were allocated to the plots of each field at random. The yields (kg per plot) were
| field | \(A\) | \(B\) | \(C\) | \(D\) | total \(y_{\cdot j}\) |
| \(1\) | \(45\) | \(47\) | \(50\) | \(45\) | \(187\) |
| \(2\) | \(41\) | \(43\) | \(47\) | \(45\) | \(176\) |
| \(3\) | \(46\) | \(48\) | \(53\) | \(48\) | \(195\) |
| \(4\) | \(41\) | \(42\) | \(46\) | \(43\) | \(172\) |
| \(5\) | \(43\) | \(46\) | \(48\) | \(44\) | \(181\) |
| total \(y_{i\cdot }\) | \(216\) | \(226\) | \(244\) | \(225\) | \(911\) |
| mean \(\bar y_{i\cdot }\) | \(43.2\) | \(45.2\) | \(48.8\) | \(45.0\) | \(45.55\) |
and \(\sum \sum y_{ij}^2 = 41671\). Field \(3\) is the best in every column and field \(4\) the worst: the fields differ, and a design that ignored them would put that variation into the error.
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