Re: Schall method and ASREML

# Re: Schall method and ASREML


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
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