Corrigendum. Trinca and Gilmour (2000) stated that, for model (1), S = (X X) 1 X B

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1 Corrigedum Trica Gilmour (000) preseted a algorithm for arragig respose surface desigs i blocks The polyomial respose surface model for blocked experimets is y = Bα + Xβ + ɛ, () where y is the respose vector, B is the b matrix whose colums are idicators for blocks, α is the b vector of block effect parameters, X is the p treatmet combiatios matrix exped to icorporate the polyomial model, where for example p = (q + )(q + )/ for the secod order model for q factors, β is the p vector of treatmet parameters ɛ N(0, σ I) is the rom error vector The rak of the matrix (B : X) is less tha b + p, the umber of parameters i the model, so, i order to obtai least squares estimates of the parameters, a costrait is imposed that the block effects, weighted by the correspodig block sizes, add up to zero, ie b α = 0, where b is the b vector of block sizes Trica Gilmour (000) stated that, for model (), where V(ˆβ) = (X X) + SA S σ, () S = (X X) X B A = B B + b b B X(X X) X B They the used this equality to develop their blockig algorithm usig rows to p of the scores matrix, S I fact the stated result () is icorrect, although this does ot affect the procedure used The validity of the procedure used is proved i this ote I what follows, we will use the followig result (Searle, 98, page 6) several times: (D CA B) = D + D C(A BD C) BD (3) Let Z be the cetred versio of the matrix X, ie Z = X x, where X = X x = x x p Note that we are ceterig each colum ot the origial explaatory variables The model () ca be writte as y = Bα + Zθ + ɛ,

2 where θ i = β i for i =,, p, ie except for the itercept the parameters do ot chage Usig a result ( o) iverses of partitioed matrices (Searle, 98, page 60) o α the vector, we ca show that the least squares estimator of θ, after θ allowig for blockig, is ˆθ = (Z RZ) Z Ry, ( ) B where R = I B(B B ) B B =, ie B is the matrix B augmeted to allow for the required costrait The variace of ˆθ is V(ˆθ) = (Z RZ) Z RV(y)R Z(Z RZ) sice V(y) = σ I R is symmetric b = (Z RZ) Z RRZ(Z RZ) σ, (4) We will evaluate the parts of (4) separately First, otig that we obtai from (3), where B B = B B + b b, (5) (Z RZ) = Z Z Z B(B B + b b) B Z = (Z Z) + S A S, (6) S = (Z Z) Z B A = B B + b b B Z(Z Z) Z B Secodly, usig (3) (5), we obtai (B B ) = (B B b ( ) b) = (B B) + (B B) b b (B B) b b (B B) = (B B) (B B) b ( + ) b(b B), sice (B B) b = b Hece, (B B ) = (B B) + J b b

3 sice BJ b b B = J Thus B(B B ) B = B(B B) B + J, RR = I B(B B ) B + B(B B ) B B(B B ) B = I B(B B ) B + B(B B ) B B(B B) B + J = I B(B B ) B + B(B B ) B B(B B) B + B(B B ) B J = I B(B B ) B B(B B) B + + J J = R + J + ( + ) J J = R + J + ( + ) J = R ( + ) J (7) Substitutig (7) ito (4) we have σ V(ˆθ) = (Z RZ) Z R ( + ) J Z(Z RZ) = (Z RZ) Z RZ(Z RZ) ( + ) (Z RZ) Z J Z(Z RZ) = (Z RZ) Substitutig (6) ito this expressio, we obtai ( + ) (Z RZ) Z J Z(Z RZ) σ V(ˆθ) = (Z Z) + S A S ( + ) (Z RZ) Z J Z(Z RZ) Now Z J Z = 0 Let (Z RZ) = C partitio it as c c c C 3 0 0

4 The (Z RZ) Z J Z(Z RZ) = c c c c c 0 Hece the lower right (p ) (p ) portio of σ V (ˆθ) is equal to the lower right portio of (Z Z) + S A S Hece the methods developed i Trica Gilmour (000) are valid for the cetred form of the model, usig the modified scores matrix, S We kow that (Z Z) 0 =, 0 D where (X X) = d d d D Hece if we are usig the origial form of the model, the oly ew calculatio required is of S We will ow show that the last p rows of this are equal to the last p rows of S, so that the algorithm used i the origial paper is i fact valid We write We also have Z B = S = S = s S s S b X x b Hece S = D X B D x b Similarly so S = D X B + d b X B = b X B We will ow show that D x = d We have X X = x x X X 4

5 Usig the result o partitioed matrices i Searle (98, page 60) we get d = (X X x x ) x D = (X X x x ) Hece S = S basig our algorithm o the ucetred model gives exactly the same results as basig it o the cetred model Hece all of the practical results obtaied by Trica Gilmour (000) are correct Ackowledgmet The authors thak Peter Goos for poitig out that the proof give i Trica Gilmour (000) is wrog Refereces Searle, S R (98) Matrix Algebra Useful for Statistics New York: Wiley Trica, L A Gilmour, S G (000) A algorithm for arragig respose surface desigs i small blocks Computatioal Statistics Data Aalysis, 33,