ASReml-S reference manual - VSN International

2.2 Estimation 112.2.1 Variance parametersEstimation of the variance parameters is carried out using residual or restricted maximumlikelihood (REML), developed by Patterson and Thompson [1971]. Note firstly thaty ∼ N(Xτ , H). (2.3)where H = R + ZGZ ′ . REML does not use (2.3) for estimation of variance parameters,but rather uses a distribution free of τ , essentially based on error contrasts or residuals.The derivation given below is presented in Verbyla [1990].We transform y using a non-singular matrix L = [L 1 L 2 ] such thatL ′ 1X = I p , L ′ 2X = 0.If y j = L ′ jy, j = 1, 2,[y1]∼ Ny 2([ ] [τ L′, 1 HL 1 L ′ ])1HL 20 L ′ 2HL 1 L ′ .2HL 2The full distribution of L ′ y can be partitioned into a conditional distribution, namelyy 1 |y 2 , for estimation of τ , and a marginal distribution based on y 2 for estimation of γand φ; the latter is the basis of the residual likelihood.The estimate of τ is found by equating y 1 to its conditional expectation, and after somealgebra we find,ˆτ = (X ′ H −1 X) −1 X ′ H −1 yEstimation of κ = [γ ′ φ ′ ] ′ is based on the distribution of y 2 ,wherel R = − 1 2 (log det L′ 2H −1 L 2 + y ′ 2(L ′ 2HL 2 ) −1 y)= − 1 2 (log det X′ H −1 X + log det H + y ′ P y) (2.4)P = H −1 − H −1 X(X ′ H −1 X) −1 X ′ H −1 .Note that y ′ P y = (y − X ˆτ ) ′ H −1 (y − X ˆτ ). The log-likelihood (2.4) depends on X andnot on the particular non-unique transformation defined by L.The log residual likelihood (ignoring constants) can be written asWe can also writel R = − 1 2 (log det C + log det R + log det G + y′ P y). (2.5)P = R −1 − R −1 W C −1 W ′ R −1with W = [X Z]. Letting κ = (γ, φ), the REML estimates of κ i are found by calculatingthe 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