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*To*: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>*Subject*: Re: Schall method and ASREML*From*: averbyla@biometricssa.adelaide.edu.au*Date*: Thu, 26 Aug 1999 16:27:29 +0930 (CST)*cc*: asreml@chiswick.anprod.csiro.au*In-Reply-To*: <199908260611.QAA21474@ornsun.agric.nsw.gov.au>*Reply-To*: averbyla@biometricssa.adelaide.edu.au*Sender*: asreml-owner@ram.chiswick.anprod.csiro.au

Dear Arthur Shall's method is based on linearisation, and uses the standard REML estimate of the "residual variance", which is actually a scale factor. In detail, if g(mu_i) = eta_i = x_i'\beta + z_i'u, i=1,2,...n, g is the link function and eta the linear predictor, u is a vector of random effects so that we have a mixed model on the scale of the link function, a Taylor Series expansion is m_i = g(y_i) = g(mu_i) + (y_i-mu_i) g'(mu_i) = x_i'\beta + z_i'u + e_i g'(mu_i) m_i is a working variate and e_i has mean zero and variance \phi v(\mu_i), in GLM terms. The \phi is the scale parameter. It is 1 for standard logit and Poisson log-linear modelling, but can be introduced here in what is really a moment or quasi-likelihood modelling approach. Thus conditional on u E(m_i|u) = x_i'\beta + z_i'u and var(m_i|u) = \phi v(\mu_i) [g'(mu_i)]^2 = \phi w_i^{-1} where w_i is the usual glm weight (but evaluated using the mixed model, that is including BLUPs). Thus marginally (if u ~ N(0,G)) E(m_i) = x_i'\beta and var(m_i) = \phi w_i^{-1} + z_i'Gz_i = h_i I think Schall now simply uses standard REML so that \hat\phi = (M-X\hat\beta-Z\tilde u)'W(M-X\hat\beta-Z\tilde u)/df where the upper case letters are now vectors and matrices, but the df are not integer. This actually is \hat\phi = residual'W residual / trace(\phi P) which is the EM algorithm equivalent (I think) and where \phi appears on the right(!)as well as the keft side, P = H^{-1} - H^{-1}X(X'H^{-1} X)^{-1}X'H^{-1}. Does this help???? The AI algorithm needs to incorporate known weights (ie W) at each iteration .... that is the R stucture is a scaled diagonal matrix, with diagonal entries known, R = \phi W^{-1}. Ari ________________________________________________________________________ Dr Arunas (Ari) Verbyla, Email: Ari.Verbyla@adelaide.edu.au Director, International: BiometricsSA, Phone: +61 8 8303 6760 University of Adelaide/SARDI Fax: +61 8 8303 6761 Private Mail Bag 1, Glen Osmond, Australia: South Australia, 5064, Phone: 08 8303 6760 AUSTRALIA. Fax: 08 8303 6761 ________________________________________________________________________ -- Asreml mailinglist archive: http://www.chiswick.anprod.csiro.au/lists/asreml

**References**:**Schall method and ASREML***From:*Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>

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