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:
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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
Yxp(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
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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" <ahmad.abuhelwa_at_mymail.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-transform data
#yiv5739009609 #yiv5739009609 -- filtered {panose-1:2 4 5 3 5 4 6 3 2 4;}#yiv5739009609 filtered {font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;}#yiv5739009609 p.yiv5739009609MsoNormal, #yiv5739009609 li.yiv5739009609MsoNormal, #yiv5739009609 div.yiv5739009609MsoNormal {margin:0cm;margin-bottom:.0001pt;font-size:11.0pt;}#yiv5739009609 a:link, #yiv5739009609 span.yiv5739009609MsoHyperlink {color:#0563C1;text-decoration:underline;}#yiv5739009609 a:visited, #yiv5739009609 span.yiv5739009609MsoHyperlinkFollowed {color:#954F72;text-decoration:underline;}#yiv5739009609 p.yiv5739009609MsoListParagraph, #yiv5739009609 li.yiv5739009609MsoListParagraph, #yiv5739009609 div.yiv5739009609MsoListParagraph {margin-top:0cm;margin-right:0cm;margin-bottom:0cm;margin-left:36.0pt;margin-bottom:.0001pt;font-size:11.0pt;}#yiv5739009609 span.yiv5739009609EmailStyle17 {color:windowtext;}#yiv5739009609 .yiv5739009609MsoChpDefault {font-size:10.0pt;}#yiv5739009609 filtered {margin:72.0pt 72.0pt 72.0pt 72.0pt;}#yiv5739009609 div.yiv5739009609WordSection1 {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 filtered {}#yiv5739009609 ol {margin-bottom:0cm;}#yiv5739009609 ul {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