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Re: Very small P-Value for ETABAR

From: Nick Holford <n.holford>
Date: Mon, 17 Nov 2008 20:00:17 +1300

Xia,

I wrote:
> ETABAR=THETA(3)
> Y=F + EPS(1)*SD*EXP(ETA(1)) * ETABAR*EXP(ETA(2))
>
> It seems to me that the variance of ETA(1) will be confounded with the
> variance of ETA(2). Would you please explain more clearly (with an
> explicit NM-TRAN code fragment if possible) what you are suggesting?
>
Leonid added:
> CL = THETA(1) exp(THETA(2)*exp(ETA(1))) (2)
>
> But not every transformation is reasonable. I hardly can imagine the
> case when you may want to use (2). Could you give some more realistic
> examples, please, and situation when they were useful?
You replied but between

"Sorry, I did make myself clear."
and
"Sorry for any confusion!"

I only found unclear and confusing remarks (e.g. where is ETABAR actually used?)

Would you please focus more on answering our specific requests for an explicit NM-TRAN code fragment and justification for an apparently bizarre transformation and spend less time offering meaningless apologies?

Nick

XIA LI wrote:
> Leonid,
> Sorry, I did make myself clear.
>
> CL=THETA(1)*EXP(ETA(1)) (1)
> where ETA(1) is Normal( 0, omega^2) or
> log Normal(Eta_bar,omega^2)
>
> Adding one more stage means giving some functions for the MEAN and VARIANCE of ETA(1), say:
>
> Eta_bar=THETA(2)
> omega^= THETA(3)*EXP(ETA(2)) (2)
>
> Sorry for any confusion!
> Best,
> Xia
>
>
> ---- Original message ----
>
>> Date: Fri, 14 Nov 2008 18:37:22 -0500
>> From: Leonid Gibiansky <LGibiansky
>> Subject: Re: [NMusers] Very small P-Value for ETABAR
>> To: Xia Li <lix4
>> Cc: "'Nick Holford'" <n.holford
>>
>> Xia,
>> I could be missing something but this
>> ETA(1)= THETA(2)*exp(ETA(2)) (Eq. 1)
>> does not make sense to me. In the original definition, ETA(1) is the
>> random variable with normal distribution. Even if posthoc ETAs are not
>> normal, they are still random. For example, it can be either positive or
>> negative (unlike ETA1 given by (1)). If I the understood intentions
>> correctly, this is an attempt to describe a transformation of the random
>> effects to make it normal:
>>
>> CL = THETA(1) exp(ETA(1)) is replaced by
>> CL = THETA(1) exp(THETA(2)*exp(ETA(1))) (2)
>>
>> But not every transformation is reasonable. I hardly can imagine the
>> case when you may want to use (2). Could you give some more realistic
>> examples, please, and situation when they were useful?
>>
>> On the separate note, mean of THETA(2)*exp(ETA(2)) is not equal to
>> THETA(2): geometric mean of THETA(2)*exp(ETA(2)) is equal to THETA(2)
>>
>> Thanks
>> Leonid
>>
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web: www.quantpharm.com
>> e-mail: LGibiansky at quantpharm.com
>> tel: (301) 767 5566
>>
>>
>>
>>
>> Xia Li wrote:
>>
>>> Hi Nick,
>>> My pleasure!
>>>
>>> This is a topic from Bayesian Hierarchical Model(BHM). If we look at the
>>> simplest PK statement: CL=THETA(1)*EXP(ETA(1)), where ETA(1) is the between
>>> subject random effect. We assume the "similarity" among the subjects may be
>>> modeled by THETA(1) and ETA(1).
>>>
>>> Now here, if we observe that there is an underlying pattern between
>>> ETA(1)'s, i.e. deviation from zero or no longer normal and we assume that
>>> there is a similarity among those patterns.
>>>
>>> Since ETA(1)'s are assumed similar, it is reasonable to model the
>>> "similarity" among the ETA(1)'s by THETA(2) and ETA(2): ETA(1)=
>>> THETA(2)*exp(ETA(2)). Hence we have one more stage, ETA(1) now is
>>> lognormal(nonsymmetrical) with mean THETA(2) (doesnt have to be zero).
>>>
>>> We will not say the variance of ETA(1) is confounded with the variance of
>>> ETA(2), we say it is a function of variance of ETA(2).In statistics,
>>> confounding means hard to distinguish from each other. Here, it is a direct
>>> causation.
>>>
>>> Sorry I don't have a NM-TRAN code for this now. I usually use SAS and Win
>>> bugs to do modeling and haven't tried this BHM in NONMEM. I will figure out
>>> can I do it in NONMEM later.
>>>
>>> Best,
>>> Xia
>>>
>>> -----Original Message-----
>>> From: owner-nmusers
>>> Behalf Of Nick Holford
>>> Sent: Friday, November 14, 2008 3:34 PM
>>> To: nmusers
>>> Subject: Re: [NMusers] Very small P-Value for ETABAR
>>>
>>> Jakob, Mats,
>>>
>>> Thanks very much for your careful explanations of how asymmetric EBE
>>> distributions can arise. That is very helpful for my understanding.
>>>
>>> Xia,
>>>
>>> I am intrigued by your suggestion for how to estimate and account for
>>> the bias in the mean of the EBE distribution.
>>>
>>> In the usual ETA on EPS model I might write:
>>>
>>> ; SD of residual error for mixed proportional and additive random effects
>>> PROP=THETA(1)*F
>>> ADD=THETA(2)
>>> SD=SQRT(PROP*PROP + ADD*ADD)
>>> Y=F + EPS(1)*SD*EXP(ETA(1))
>>>
>>> where EPS(1) is distributed mean zero, variance 1 FIXED
>>> and ETA(1) is the between subject random effect for residual error
>>>
>>> You seem to be suggesting:
>>> ETABAR=THETA(3)
>>> Y=F + EPS(1)*SD*EXP(ETA(1)) * ETABAR*EXP(ETA(2))
>>>
>>> It seems to me that the variance of ETA(1) will be confounded with the
>>> variance of ETA(2). Would you please explain more clearly (with an
>>> explicit NM-TRAN code fragment if possible) what you are suggesting?
>>>
>>> Best wishes,
>>>
>>> Nick
>>>
>>> Xia Li wrote:
>>>
>>>> Hi Jakob,
>>>> Thank you very much for the information adding an "eta on epsilon". This
>>>>
>>> is
>>>
>>>> what I did in my research and I am glad to see people in Pharmacometrics
>>>>
>>> is
>>>
>>>> using it.
>>>>
>>>> And in Bayesian analysis, adding one more stage for ETA, i.e
>>>> ETA=ETABAR*exp(eta2), eta2~N(0,omega2) will allow the deviation from zero
>>>> and shrinkage of ETA.
>>>>
>>>> Again, thanks all for your input.:)
>>>>
>>>> Best Regards,
>>>> Xia
>>>>
>>>> Xia Li
>>>> Mathematical Science Department
>>>> University of Cincinnati
>>>>
>>>>
> ======================================
> Xia Li
> Mathematical Science Department
> University of Cincinnati
>

--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
n.holford
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
Received on Mon Nov 17 2008 - 02:00:17 EST

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