- On 10 May 1999 at 20:47:15, "Takimoto, Chris (NCI)" (takimotc.aaa.navmed.nci.nih.gov) sent the message

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W.L. Chiou has proposed using harmonic mean values instead of arithmetic means

for summarizing PK parameters for study groups when analyzing clearance, volume

of distribution and half-lives. In my survey of the PK literature, harmonic

means and pseudo standard deviations are commonly used for half-lives, but I

have not seen widespread use for CL and Vd. I was wondering if the group

had any

broad recommendations for the presentation of this type of summary data?

References

1. Chiou WL, New calcualtion method of mean total body clearance of drugs

and its application to dosage regimens. J Pharmaceut Sci 1980;69:90

2. Chiou WL, New Calculation method for mean apparent drug volume of

distribution and appliation to rational dosage regimens. J Pharmaceutic Sci

1979;68:1067.

3. Law FC, et al. Estimation of variance for harmonic mean half-lives. J

Pharmaceutical Sci 1985;74:229.

TIA

Chris H. Takimoto, M.D., Ph.D., FACP

Senior Investigator

National Cancer Institute

Building 8, Room 5101

National Naval Medical Center

8901 Wisconsin Avenue

Bethesda, MD 20889-5105

(301) 435-5369

Fax: (301) 435-8695

E-mail: ctakim.at.helix.nih.gov - On 11 May 1999 at 20:24:37, David_Bourne (david.at.pharm.cpb.uokhsc.edu) sent the message

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[A few replies about means - db]

Date: Tue, 11 May 1999 16:19:20 +1200

From: Nick Holford

Organization: University of Aucklandr

X-Accept-Language: en

MIME-Version: 1.0

To: PharmPK.-a-.pharm.cpb.uokhsc.edu

Subject: Re: PharmPK Arithmetic vs Harmonic Means for PK parameters

I suppose it depends on what you are trying to describe. If you have a set

of PK parameters, say CL or Vd, you might first of all see if they are

compatible with some common distribution e.g. normal or log-normal. If

normal then the arithmetic mean and SD would seem to be reasonable to

describe them. If log-normal then the geometric mean and CV would seem to

be reasonable. If you have enough data then the most honest presentation

would be a frequency distribution plot.

Can you explain more clearly why you are concerned about this?

--

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, Private Bag 92019, Auckland, New Zealand

email:n.holford.at.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556

http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

---

Reply-To: "Stephen Duffull"

From: "Stephen Duffull"

To:

Subject: Re: PharmPK Arithmetic vs Harmonic Means for PK parameters

Date: Tue, 11 May 1999 09:23:25 +0100

MIME-Version: 1.0

X-Priority: 3

Hi Chris

Why use a/g/h mean at all - why not median? It is likely

that any single descriptor of the central tendancy of a

parameter such as CL or V within a population will be wrong

(although many may be close to the true but unknown value)

and hence undue concern about which descriptor to use seems

unnecessary. I for one therefore prefer simple descriptions

such as medians and percentiles as these are less

presumptive about the underlying distribution of the

parameter within a study group than are means and SDs.

Regards

Steve

=====================

Stephen Duffull

School of Pharmacy

University of Manchester

Manchester, M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396

---

From: "Bonate, Peter, Quintiles"

To: "'PharmPK.at.pharm.cpb.uokhsc.edu'"

Subject: RE: PharmPK Arithmetic vs Harmonic Means for PK parameters

Date: Tue, 11 May 1999 07:20:19 -0500

Mime-Version: 1.0

In regards to the comments by Takimoto:

There is a good deal of theory behind the use of harmonic or geometric means

in reporting of clearance and Vd and technically this is the most

informative way to present the data, but more people understand arithmetic

mean and variance so that is the way pk data is usually presented.

Unfortunately, arithmetic means present a biased view of the data because

often pk variables are log-normal, not normally distributed, so the

population mean is often underestimated.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9708 (L4-M2828)

Kansas City, MO 64134

phone: 816-767-6084

fax: 816-767-3602

---

Date: Tue, 11 May 1999 09:02:11 -0400

From: Harold Boxenbaum

Organization: Z

X-Accept-Language: en

MIME-Version: 1.0

To: PharmPK.aaa.pharm.cpb.uokhsc.edu

Subject: Harmonic Means

If PK parameters are normally distributed, conventional means are

satisfactory. If

terminal exponential rate constants are normally distributed, then the terminal

exponential half-lives will not be so, but reciprocals of the half-lives

will be

(because of the inverse relationship of the two parameters). In theory, if

reciprocals of any PK parameter are normally distributed (and the parameter

itself

is not), one could use harmonic means. I have an excel program which

calculates

harmonic means and pseudo standard deviations. If interested, let me know,

and I

will e-mail a copy. I've sent a copy to David with this e-mail. Harold

Boxenbaum - On 12 May 1999 at 22:35:33, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

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[Two replies - db]

Reply-To: "Stephen Duffull"

From: "Stephen Duffull"

To:

Subject: Re: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

Date: Wed, 12 May 1999 09:03:34 +0100

MIME-Version: 1.0

X-Priority: 3

PETER BONATE wrote:

>There is a good deal of theory behind the use of harmonic

or geometric means

>in reporting of clearance and Vd and technically this is

the most

>informative way to present the data,

I would contend that this is *not* true. The most

informative way to present the data would be to present the

whole pdf. [Nick also commented on this point previously.]

>but more people understand arithmetic

>mean and variance so that is the way pk data is usually

presented.

I'm not sure this is a good reason to report an

inappropriate statistic (see comment on geometric mean

below).

>Unfortunately, arithmetic means present a biased view of

the data because

>often pk variables are log-normal, not normally

distributed, so the

>population mean is often underestimated.

I'm sure PK parameters are often said to be log normally

distributed - but are they really?

When I have performed popn analyses and obtained contiguous

frequency histograms of the parameters it seems to me that

neither normal or lognormal distributions would have

provided an adequate summary of the data. I'm not convinced

that the assumption that PK parameters are lognormaly

distributed isn't just another easy way out - hence we may

be perpetuating more inappropriate statistics (see comment

above etc).

An additional note: the arithmetic mean will overestimate

the central tendency of a distribution if the distribution

is positively skewed.

Regards

Steve

=====================

Stephen Duffull

School of Pharmacy

University of Manchester

Manchester, M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396

---

From: "Takimoto, Chris (NCI)"

To: "'PharmPK.-a-.pharm.cpb.uokhsc.edu'"

Subject: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

Date: Wed, 12 May 1999 07:53:42 -0400

MIME-Version: 1.0

Thanks for all of the thoughtful replies. In response to Dr. Holford's inquiry

on why I am concerned about this issue, it is because I was looking for some

rational guidelines on how to summarize pharmacokinetic parameters from small

clinical studies. I have seen both arithmetric means and SD and harmonic means

and pseudo standard deviations used for half-life values, but rarely or never

for CL or Vd. Some type of summary statistic for presenting the PK parameters

for groups of 20 to 25 patients is helpful and frequency distribution plots are

often not an option in journal publications due to space considerations. The

suggestion to use medians seems logical and I will definitely look at the

underlying distribution of our PK parameters before deciding which

statistics to

employ. Thanks again to all.

Chris H. Takimoto, M.D., Ph.D., FACP

Senior Investigator

National Cancer Institute

Building 8, Room 5101

National Naval Medical Center

8901 Wisconsin Avenue

Bethesda, MD 20889-5105

(301) 435-5369

Fax: (301) 435-8695

E-mail: ctakim.at.helix.nih.gov - On 13 May 1999 at 22:19:42, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

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[A few replies - db]

Date: Thu, 13 May 1999 18:15:33 +1200

From: Nick Holford

Organization: University of Aucklandr

X-Accept-Language: en

MIME-Version: 1.0

To: PharmPK.-at-.pharm.cpb.uokhsc.edu

Subject: Re: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

> From: "Takimoto, Chris (NCI)"

> To: "'PharmPK.aaa.pharm.cpb.uokhsc.edu'"

> Subject: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

> Date: Wed, 12 May 1999 07:53:42 -0400

> on why I am concerned about this issue, it is because I was looking for some

> rational guidelines on how to summarize pharmacokinetic parameters from small

> clinical studies.

But the key thing is what is the summary statistic to be used for? If it

just so you can list some numbers in a journal article as an end in itself

then maybe it doesn't matter what you use. However, if the numbers are

there to guide rational dose selection then using the parameter estimates

from a population PK model would seem to be the most reasonable thing. Your

problem of reporting arithmetic, geometric, harmonic mean is a consequence

of using the standard two stage perspective of population analysis. If you

do a full blown popln analysis then the population parameters and their

variability parameters and any covariate relationships become the natural

numbers to report in a journal article.

> Some type of summary statistic for presenting the PK parameters

> for groups of 20 to 25 patients is helpful and frequency distribution

>plots are

> often not an option in journal publications due to space considerations.

I think the journal space consideration is a weak argument. If you really

want to describe how the PK parameters occurred in these small groups then

a plot of the distribution of the values is the most unambigous way of

doing that. If you sort the values and plot them as vertical lines

proportional to the size of the parameter it is very easy to see the

distribution of all the values (this is very easy to do using Excel). If

the journal editor thinks differently (and you should ask before assuming

there is a problem) then maybe you should submit to a journal that it is

more friendly towards clinical pharmacokinetics.

> The suggestion to use medians seems logical

I cannot see any logic. If you have 25 values for CL and you *only* report

the median you have effectively thrown away the information from 24 people.

I think that is unethical use of their data. You should do everything

possible to use and report the information from all the subjects.

--

Nick Holford, Dept Pharmacology & Clinical Pharmacology

University of Auckland, Private Bag 92019, Auckland, New Zealand

email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556

http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm

---

Alternate-recipient: prohibited

Date: Thu, 13 May 1999 09:43:44 -0500 (EDT)

From: Jogarao Gobburu 301-594-5661 FAX 301480-8329

Subject: Harmonic Vs. Arithmetic Mean !

To: PharmPK PharmPK

MIME-version: 1.0

Posting-date: Thu, 13 May 1999 09:43:45 -0500 (EDT)

Importance: normal

Priority: normal

Sensitivity: Company-Confidential

A1-type: MAIL

Hello, I am re-posting my response I sent a couple of days ago which

bounced back.

======

The decision about the appropriate distribution (Normal vs. Lognormal

etc..) of PK/PD parameters can be made based on graphical display and

fitting appropriate density functions to the individual PK/PD

parameters. Where sparse data is available and modelling individual data

is not possible, exploring the available distribution functions based on

some statistical criteria is probably the only method. Most PK/PD

parameters have physiological boundaries there by making the

distribution skewed to one or the other side. Hence lognormal

distribution may be thought of as more appropriate. However, selecting

density functions is empirical.

With regards,

Joga

Joga Gobburu

Pharmacometrics,

CDER, FDA

---

From: "Bonate, Peter, Quintiles"

To: "'PharmPK.-at-.pharm.cpb.uokhsc.edu'"

Subject: RE: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

Date: Thu, 13 May 1999 12:03:56 -0500

Mime-Version: 1.0

Stephen Duffull recently responded to some comments I made regarding

arithmetic and geometric means. First, the original question was in regards

to small clinical studies, not population studies. There is a world of

difference in summarizing data from these types of studies. Second, two

reports deal with the theoretical and practical aspects of the distribution

of PK parameters in small sample studies:

E. Mizuta and A. Tsubotani: Preparation of mean drug

concentration-time curves in plasma. A study on the frequency distribution

of pk parameters. Chem Pharm Bull 33, 1620-1623, 1985.

LF Lacey: Common noncompartmental pharmacokinetic variables: are

they normally or log-normally distributed? J Biopharm Stat. 7, 171-178,

1997.

In both papers the authors conclude that pk parameters have a theoretically

log-normal distribution. Also, if you assume a true normal distribution, it

is possible for pk parameters to take on negative values, which is

impossible.

I agree that if the geometric mean more accurately describes the shape of

the data then one should report it. But how data is presented from a

clinical study in a journal is ultimately the responsibility of the

reviewers and editors. Until they start suggesting more rigorous summary

statistics of data, this practice will not change.

Stephen is also correct in the mean will overestimate the median for a

log-normal distribution. That was a typo on my part.

PETER L. BONATE, PhD.

Clinical Pharmacokinetics

Quintiles

POB 9708 (L4-M2828)

Kansas City, MO 64134

phone: 816-767-6084

fax: 816-767-3602 - On 17 May 1999 at 14:38:02, "Hans Proost" (J.H.Proost.-at-.farm.rug.nl) sent the message

Back to the Top

Dear Colleagues,

Many good points have been raised by others. I fully agree with Nick

Holford that it is typically a problem in the 'Standard Two-Stage

approach' (STS). STS overestimates the variability between

individuals considerably, and thus increases the problem of

choosing the most appropriate averaging procedure.

So, don't use STS!

I would like to make a final comment on distributions of PK

parameters. Several repliers have stated that parameters may be non-

normal, non-log-normal et cetera.

How did they come to these conclusions?

Such a conclusion should not be drawn too easily:

1) You need many, many patients to make such a conclusion. E.g., in

general, 20 subjects is not sufficient. This can be concluded from

Monte Carlo simulations. Starting with a normal distribution, you

will see many 'strange' distributions when repeating this several

times.

2) Estimates of PK parameters are often confounded by relatively

large standard errors; it is not uncommon that these standard errors

are of the same order of magnitude as the interindividual

variability. This implies that conclusions about the distribution of

these point estimates are confounded; to make conclusions, you need

much more subjects.

3) Don't forget the correlation between the point estimates of the

parameters of each subjects, and the correlation between the

population parameters.

Of course, using the whole pdf is the best thing to do. However, it

is not practical for most purpose. Whenever possible, one parameter

for the central tendency and one for the dispersion is preferred for

practical applications.

Therefore, my suggestion is: assume a log-normal distribution,

unless you have really good evidence that this is not warranted.

Johannes H. Proost

Dept. of Pharmacokinetics and Drug Delivery

University Centre for Pharmacy

Groningen, The Netherlands

tel. 31-50 363 3292

fax 31-50 363 3247

Email: j.h.proost.-at-.farm.rug.nl - On 20 May 1999 at 21:33:27, "Takimoto, Chris (NCI)" (takimotc.-a-.navmed.nci.nih.gov) sent the message

Back to the Top

Again, thanks for the always thoughtful, and typically stimulating suggestions

from Prof. Holford. I certainly agree that is important to try to elevate the

presentation of pharmacokinetic data in journals not principally focused on

clinical pharmacokinetics and that was the main point of my initial question.

Furthermore, the suggestion to use a population analysis is perhaps the most

elegant solution to this dilemma. Nonetheless, I do promise to try educate the

uneducated (i.e., journal editors) in this regard. And finally, I will never

EVER use AUC when clearance can be presented instead! ;) - On 10 Jun 1999 at 14:37:48, Roger Jelliffe (jelliffe.at.usc.edu) sent the message

Back to the Top

Dear Chris:

Thanks for your input. I think it is most important to spread the word

outside the area of our specialty, as you suggest. I agree that presenting

results in the form of population models is important. However, I would

suggest that the reason for this is to give the most informed dosage

regimen possible to achieve AND MAINTAIN a desired target goal. Population

modeling, then, is only the beginning, but it has received the most recent

attention. It helps us plan the most informed initial regimen.

However, just as important is what we do after that, and recently

this has

not had as much attention. Getting feedback and making a really

individualized PK model, tuning the model to best guide subsequent therapy,

is just as important. The really important thing is truly individualized

therapy, not just getting started.

Population models are the storage of past experience. As such, they are

always a bit like yesterday's newspaper. The important other thing is to

supplement the population model with feedback from the individual patient,

to remove the diversity present in the population model, to get the best

individualized model and regimen.

How to do that, and how to achieve desired targets with maximum

precision,

is what is important. Models that have only a single value for each

parameter provide no way to estimate the precision with which it is

possible to achieve a stated target goal. This is the problem with

parametric models, those based on one measure of the central tendency, plus

another for the dispersion. Witness all the arguments about what is the

"best" estimator of a distribution - mean, harmonic mean, median, etc. The

real problem is that only a single estimate is SOUGHT, and often very

important assumptions are made about the assumed shape of the population

parameter distributions - witness normal, lognormal, etc, in the recent

discussions. Since many drugs are metabolized fast by some and slowly by

others, some other way is needed to see the actual distribution of values

in a population, and to use that information with maximum precision.

One might get the conventional Bayesian posterior parameter values for

each subject in a population. Then the expected frequency of each of the N

subjects parameter values will be 1/N. If one wants to try a candidate

regimen, he can "give" it to each subject's model values and predict the N

future levels that will result. Then, at a desired target time, one can

compare the predictions with the goal. NOW, there is a tool to evaluate the

expected precision of the achievement of the desired target, by computing

the weighted squared error of its achievement, for example. This is the

essence of "multiple model" dosage design. One can then examine other

regimens, and find the one which specifically minimizes the expected value

of the weighted squared error in the failure to hit the target.

A more likely starting point, instead of starting with the

collection of

each patient's Bayesian posterior values, is starting with the essentially

N sets of parameter estimates given by making a nonparametric population

model. It is the most likely set of parameter values, each with its own

estimated probability, that results from the raw data in the population.

This is the strength of nonparametric population models. This does NOT mean

NONCOMPARTMENTAL, as some have confused noncompartmental models with

nonparametric ones. NONPARAMETRIC means that the ENTIRE population

parameter DISTRIBUTION is estimated, but just single values for parameters.

This results in essentially N discrete sets of parameter values, each with

its own estimated probability. Conventional summaries such as mean, median,

mode, etc, can also be obtained, but the strength of the method is that it

is not subject to any such parametric assumptions as to the actual shape of

the parameter distributions.

The important thing is not what is the best summary estimate of the

population parameter values, but rather how can we develop the regimen,

based on the raw population data from each past subject in the population,

which will achieve a desired target goal MOST PRECISELY in the future. The

"separation" or "certainty equivalence" principle (Bertzekas D. Dynamic

programming: deterministic and stochastic models. Englewood Cliffs NJ:

Prentice-Hall, 1987, pp 144-146) states that whenever one seeks to control

a system by separating the process into the 2 phases, first, of getting

single parameter estimates (the current PK/PD cultural paradigm) and then

of using these values to control the system, then the control inevitably is

done suboptimally. This is the problem when using parametric population

models to guide the development of dosage regimens. This is avoided by

using multiple models, having multiple sets of parameter values, which

permit evaluation of therapeutic precision and fine tuning the regimen to

optimize it. This is the way the modern aircraft fly and modern missiles

are guided, to achieve target goals with maximum precision.

Further discussion is available in Clin Pharmacokinetics, 34:

57-77, 1998.

Sincerely,

Roger Jelliffe

Roger W. Jelliffe, M.D. Professor of Medicine, USC

USC Laboratory of Applied Pharmacokinetics

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.aaa.hsc.usc.edu

Our web site= http://www.usc.edu/hsc/lab_apk

*************************** - On 13 Jun 1999 at 19:11:11, David_Bourne (david.aaa.boomer.org) sent the message

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X-Sender: mentor.at.hardlink.com

Mime-Version: 1.0

Date: Sun, 13 Jun 1999 15:20:30 -0400

To: PharmPK.at.pharm.cpb.uokhsc.edu

From: Daro Gross

Subject: Re: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

At 2:38 PM -0500 5/17/99, Hans Proost wrote:____________________

> PharmPK - Discussions about Pharmacokinetics

> Pharmacodynamics and related topics

>

> Dear Colleagues,

>

> Many good points have been raised by others. I fully agree with Nick

> Holford that it is typically a problem in the 'Standard Two-Stage

> approach' (STS). STS overestimates the variability between

> individuals considerably, and thus increases the problem of

> choosing the most appropriate averaging procedure.

> So, don't use STS!

Correct. Averaging makes undlying assumptions that cannot be substantiated.

> 1) You need many, many patients to make such a conclusion. E.g., in

> general, 20 subjects is not sufficient. This can be concluded from

> Monte Carlo simulations. Starting with a normal distribution, you

> will see many 'strange' distributions when repeating this several

> times.

One needs a general description of the macro-biological phenomenom in order

to avoid improper statistical analysis. Study size helps in mapping out

this macro-biological phenomenom: 20 subjects is rarely sufficient. Several

hundred subjects is usually necessary to eliminate erroneous assumptions.

> 3) Don't forget the correlation between the point estimates of the

> parameters of each subjects, and the correlation between the

> population parameters.

Try establishing a normal function and then correlating patients to the

normal function. It works much better at eliminating errors in measurement

and incomplete data.

> Therefore, my suggestion is: assume a log-normal distribution,

> unless you have really good evidence that this is not warranted.

Log-normal distributions are rarely warranted, unless it can be established

that there is a linear correlation between all the variables: a very rare

occurance.

---

X-Sender: mentor.aaa.hardlink.com

Mime-Version: 1.0

Date: Sun, 13 Jun 1999 15:30:54 -0400

To: PharmPK.at.boomer.org

From: Daro Gross

Subject: Re: PharmPK Re: Arithmetic vs Harmonic Means for PK parameters

At 2:37 PM -0500 6/10/99, Roger Jelliffe wrote:____________________

> is just as important. The really important thing is truly individualized

> therapy, not just getting started.

The PK model need not be so highly individualized to work, but I applaud a

return to basic medical values: treatment of patients, not publication.

> Population models are the storage of past experience. As such, they are

> always a bit like yesterday's newspaper. The important other thing is to

> supplement the population model with feedback from the individual patient,

> to remove the diversity present in the population model, to get the best

> individualized model and regimen.

Past experience can be informative and confusing. Just like gossip, it

tends to become distorted over time as the actual data is replaced by

statistical measurements of the data that are often more confusing than

elucidating.

> discussions. Since many drugs are metabolized fast by some and slowly by

> others, some other way is needed to see the actual distribution of values

> in a population, and to use that information with maximum precision.

Keep in mind that populations are often made up of groups with highy

disparate chemistries.

> regimens, and find the one which specifically minimizes the expected value

> of the weighted squared error in the failure to hit the target.

Defining "target populations" is more difficult than it seems and may be

the undlying problem.

> This results in essentially N discrete sets of parameter values, each with

> its own estimated probability. Conventional summaries such as mean, median,

> mode, etc, can also be obtained, but the strength of the method is that it

> is not subject to any such parametric assumptions as to the actual shape of

> the parameter distributions.

One basic assumption must be made: that any population will be

noncompartmental. However, it is often useful to compartmentalize

populations in order to identify normal functions in the data.

> models to guide the development of dosage regimens. This is avoided by

> using multiple models, having multiple sets of parameter values, which

> permit evaluation of therapeutic precision and fine tuning the regimen to

> optimize it. This is the way the modern aircraft fly and modern missiles

> are guided, to achieve target goals with maximum precision.

Agreed. - On 27 Jul 1999 at 21:41:38, Roger Jelliffe (jelliffe.aaa.usc.edu) sent the message

Back to the Top

Dear All:

Perhaps this is beating a dead horse, but I don't quite think so. The

basic statements have been made below these more recent ones of mine. There

is much discussion (see below) of harmonic mean, etc, and of other points

relating to optimal estimation of population parameter values and their

variances.

In my view, there are better ways to do all this. Let me

begin first, by

asking WHY we wish to do all this? I think it is usually so we can take the

optimal course of action based on the information contained in the raw data

of the population studied. Usually this course of action is to develop a

dosage regimen that will achieve a desired target goal most precisely.

There has been all this argument and discussion about what

constitutes the

best parameter estimate. However, the separation principle used in

stochastic control of systems states that whenever the process of

controlling a system is separated into the 2 steps of:

1. Getting the best single point parameter estimates (what everybody is

discussing), and then

2. Using these estimates to control the system (design the

dosage regimen),

that the control is done SUBOPTIMALLY.

That is the problem with all this discussion. With such a

strategy, there

is no performance criterion to be optimized. There is no way to compute, in

advance, the expected error which which the goal will be achieved. The

regimen is simply computed, and it is assumed that it will in fact achieve

the desired goal.

Talk to Alain Mallet, and read his stuff. Nonparametric PK/PD

modeling is

NOT noncompartmental modeling, as many still appear to think. It is

compartmental modeling in which there are no parametric assumptions such as

mean, SD, etc, made about the shape of the distributions of the parameters

in the population.

The truly best population model one could ever make is to get

the entire

collection of each subject's exactly known parameter values, with the

correct structural model. You never can do better that that. There is no

need to summarize it further by getting means, medians, modes, or whatever,

although you can if you want to. NP pop modeling, whether Mallet's NPML or

Alan Schumitzky's NPEN approach, gets you essentially one set of parameter

estimates for each subject studied, each with an estimate of the

probability of each set of estimates. As Mallet has shown, this, and not

some assumed normal or lognormal distribution, is the most likely solution

to the population analysis problem short of, perhaps, the hierarchical

Bayesian or Gibbs sampling approach, which is a further improvement.

So what? Now, instead of having only ONE estimate to the population

parameter values, you have one for each subject. Now, you can examine the

time course of the predicted serum levels, or other responses, from any

candidate regimen. You can compute the weighted squared error in the

achievement of the desired goal at that desired time. A real simulated

clinical trial! Then, you can examine other regimens, and can find the

regimen which specifically minimizes the weighted squared error. This is

the multiple model strategy of dosage design. It develops regimens which

specifically maximize the predicted precision with which it is possible to

achieve the desired target goal at the desired time.

Precise control of this type is what is really important.

Let's talk about

this as the goal, NOT what is the "best" single point population parameter

estimate, or some estimate of the variance within the population. That does

NOT lead to optimally precise control, to optimally precise regimens. The

strength of nonparametric population modeling is that it gets you N

parameter estimates from N subjects, and the shape of the distribution

simply IS whatever it IS. NP modeling is what lets you estimate the

precision of with which a regimen will achieve a goal, and multiple model

dosage design will get the regimen which achieves the goal most precisely.

The flight control systems for the Airbus, the 777, the modern fighters,

and the missiles and spacecraft are all multiple model Bayesian adaptive

controllers, I am told. They do the job most precisely because they are

designed specifically to do so.

Hope this at least stirs up the dust a bit. Be very clear -

nonparametric

population models of this type are NOT, repeat NOT, concompartmental.

Best regards to all,

Roger Jelliffe

Roger W. Jelliffe, M.D. Professor of Medicine, USC

USC Laboratory of Applied Pharmacokinetics

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.at.hsc.usc.edu

Our web site= http://www.usc.edu/hsc/lab_apk

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