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Thus for the 5 X 5 square the first row con ists of 1111, 2222, 3333, 4444, 5555; the first element of the second row is 2345, whi h is followed by 3451 and the other cyclic permutations within this row j the first I ment · of the r maining rows are 3524, 4253, 5432 , each row again being completed by cyclic permutation. The reader should write out this square in full, ee the manner of its construction, and himself con truct a 7 X 7 square. Completely orthogonalized squares an also be constructed when t is any power of any prime (Bose, 1939; Stevens, 1939).

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"~. 12) is 2 (] 2 2+N -1:r1' 2 N(t-l) O"~. (2 . 14) The multiplier of ~ simplifies to , for a symmetrical design in which all the '1' are equal, and is not very different from the mean value of ' 1' unless the separate '1' differ widely.

1). ), and error exa tly as for an ordinary randomized block design. S / Blocks .............. Temperatures (T) . . Quantities (Q) . . . . Periods (P) . . . . . TQ ................. TP ........ ..... QP ...... . TQP ........ l. Variation 3 X 2 Mean Square 5 2 3 1 6 2 3 6 Treatments . . . . . Error ...... .... " . 23 115 Total .... .... 1, the treatment component could be subdivid d into 23 separate squares for single and mutually orthogonal contrasts. 1 uffices. Yields are classified according to temperature, and totals of the groups of 48 formed irrespective of quantity of material and period.

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An Introduction To The Theory Of Experimental Design by D. J. Finney

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