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 <[EMAIL PROTECTED]> >Subject: Re: [NMusers] Very small P-Value for ETABAR >To: Xia Li <[EMAIL PROTECTED]> >Cc: "'Nick Holford'" <[EMAIL PROTECTED]>, "'nmusers'" <[email protected]> > >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: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On >> 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
