Re: Additive plus proportional error model for log-transform data
Hi Ahmad,
This issue hasbeen discussed a lot and I'm afraid there's no consensus yet.
To your question:
1. Is there away to solve this problem for the indicated formulas?
As you said, thisproblem occurs at the early/later time points. In other words,
it happenswhen prediction is relatively low. This is because there's an
underlyingapproximation in the derivation of the two formulas:
----------------------------------------------------------------------------------------------------------------------------------------
Innon-transformed terms
Conc=F*EXP(SQRT(THETA(x)**2+THETA(y)**2/F**2)*EPS(1))
Assuming thatEXP(x)=1+x (for small x), you get
Conc=F*(1+SQRT(THETA(x)**2+THETA(y)**2/F**2)*EPS(1))
Variance of thisexpression
Var(conc)=F**2*(THETA(x)**2+THETA(y)**2/F**2)=
= F**2*THETA(x)**2+THETA(y)**2
On the otherhand, for the error model
Y=Fexp(EPS1)+EPS2=F(1+EPS1)+EPS2
variance isequal to
F**2*OMEGA1+OMEGA2
Thus, thesemodels are similar if not identical with
OMEGA1=THETA(x)**2,
OMEGA2=THETA(y)**2
-------------------------------------------------------------------------------------------------------------------------------------------
So when F is rathersmall, the approximation of exp(x)=1+x doesn't workanymore.
This may not be a problem when fitting your data.It may only occurwhen the
prediction is extremely low, let say 10^-4. In opinion it's safe touse this
formula in most cases. But if you are seeking a perfect answer,Jacob's
suggestion may be the one.
2. Are the twoformulas below equally valid?
As far asI'm concerned, these two are equally valid. I question the result
that there'sa difference in OFV and estimates between the two.
3. Is there analternative formula that I can use which does not have
this numerical problem?
Maybe youcan try Stu's "double exponential error model": Y = LOG(F+M)
+(F/(F+M))*ERR(1) + (M/(F+M))*ERR(2).
Best regards,
Rong Chen
School ofPharmaceutical Science
Peking University
Beijing, China
Quoted reply history
From: "Abu Helwa, Ahmad Yousef Mohammad - abuay010"
<[email protected]>
To: "[email protected]" <[email protected]>
Sent: Thursday, 2 June 2016, 10:27
Subject: [NMusers] Additive plus proportional error model for log-transform
data
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{margin-bottom:0cm;}#yiv5739009609 Dear NMusers, I am developing a PK model
using log-transformed single-dose oral data. My question relates to using
combined error model for log-transform data. I have read few previous
discussions on NMusers regarding this, which were really helpful, and I came
across two suggested formulas (below) that I tested in my PK models. Both
formulas had similar model fits in terms of OFV (OFV using Formula 2 was one
unit less than OFV using Formula1) with slightly changed PK parameter
estimates. My issue with these formulas is that the model simulates very
extreme concentrations (e.g. upon generating VPCs) at the early time points
(when drug concentrations are low) and at later time points when the
concentrations are troughs. These simulated extreme concentrations are not
representative of the model but a result of the residual error model structure.
My questions: 1. Is there a way to solve this problem for the indicated
formulas? 2. Are the two formulas below equally valid? 3. Is there an
alternative formula that I can use which does not have this numerical problem?
4. Any reference paper that discusses this subject? Here are the two
formulas: 1. Formula 1: suggested by Mats Karlsson with fixing SIGMA to 1:
W=SQRT(THETA(16)**2+THETA(17)**2/EXP(IPRE)**2) 2. Formula 2: suggested
by Leonid Gibiansky with fixing SIGMA to 1: W = SQRT(THETA(16)+
(THETA(17)/EXP(IPRE))**2 ) The way I apply it in my model is this:
FLAG=0 ;TO AVOID ANY CALCULATIONS OF LOG (0) IF
(F.EQ.0) FLAG=1 IPRE=LOG(F+FLAG)
W=SQRT(THETA(16)**2+THETA(17)**2/EXP(IPRE)**2) ;FORMULA 1 IRES=DV-IPRE
IWRES=IRES/W Y=(1-FLAG)*IPRE + W*EPS(1) $SIGMA 1. FIX Best regards,
Ahmad Abuhelwa School of Pharmacy and Medical Sciences University of South
Australia- City East Campus Adelaide, South Australia Australia