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By D. J. Finney

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5 . . ... .. .... BLOCX D E C A B . II III IV D A C V VI B E A D D B C A C A B D B E E D E C B E C A The treatments are denoted by the letters A, B, C, D, E . i, and (77 + f'Ji) is the average for all treatments on block j alone. The residual t ii is the amount by which Yii differs from a value predicted by adding average block and treatment "effects" to 77; it is compounded of any "interaction" between the relative effects of different treatments and the block classification (d. 2) and of experimental error.

Columns .............. Latin treatments .. . . Greek treatments ...... Error . . . . 1- 1 1- 1 I- I I-I Total. . . ...... tX I GRECO-LATIN SQUARE Sum of quur .. n Square (1-1)(1 - 3) 1'- 1 letter of the new square occur once in each row and once in each column, but also it occurs once with each letter of the first square. 2 might have wi hed to test simultaneously inoculation by five different techniques Ca, (3, ... , E). 4. 3 also has no orthogonal square. 4 is both permissible and easily computed; thus the two 33 Randomized Blocks and Orthogonal Squares sets of treatments are examined simultaneously in the one experiment.

For the completely randomized design, on the normal error model, these may be derived by use of J E (y j1') = 11 + 7"11 , (2. 11) It is then easily shown that the mean square "within treatments" in the analysis of variance ha~ expectation 0"2, and is thus independent of the 7"1" For this reason, the name "error" is commonly used for it. 13) 1 1'=---:-- N Hence the expectation of this mean square necessarily exceeds 0"2 unless all the 1"1' are equal. Sometimes the t treatments can reasonably and usefully be regarded as themselves a random sample from a large population of possible treatments, in which population the variance of 7"1' is O"~.

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