By Klaus Hinkelmann, Oscar Kempthorne
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91) is symmetric and idempotent. Furthermore, SX = 0, and hence Sy is distributed N (0, SV Sσe2 ) independently of X V −1 y. 91) that S is singular. 90) a matrix, T say, which is derived from S by deleting its r1 th, (r1 + r2 )th, (r1 + r2 + r3 )th, . , (r1 + r2 + · · · + rt )th rows, thereby reducing an n × n matrix to an (n − t) × n matrix (with n − t representing the number of linearly independent error contrasts). More explicitly, we can write T as t T = + i=1 t = i=1 + . 1 I ri −1 .. 0ri −1 − Iri −1 Iri ri I ri −1 − .
Tuv − T .. 31) where µ = U .. + T .. is the overall mean bj = U j. − U .. is the effect of the j th block (j = 1, 2, . . , b) τuv = tuv − T .. is the effect of the uv treatment (u = 1, 2, . . , b; v = 1, 2, . . , k) uj = Uj − U j. is the unit error ( = 1, 2, . . , k) with j bj = 0 = uv τuv = for the uv treatment, yuv , as yuv = uj . 34) 18 GENERAL INCOMPLETE BLOCK DESIGN is a random variable with E(ωuv ) = 0 2 E(ωuv )= E(ωuv ωuv ) = − E(ωuv ωu v ) = 0 1 bk(k − 1) 1 bk u2j j u2j (v = v ) j (u = u ) In deriving the properties of the random variables βu and ωuv we have used, of course, the familiar distributional properties of the design random variables αju and δjuv , such as 1 P (αju = 1) = b P (αju = 1 | αju = 1) = 0 (j = j ) 1 b(b − 1) 1 = 1) = k = 1) = 0 P (αju = 1 | αju = 1) = P (δjuv P (δjuv = 1) | (δjuv (u = u , j = j ) ( = ) P (δjuv = 1) | (δjuv = 1) = ( = ,v = v ) P (δjuv = 1) | (δju v (j = j , u = u ) 1 k(k − 1) 1 = 1) = 2 k and so on.
65) looks deceptively simple, but the reader should keep in mind that the elements of A−1 depend on σβ2 and σe2 . 10. 60) is of full rank and the elements of its coefficient matrix depend on the unknown parameters σβ2 and σe2 . 4 Some Special Cases As a special case of the above derivations we mention explicitly the equireplicate, proper design, that is, the design with all ri = r and all kj = k. 68) for two special cases: 1. 7) for the intrablock analysis. This means, of course, that in this case no interblock information is available.