From: "Bachman, William" 
Subject: [NMusers] NONMEM Tips #16 - April 2, 2003 - Modeling a "baseline" component and an additive "drug" component
Date: Wed, 2 Apr 2003 09:15:33 -0500

This tip was contributed by Lewis Sheiner:

I'm not sure if anyone has written this up for NMUSERS (perhaps this is 
simply a duplicate of stuff that's already been sent around -- the fact 
that I can't recall it provides no valid evidence on this point), or 
maybe it is so well known as not to be worth a note, but if neither of 
those are true, then I have found the following of occasional use, and 
you might want to pass it on to the group.

Set-up: Usually PD, but could be PK with endogenous production.  The 
model will have a "baseline" component and an additive "drug" component. 
The data consist of a baseline(pre-drug) measurement, and then serial 
measurements after the drug.  The goal is  the drug model, and the 
baseline is a nuisance variable.  

The obvious model (in NONMEM speak), illustrated for the simplest 
possible case (clearly the model for both IPRED and Y can be elaborated 
at will),

(1) $ERROR
        IPRED = THETA(1)+ETA(1) + F
        Y = IPRED + ERR(1)

where F(time=0) = 0, and the data includes DV at time zero (the observed 
baseline value), involves a model for the baseline  (in this simple 
example, a normally distributed r.v., with mean THETA(1))), and if this 
model has a problem (e.g., baseline is not symmetrically distributed), 
then some power or precision will be lost in making inferences or 
estimating more interesting parameters, say the influence of a covariate 
on the drug response model (F).

On the other hand, deleting the baseline DV, but including its value as 
a covariate in the data, say BSL, present in every record),  and modeling

(2) $ERROR
        IPRED = BSL + F
       Y = IPRED + ERR(1)

avoids the problem of model (1)  by conditioning on the baseline (while 
making no modeling assumptions about it), but has the same problem that 
'subtracting the baseline' always has, namely that the baseline is 
measured with error and that error is also being conditioned upon.

A conditional model, in the spirit of (2), but which avoids conditioning 
on the error, uses the same (reduced) data set as for model (2), assumes 
that the baseline is measured with the same noise model as all 
subsequent measurements, and uses

(3) $ERROR
        IPRED = BSL + THETA(2)*ETA(1) + F
        Y = IPRED + THETA(2)*ERR(1)
      $OMEGA 1 FIX
      $SIGMA  1 FIX

The "trick" here is that the error in BSL (that is, "true" BSL minus 
observed value, "persists" throughout the individual record, and hence 
an ETA must be used, but it must have the same variance as the epsilon 
error.  Model (3) accomplishes this.

Best,
Lewis.
     
 
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From:Mats Karlsson 
Subject:Re: [NMusers] NONMEM Tips #16 - April 2, 2003 - Modeling a "baseline" componentand an additive "drug" component
Date:Fri, 04 Apr 2003 20:45:11 +0200

Dear all,

Just a couple of thoughts on Lewis alternatives 1 and 3 (2 seems to have no
advantages over 3).

Alt. 3 will differ from 1 in at least a few aspects:

1) For the FO method, Model 3 seems advatageous as the linearization would occur
closer to the individuals' predictions. This could be of value whenever residual
error is heteroscedastic (instead of the homoscedastic error in the simple
example below). Also when BSL comes into the model in a more complex form such
that it influences the predicted profiles in another way than mere scaling, this
could be advantageous.

2) If there is a correlation between baseline and any other parameter, this
would be incorporated into model 1 as an estimated covariance, whereas in model
3 as a covariate relationship between BSL and the parameter in question. The
latter will provide a wider range of shapes for the relation than a mere
correlation. This difference may well also have other implications.

3) Model 3 does not assume any distribution of BSL in the population (as Lewis
points out). On the other hand simulation from the final model will rely on the
empirical distribution of BSL values. Model 1 on the other hand will estimate
(with imprecision estimates) the features of BSL and any covariate relations
that influence BSL. If Model 3 is used, such a model could of course be
developed separately but with less information speaking to BSL (all data point
speak to some extent to BSL) and with additional/other assumptions about error
structure.

4) Some modellers like, for indirect effect models, to estimate Kin and Kout, as
true physiological parameters, rather than BSL. Model 3 would not allow such
parametrisation.

Best regards,
Mats


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