From: Jean-Marie MARTINEZ <Jean-Marie.Martinez@sanofi-synthelabo.com>
Subject: [NMusers] Calculation of WRES...
Date: 8/12/2003 10:34 AM

Hello NM Users,

I'm trying to understand how NonMem computes WRES, and I'm in trouble with
this topic...

Indeed, one can read in several documentations that "WRES are the RES
expressed in (i.e. as fractions of) population standard deviation units",
or that "The WRES is a simple normalization of the RES, viz. the RES
divided by the population standard deviation of the observation." (NonMem
topic 6).

The problem is that I cannot recalculate WRES 'by hand', due to the fact
that the expression of PopSD remains unclear to me: while IWRES is easily
obtained by dividing IRES by W, the same applied to RES does not give WRES
!

Thank you for your help.

Regards,

Jean-Marie MARTINEZ
Clinical Pharmacokineticist
Department of Clinical Metabolism and Pharmacokinetics
SANOFI-SYNTHELABO RECHERCHE
371 rue du Professeur Joseph Blayac
34184 MONTPELLIER Cedex 04 - FRANCE
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From: William Bachman <bachmanw@globomax.com>
Subject: RE: [NMusers] Calculation of WRES...
Date: 8/12/2003 1:26 PM

WRES's for an individual are given by the vector Ri(THETA,OMEGA,SIGMA)
described at the bottom of page 37 of NONMEM Users Guide I.

Also, from NONMEM Users Guide VIII:

 WRES The weighted residuals for an individual are formed by transform-
      ing  the  individual's  residuals  so  that  under the population
      model, assuming the true values of the population parameters  are
      given  by the estimates of those parameters, all weighted residu-
      als have unit variance and are uncorrelated. As with the  predic-
      tion and residual, the weights are also computed at eta = 0.

nmconsult@globomaxnm.com
GloboMax LLC
7250 Parkway Drive, Suite 430
Hanover, MD 21076
Voice: (410) 782-2205
FAX: (410) 712-0737
_______________________________________________________

From: Matthew Hutmacher <matthew.hutmacher@pharmacia.com>
Subject: RE: [NMusers] Calculation of WRES...
Date: 8/12/2003 4:52 PM

Jean-Marie,

In a heuristic sense:

Let y(i) represent the data vector for individual i.
Let f(i) represent the mean of y(i).
Let V(i) represent the estimated (and approximated) variance of y(i)
Let H(i) represent the inverse of V(i).

In a similar fashion to univariate standardization, standardization of RES
is accomplished:

WRES(i)=(y(i)-f(i)*K(i)

where K(i) is the "root" matrix of H(i).  This "root" matrix is not unique.
Two popular methods of computing K(i) are:

1) The Cholesky decomposition.
2) (PD)(PD)' where P is the matrix of eigenvectors and D is a diagonal
matrix of the square roots of the eigenvalues. 

K(i) from each method will not be the same.  I had the same question you
have some years ago, and far as I can remember, I pursued it to the point of
determining that NONMEM does not use the Cholesky decomposition (I think).
My guess is to try the eigenvalue method and see if you can compute it from
there, but I would suggest that whenever possible, try to have NONMEM
calculate them.

Matt
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