From: laurent.nguyen@pierre-fabre.com
Subject: Dble exponential error model
Date: Thu, 7 Jan 1999 09:23:29 +0100

Dear NONMEM users,
I need to calculate the Individual Weight residuals (IWRES) with a double exponential intra-individual error model. Could anyone show me how to obtain them? The intra-individual error model I'm using is:

\$ ERROR
Y=F*EXP(EPS(1))+THETA(10)*EXP(EPS(2)) ; THETA(10) near L.O.Q.
IPRED=F
IRES=DV-IPRED
W= ?
IWRES = IRES/W

happy new year,

Laurent Nguyen

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From: "Steve Duffull" <sduffull@fs1.pa.man.ac.uk>
Subject: Re: Dble exponential error model
Date: Thu, 7 Jan 1999 09:53:45 GMT

Hi Laurent

> I need to calculate the Individual Weight residuals (IWRES) with a double
> exponential intra-individual error model.

It would seem to me that you should obtain the weighting the same as usual. Therefore

W=F*EXP(EPS(1))

the rest of the error term "+THETA(10)*EXP(EPS(2))" is independent of the model predicted concentration and is constant for any given individual.

Perhaps the question might be looked at in terms of, since "THETA(10)*EXP(EPS(2))" will always be positive should:

IWRES=IRES/W-C
where C=THETA(10)*EXP(EPS(2))

However I have 2 points:
I do not understand why you would want to use this model.
I do not understand how your model can identify the difference between THETA(10) and EPS(2).

Regards

Steve

Stephen Duffull
School of Pharmacy
University of Manchester
Ph +44 161 275 2355
Fx +44 161 275 2396

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From: LSheiner <lewis@c255.ucsf.edu>
Subject: Re: Dble exponential error model
Date: Thu, 07 Jan 1999 10:11:37 -0800

In reply to Laurent Nguyen's query:

1. His model as written is not identifiable:
The use of the exponential in the intra-individual error terms is misleading since under all methods of analysis, NONMEM linearizes in EPS Thus, Y is actually (as implemented)

Y=F*(1+EPS(1))+THETA(10)*(1+EPS(2))

which is equivalent to:

Y=F*(1+EPS(1))+THETA(10)*(1+EPS(2))

The variance of this is:

F**2*`SIGMA(1,1) + THETA(10)**2*SIGMA(2,2).

As you can see, THETA(10) and SIGMA(2,2) are unidentifiable.

2. Writing the identifiable model,

Y=F*(1+EPS(1))+ EPS(2)

one still has the problem he noted; one cannot compute W in \$ERROR. This can be dealt with by making the variances of EPS(1) and EPS(1) be thetas. Thus, assuming that THETA(10) and above are available, one writes

W = (F*F*THETA(10)*THETA(10) + THETA(11)*THETA(11))**.5
Y = F + F*THETA(10)*EPS(1) + THETA(11)*EPS(2)
etc.,

along with

\$THETA .... (0,.1), (0,<low lim quantif>); ... THETA(10), THETA(11)
\$SIGMA 1 FIX 1 FIX

LBS.

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From: LSheiner <lewis@c255.ucsf.edu>
Subject: Re: Dble exponential error model
Date: Thu, 07 Jan 1999 10:24:14 -0800

See my previous note regarding the original query;
In answer to Steve's comments, see below ...

Steve Duffull wrote:

>

> Hi Laurent
>
> > I need to calculate the Individual Weight residuals (IWRES) with a double
> > exponential intra-individual error model.
>
> It would seem to me that you should obtain the weighting the same as usual.
> Therefore
>
> W=F*EXP(EPS(1))
>
> the rest of the error term "+THETA(10)*EXP(EPS(2))" is independent
> of the model predicted concentration and is constant for any given
> individual.

While it may be constant, it none-the-less influences the variance. The model (corrected as in my last note, so as to linearize in EPS) says that the variance of an observation, plotted against the squared (true) value of the observation is a straight line with a non-zero intercept. The model above says that this regression goes through the origin. These are different variance models with quite different consequences for the relative weight assigned to different observations.

>
> However I have 2 points:
> I do not understand why you would want to use this model.

Steve may be refering to his next point, below, in which case I agree, but just in case there is a question in anyone's mind about the usefulness of the (corrected) model,

Y = F + F*EPS(1) + EPS(2),

let me opine that, as corrected, the model is perhaps the most useful single model for residual error that I know of: it says that there is a proportional component of error, but also a fixed component, for observations near zero, such that those observations are not associated with large weights, as they would be using a pure proportional (or power) error model.

> I do not understand how your model can identify the difference
> between THETA(10) and EPS(2).

Actually, THETA(1) and SIGMA(2,2); this was the first point in my last note.

LBS.

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From: Mats Karlsson <mats.karlsson@biof.uu.se>
Subject: Re: Dble exponential error model
Date: Thu, 07 Jan 1999 23:50:01 +0100

Dear Laurent and others,

Although the answer I (and Steve and Lewis) gave on the problem below offered correct slope-intercept models, they didn't realize that the model suggested actually was a valid (although unusual) one. It is not unidentifiable. One obvious example is if there are measured concentrations where the PK model (via F) predicts none, e.g. with an endogeneous substance. However, from what he writes I don't think Laurent meant such a situation, but rather one where the additive component (THETA(10)) should take care of a positive bias in the assay and improve the error model for low observations. To get IWRES for such a model, the following code could be used:

W=SQRT(THETA(11)**2 + F**2*THETA(12)**2)
;THETA(11)=SD(additive) THETA(12)=CV(proportional)
IWRES=(DV-RES)/W
Y=F+THETA(10)+W*EPS(1)
\$SIGMA 1 FIX

However, I agree with the misgivings of Steve and Lewis earlier that for an exogeneous substance it is an odd error model. The risk is that the constant (THETA(10)) will intefere with the estimation of e.g. the terminal slope.

Best regards,
Mats

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From: LSheiner <lewis@c255.ucsf.edu>
Subject: Re: Dble exponential error model
Date: Thu, 07 Jan 1999 17:17:41 -0800

As usual, Mats is right -

The original model is equivalent to

Y = F + F*EPS(1) + THETA(10) + THETA(10)*EPS(2)

so indeed, THETA(10) acts as a bias offset ...What I wrote assumes taht the intended model was

Y = F + F*EPS(1) + THETA(10)*EPS(2).

LBS.

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From: laurent.nguyen@pierre-fabre.com
Subject: Dble exponential error model
Date: Fri, 8 Jan 1999 17:50:46 +0100

Dear NONMEM users,

Thanks a lot for these useful informations regarding this odd and unusual dble exponential intra-individual error model for exogeneous substance (that I'm modelling).

As Mats or L Sheiner wrote : effectively, I used the following model: Y=F+F.EPS(1) + THETA(10) + THETA(10) .EPS(2) to correct a bias in the lowest observations. I used this model with rich PK data varying over 3 log; and with the lowest values of DV very closed of the limit of assay. As compared with a simple CCV or a slope-intercept intra-individual error model, data seemed better fitted using FO method; especially in the lowest values (less bias in WRES vs Time or PRED plot).In addition, significant but artefactual decrease in objective function (more than 150 points as compared with the slope-intercept error model) resulted using FO approximation. Howewer, as Mats suggested again, I have noted the fact that the constant THETA(10) interferes with other parameters. As a result, residual plots of posthoc estimates (IWRES vs IPRED, etc..) showed largest bias as compared with an intercept-slope error model. Then, not a good model for my data.

Best regards,

Laurent Nguyen