# Re: Very small P-Value for ETABAR

From: XIA LI <lix4>
Date: Mon, 17 Nov 2008 00:28:25 -0500 (EST)

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
>> 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
Received on Mon Nov 17 2008 - 00:28:25 EST

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