Tip # 16 -
Modeling a "Baseline" Component and an Additive "Drug" Component:
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.
PRODUCTS:
NONMEM® USER TIPS