By R. A. Bailey

ISBN-10: 0511393407

ISBN-13: 9780511393402

ISBN-10: 0521683572

ISBN-13: 9780521683579

ISBN-10: 0521865069

ISBN-13: 9780521865067

The coherent framework at the back of reliable perform; for operating statisticians, complicated undergraduates, starting graduate students.

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8. Sums of squares 27 for either sense of sum of squares. The expected mean square for W , written EMS(W ), is the expectation of the mean square in the random sense; that is EMS(W ) = E( PW Y 2 ) . dimW First we apply these ideas with W = VT . Since t SUMT =i ui , ri i=1 PVT Y = ∑ the sum of squares for VT is equal to t t SUMT = j uj . rj j=1 SUMT =i ui · ri i=1 ∑ ∑ Now, ui · u j = 0 whenever i = j, so this sum of squares t (SUMT =i )2 (SUMT =i )2 u · u = . i i ∑ ri ri2 i=1 i=1 t =∑ The quantity ∑i (sum2T =i /ri ) is called the crude sum of squares for treatments, which may be abbreviated to CSS(treatments).

Z . . . .. . . . . . . . . . . ✻ . . . ... ✚ .. ... ...... ✚ ... ... ...... ✚ ..... ✲ ...... 0 ✚ ..... W . . .......... x . . . . ... . ................ ... .................................................................. ..... . . .. ........ . Fig. 3. 5 Linear model For unstructured plots we assume that Y = τ + Z, where τ ∈ VT , E(Z) = 0, Var(Zω ) = σ2 for all ω in Ω, and cov(Zα , Zβ ) = 0 for different plots α and β.

V0 .. ..................................................... . 32 Chapter 2. Unstructured experiments (i) dimWT = dimVT − dimV0 = t − 1; (ii) PWT v = PVT v − PV0 v for all v in V ; (iii) PWT v 2 2 + PV0 v = PVT v 2 for all v in V . 5) i=1 and t ∑ ri (τi − τ)2 = PWT τ 2 2 = τ − τu0 2 t = ∑ ri τ2i − N τ2 , i=1 i=1 which is zero if and only if all the τi are equal. Applying (ii) with v = y gives t PWT y = ∑ i=1 sumT =i ri ui − sum u0 N = fitted values for treatments − fit for null model t = ∑ (ˆτi − y) ui .

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