# Re: [NMusers] Additive plus proportional error model for log-transform data

From: Rong Chen <rongchenchn_at_yahoo.com>
Date: Thu, 2 Jun 2016 07:07:05 +0000 (UTC)

This issue hasbeen discussed a lot and I'm afraid there's no consensus yet.=
В

1.В В В В В В В Is there away to solve this pro=
blem for the indicated formulas?

As you said, thisproblem occursВ at the early/later time points. In oth=
er words, it happenswhen prediction is relatively low. This is because ther=
e'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))

В Variance of thisexpression

Var(conc)=F**2*(THETA(x)**2+THETA(y)**2/F**2)=

В  В В  В  В  В  В  В  = F**2*THETA(x)**2=
+THETA(y)**2

В On the otherhand, for the error model

Y=Fexp(EPS1)+EPS2=F(1+EPS1)+EPS2

variance isequal to

F**2*OMEGA1+OMEGA2

В Thus, thesemodels are similar if not identical with

OMEGA1=THETA(x)**2,

OMEGA2=THETA(y)**2

---------------------------------------------------------------------------=
----------------------------------------------------------------

So when F is rathersmall, the approximation ofВ exp(x)=1+xВ doesn=
't workanymore. This may not be a problem when fitting your data.It may onl=
y 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 an=
swer,Jacob's suggestion may be the one.

В 2.В В В В В В В Are the twoformulas below=
equally valid?

В As far asI'm concerned, these two are equally valid. I question the r=
esult that there'sa difference in OFV and estimates between the two.

В 3.В В В В В В В Is there analternative fo=
rmula that I can use which does not have this numerical problem?

В Maybe youcan try Stu's "double exponential error model": Y = LOG(F+=
M) +(F/(F+M))*ERR(1) + (M/(F+M))*ERR(2).В

Best regards,

Rong Chen

School ofPharmaceutical Science

Peking University

Beijing, China

ymail.unisa.edu.au>
To: "nmusers_at_globomaxnm.com" <nmusers_at_globomaxnm.com>
Sent: Thursday, 2 June 2016, 10:27
Subject: [NMusers] Additive plus proportional error model for log-transfor=
m data

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v5739009609 Dear NMusers, В  I am developing a PK model using log-tran=
sformed single-dose oral data. My question relates to using combined error =
model for log-transform data. В  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. В Both formu=
las had similar model fits in terms of OFV (OFV using Formula 2 was one uni=
t less than OFV using Formula1) with slightly changed PK parameter estimate=
s. My issue with these formulas is that the model simulates very extreme co=
ncentrations (e.g. upon generating VPCs) at the early time points (when dru=
g 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. В  My=
questions: 1.В В В В В В Is there a way to solve t=
his problem for the indicated formulas? 2.В В В В В В=
Are the two formulas below equally valid? 3.В В В В В=
В Is there an alternative formula that I can use which does not have=
this numerical problem? 4.В В В В В В Any referenc=
e paper that discusses this subject? В  Here are the two formulas: 1.=
В В В В В В Formula 1: suggested by Mats Karlsson w=
ith fixing SIGMA to 1: W=SQRT(THETA(16)**2+THETA(17)**2/EXP(IPRE)**2) =
В  2.В В В В В В Formula 2: suggested by Leonid=
Gibiansky with fixing SIGMA to 1: W = SQRT(THETA(16)+ (THETA(17)/EXP(IPR=
E))**2 В ) В  The way I apply it in my model is this: В  FLAG=
=0В В В В В В В В В В В В=
В В В В В В В В В В В В =
В В В В  ;TO AVOID ANY CALCULATIONS OF LOG (0) IF (F.EQ.0) =
FLAG=1В В В В В  IPRE=LOG(F+FLAG) В  W=SQRT(=
THETA(16)**2+THETA(17)**2/EXP(IPRE)**2) ;FORMULA 1 В  IRES=DV-IPRE I=
WRES=IRES/W Y=(1-FLAG)*IPRE + W*EPS(1) В  \$SIGMA 1. FIX В  Be=
st regards, В  Ahmad Abuhelwa School of Pharmacy and Medical Sciences =
University of South Australia- City East Campus Adelaide, South Australia A=
ustralia В

Received on Thu Jun 02 2016 - 03:07:05 EDT

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