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2 Some theory 13Letting κ = (γ, φ), the REML estimates of κ i are found by calculating the scoreU(κ i ) = ∂l R /∂κ i = − 1 2 [tr (P H i) − y ′ P H i P y] (2.6)and equating to zero. Note that H i = ∂H/∂κ i .The elements of the observed information matrix are− ∂2 l R∂κ i ∂κ j= 1 2 tr (P H ij) − 1 2 tr (P H iP H j )where H ij = ∂ 2 H/∂κ i ∂κ j .+ y ′ P H i P H j P y − 1 2 y′ P H ij P y (2.7)The elements of the expected information matrix are( )E − ∂2 l R= 1 ∂κ i ∂κ j 2 tr (P H iP H j ) . (2.8)Given an initial estimate κ (0) , an update of κ, κ (1) using the Fisher-scoring (FS)algorithm isκ (1) = κ (0) + I(κ (0) , κ (0) ) −1 U(κ (0) ) (2.9)where U(κ (0) ) is the score vector (2.6) and I(κ (0) , κ (0) ) is the expected informationmatrix (2.8) of κ evaluated at κ (0) .For large models or large data sets, the evaluation of the trace terms in either(2.7) or (2.8) is either not feasible or is very computer intensive. To overcomethis problem ASReml uses the AI algorithm (Gilmour, Thompson and Cullis,1995). The matrix denoted by I A is obtained by averaging (2.7) and (2.8) andapproximating y ′ P H ij P y by its expectation, tr (P H ij ) in those cases whenH ij ≠ 0. For variance components models (that is those linear with respect tovariances in H), the terms in I A are exact averages of those in (2.7) and (2.8).The basic idea is to use I A (κ i , κ j ) in place of the expected information matrix in(2.9) to update κ.The elements of I A areI A (κ i , κ j ) = 1 2 y′ P H i P H j P y. (2.10)The I A matrix is the (scaled) residual sums of squares and products matrix ofy = [y 1 , . . . , y k ]