From: genxer@earthling.net
Date: Tue, 2 Mar 1999 16:10:23 -0500 (EST)
Subject: offsets - Centering covariates

Hi

Can someone explain (to me) why an expression like:

TVCL=THETA(1)+THETA(2)*(AGE-30) [1]

works better than say

TVCL=THETA(1)+THETA(2)*(AGE) [2]

Joseph

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Subject: RE: offsets
Date: Wed, 3 Mar 1999 09:44:44 +0100

Dear Joseph,
I don't know what to you mean saying "works better". My experience is that fitting Eq[1] usually creates problems whereas Eq[2] does not. With Eq [1] you have to constrain properly initial estimates for THETAs to avoid CL<=0. Of course, the interpretation of THETA(1) in Eq [1] is more transparent: it is just a typical CL in individuals of 30 yr old. If 30 yr is the median age in your (sub)population, everything is fine. On the contrary, THETA(1) in Eq. [2] is an intercept which may even be negative and thus difficult to interpret.

Hope this helps.

Best regards,
----------------------------------------------------
Clinical Pharmacokinetics, ext 5463
Janssen Research Foundation
2340 Beerse, Belgium
e-mail: vpiotrov@janbe.jnj.com

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From: KENNETH.G.KOWALSKI@monsanto.com
Subject: Re[2]: offsets
Date: 03 Mar 1999 08:51:10 -0600

Joseph,

My experience is in direct contrast to what Vladimir has suggested. When estimates of the thetas for Eq [1] or [2] are numerically stable, both should lead to identical fits (same minimum value of ELS) since they are just two different parameterizations of the same model.

However, when there is some numerical instability due to high correlations among the estimates of the thetas then centering (as in your Eq [1]) or scaling (dividing the covariate by some scale factor when covariates are included in a multiplicative form) should reduce the correlation among the estimates and lead to faster convergence.

You might try fitting both parameterizations and see which converges more quickly. Of course, to make it a valid comparison, you need to set the starting values for THETA(1) and THETA(2) for the two parameterizations such that they yield the same value for ELS at the 0-th iteration.

Ken

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From: LSheiner <lewis@c255.ucsf.edu>
Date: Wed, 03 Mar 1999 11:58:57 -0800
Subject: centering variables

As I recall there was considerable discussion on this topic sometime in the past, and perhaps someone can find that discussion in the archives at http://gaps.cpb.uokhsc.edu/nm/ although I was unsuccessful ...

Anyway, the reasons are primarily these

Example:
(A) Cl = theta(1) + theta(2)*age
vs.
(B) Cl = theta(1) + theta(2)*(age - mean_age)

Observations:
1. Numerical: the parameters are less correlated which means the search is easier, less likely to terminate with rounding errors, etc. With (A), since most of the data will be at ages around mean_age, a small change in theta(2) will sweep out a large change in theta(1) - hence high correlation of parameter errors. On the other hand, with (B) the "intercept" (theta(1)) is the CL at the mean age and the slope pivots around that point - hence slope and intercept estimation errors are uncorrelated.

2. Meaningful parameters: With (A), theta(1) is clearance at a theoretical age of zero. This is not a parameter that has much biological meaning. With (B), theta(1) is mean CL at age = mean_age, a very meaningful parameter, and one that may be of considerable importance in its own right. With (B), the covariance step gives you the SE of a meaningful parameter; with (A), it does not.

Bottom line: There is nothing to lose with centering, and much to gain. Hence, ALWAYS do it!

LBS.
--
Lewis B Sheiner, MD
Professor: Lab. Med., Biopharm. Sci., Med.
Box 0626
UCSF, SF, CA
94143-0626
voice: 415 476 1965
fax: 415 476 2796
email: lewis@c255.ucsf.edu

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From: "Bruce CHARLES" <bruce@pharmacy.uq.edu.au>
Date: Thu, 4 Mar 1999 08:07:21 +1000
Subject: Re: centering variables

--In response to Dr. Sheiner's message-----

I agree with what you say, except that the statement immediately above in 2. is not quite true, is it? I hope this is not splitting hairs but my interpretation of this always has been that the CL in such "slope-intercept" models has 2 components: Theta(1), a "base" clearance which is independent of

age, and Theta(2) a proportionality coefficient for an additional age-dependent CL. Therefore, it's not so much a lack of any "biological meaning" at age 0, its simply that there is an underlying component which contributes to the total CL, independent of age. This is analagous to the contribution of the non-renal component of aminoglycoside CL, and that explained by creatinine clearance, i.e. CL = Theta(1) + Theta(2)* ClCr. The CL reduces to Theta(1) in total renal dysfunction (provided that this pathophysiological state has no influence on non-renal elimination) .

Furthermore, when using Model (B) in a mostly elderly population (age distribution skewed to the right) isn't there the risk of getting negative CL values for the younger subjects?

Cheers,
BC
Bruce CHARLES, PhD
Senior Lecturer
School of Pharmacy
The University of Queensland
Brisbane, QLD Australia 4072
TEL: +61 7 336 53194
FAX: +61 7 336 51688
Email: Bruce@pharmacy.uq.edu.au

****

From: LSheiner <lewis@c255.ucsf.edu>
Date: Wed, 03 Mar 1999 15:18:16 -0800
Subject: Re: centering variables

--In response to Dr. Charles' message----

In the renal clearance case, indeed, the intercept is the component of clearance in the absence of renal function. What does it mean to speak about the component of clearance in the absence of age? How can there be an age-independent component if age is a surrogate (as it is in model (A))) for size as well?

The notion of the covariate-independent part of clearance is only meaningful (at least to me) for the covariates that can be "absent" or zero and be meaningful (as, for example, the deviation of age from mean age can, indeed, be zero meaningfully) Absent renal function is an observable and meaningful state. Absent age is not (unless you define age=0 as time of birth, and then the intercept has the meaning of clearance at time of birth ... making it obvious why you would never want to use such a model if all your data were from people aged 18 and over - it is absurd to imagine that one could extrapolate such data down to time of birth!).

>
> Furthermore, when using Model (B) in a mostly elderly population
> (age distribution skewed to the right) isn't there the risk of getting
> negative CL values for the younger subjects?
>

This is the problem of using linear models to extraoplate beyond the limits of one's data. All linear models will ultimately break down for parameters that reflect real (positive) biological quantities. But, technically, for any data set, you will get the identical model whether you use (A) or (B), so if you have the problem with one parameterization, you have the same problem with the other: recall, the only differences are (i) ease of finding solution, and (ii)

interpretability of the untransformed parameters, and hence direct estiamtes of SE's of interpretable parameters. The model will predict the exact same values of CL as a function of age with either parameterization.

LBS.
--
Lewis B Sheiner, MD
Professor: Lab. Med., Biopharm. Sci., Med.
Box 0626
UCSF, SF, CA
94143-0626
voice: 415 476 1965
fax: 415 476 2796
email: lewis@c255.ucsf.edu

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Date: Thu, 4 Mar 1999 08:55:43 +0000
Subject: Re: Re[2]: offsets

Dear all

sorry maybe I am totaly wrong and it is far to easy what I am suggesting: Isn't it, that if the distribution of ages is narrow around 30 the variability of the individuals is stronger represented with EQ [1] because differences between ages are bigger relative to THETA ?

Pharmazeutische Technologie und Biopharmazie
Rheinische Friedrich-Wilhelms Universität Bonn
An der Immenburg 4
53121 Bonn
Tel.: 0228 73-5117
Fax.: 0228 73-5268

****

From: "Nick Holford" <n.holford@auckland.ac.nz>
Subject: Re: centering variables
Date: Thu, 4 Mar 1999 21:02:39 +1300

>> Furthermore, when using Model (B) in a mostly elderly population
>> (age distribution skewed to the right) isn't there the risk of getting
>> negative CL values for the younger subjects?
>>
>
>This is the problem of using linear models to extraoplate beyond
>the limits of one's datae. All linear
>models will ultimately break down for parameters that
>reflect real (positive) biological quantities

One way to avoid negative clearance yet using a simple extension to the linear model without worrying about a priori constraints on THETA(age) is to use:

CLtv = CLpop * EXP(THETA(age)*(AGE-AGEstd))

For small values of THETA(age)*(AGE-AGEstd) this is essentially the same as:

CLtv = CLpop * (1+THETA(age)*(AGE-AGEstd))

If THETA(age)*(AGE-AGEstd)) becomes large and would cause a negative Cl in the linear model then the exponential model will gently decrease the clearance asymptotically towards zero. On the other hand, for small differences between Clpop and CLtv, THETA(age) has the same interpretation as with the linear model.

--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford@auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.html