Re: Epsilon shrinkage and IWRES
I spare Bob Bauer from the need to re-type the answer since we discussed it at some point, and he provided the following info (see below). The following error model was discussed:
Y=IPRED(1+EPS1)+EPS2
that is equivalent to
Y=IPRED + W*EPS(1)
where
W=SQRT(sigma1^2+sigma2^2*IPRED^2)
where sigma 1 and sigma 2 are some parameters and variance of EPS1 is 1
;;;;;;;;;;;;;;;;;;;;;;
For each data point i,
Wi=SQRT(sigma1^2+sigma2^2*IPREDi^2)
Each Wi from each data point is used in assessing sigma1 and sigma2. As
long as IPREDi is non-zero for every data point, then every Wi is used
in assessing both sigma1 and sigma2 in the optimization process.
However, when IPREDi=0, then that particular Wi is
Wi=SQRT(sigma1^2)
Which contributes to evaluation of sigma1, but does not contribute to
sigma2. How does the EPSshrink algorithm figure this out? By in fact
using the H gradient that NONMEM provides, which NONMEM uses to estimate
sigma1 and sigma2. The actual residual variance is in matrix form
evaluated as
H'SigmaH
Here H is the partial derivative of yi with respect to sigmaj. The Wi is
itself not decomposed, but the decision to add the Wi to the accumulator
for sigmaj is determined by whether partial hi with respect to sigmaj is
non-zero.
And wherever H' is not 0, the EPS shrink accumulator adds the
contribution of that Wi for a sigma, and where-ever H is zero, it does
not attribute that Wi for that sigma. This is how it is known which Wi
pertain to PK data, and which Wi pertain to PD data as well, by
assessment of the H gradient that NONMEM assesses.
The epsilon shrinkage formula honors the
H'SigmaH structure, as it should, since NONMEM's FO and FOCE methods
themselves utilize that structure to estimate Sigma1 and Sigma2.
To reiterate, normally with proportional and additive sigma, all Wi are
involved in both sigmas, and hence both sigmas have identical shrinkage,
since they are always having non-zero H together. But when some IPREDi
are zero, the additive error has contributions from all Wi, whereas the
proportion sigma collects Wi information only from non-zero IPREDi.
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
Quoted reply history
On 9/22/2011 9:26 AM, Rik Schoemaker wrote:
> Dear all,
>
> Can someone enlighten me on a NONMEM implementation?
> Since NONMEM 7, epsilon shrinkage is reported in the output, and the manual
> states that it is calculated as:
> 100%*[1-SD(IWRES)]
> in accordance with "Karlsson MO and Savic RM. Diagnosing Model Diagnostics.
> Clinical Pharmacology and Therapeutics, 2007; 82(1): 17-20" as one would
> expect.
> However, as far as I know, the user needs to manually code the IWRES bit
> using the "Uppsala implementation":
>
> IPRED = F
> IRES = DV - IPRED
> W = THETA(3)
> IWRES = IRES/W
> Y = IPRED+W*EPS(1)
>
> SIGMA 1 FIXED
>
> So how does NONMEM arrive at epsilon shrinkage? And is there by any chance
> an (undocumented?) IWRES item that can be exported in a table file as well?
>
> Kind regards,
>
> Rik
>
> Rik Schoemaker, PhD
> Exprimo NV
> Tel: +31 (0)20 4416410
> E-mail: [email protected]
> Web: www.exprimo.com
>
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