- On 17 Mar 1998 at 16:08:11, "Leon Aarons" (laarons.aaa.fs1.pa.man.ac.uk) sent the message

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It seems that the debate on weighting appears regularly in the

pharmacokinetic literature (now read web for literature). The

earliest paper in a pharmacokinetic context was that of Boxenbaum,

Riegelman and Elashoff (JPB 2: 123-148 (1974)). In a biochemical

context there are numerous papers including Atkins (Biochem.J.

138:125-127 (1974)) and Finney (Appl.Stat. 26:312-320 (1977)). There

is also a long history in the Chemometrics literature (see Garden et

al Anal.Chem 52: 2310-2315 (1980)). The contribution of Peck et al

(JPB 12: 545-558 (1984)) is also illuminating but however also see

van Houwelingen (Biometrics 44: 1073-1081 (1988)).

The theory is quite clear (Draper and Smith and numerous other

books). In a least squares context weight by the inverse of the

expected variance of the datum (not as some arbitrary function of

the observed datum). In many applications this is straightfoward as

replicate experiments can be performed. Unfortunately in

pharmacokinetics this can not be done. Note that replicating the

assay or repeating the whole experiment does not replicate the

sample. The closest you might come to it is to take simultaneous

samples from different veins, althought it is not clear that this is

a true replicate. Therefore what should you do.

Roger Jelliffe advocates using a function based on the determined

assay error. As he points out it is simple to perform and certainly

systematizes the approach. However assay is not the only source of

variability, which Roger acknowledges. Stochastic variability may be

part of the process noise but it cannot be estimate separately from

the assay error except perhaps if the assay of a sample could be

repeated 10,000 times to reduce the assay error by a factor of 100.

Simply to multiply the 'assay polynomial' by a scaling factor seems

to defeat the purpose of fitting a polynomial in the first place as

there is no theory which suggests the 'process noise' should have the

same functional form as the assay error. The idea of fixing the

residual variability to that of the assay was originally used by

Mallet in the NPML program. In this case the the residual error had

to be fully specified (including the proportionality constant). My

understanding is that in the lastest version of the program it is now

possible to estimate an 'error model'.

So what should be done? Well let me say that I do not know the

perfect solution. For pragmatic reasons people tend to opt for

pragmatic solutions such as constant variance or constant cv (note

that chosing a constant weight of 1 or 0.1 etc makes no difference

to the precision estimates in least squares as the proportionality

term (sigma^2) is estimated from the residual sum of squares. The

weights only have to be proportional to the reciprocal of the

variance). Roger is right to criticize these schemes as they are only

abstractions of reality. More complicated variance models can be

dreamt up, for example a combination of constant variance and

constant cv but the parameters of these schemes are difficult to

estimate (see Raab Appl.Stat. 30: 32-40 (1981)).

My own approach is a combination of pragmatism and experience.

Eventual having analysed a number of data sets with various

'reasonable' weighting schemes look at the residual plots and choose

the 'best' common model. Mixed effects modelling can also help here.

Then use that weighting scheme for all the data sets. This scheme may

not appear optimal for all data sets but the decision has an

empirical Bayesian feel about it in that you are using information

from all of the data sets to come to this decision and I would be

happy with that.

As I said I would not claim this to be perfect and I expect to hear

of alternatives and objections.

Leon Aarons

________________

Leon Aarons

School of Pharmacy and Pharmaceutical Sciences

University of Manchester

Manchester, M13 9PL, U.K.

tel +44-161-275-2357

fax +44-161-275-2396

email l.aarons.-at-.man.ac.uk

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