Re: Time-to-event analysis with $DES
Andy,
Thanks for this response. In fact I did not propose that particular Weibull parameterization but simply pointed out that it seemed to be what the original questioner seemed to be trying to use.
My own preference for parameterizing the Weibull is
HAZARD=LAMDA0*EXP(BETA*LOG(TIME))
This parameterization makes is easier to understand that the Weibull (ln(time)) is just a transformation of the Gompertz (time):
HAZARD=LAMDA0*EXP(BETA*TIME)
Its then easy to extend hazards to more interesting and possibly more useful hazards such as the Gombull (aka the Weipertz):
HAZARD=LAMDA0*EXP(BETAG*TIME + BETAW*LOG(TIME))
One of the features of the Weibull that I dont like is that the hazard is predicted to be zero when time is zero. This seems a particularly unlikely biological occurrence and also requires extra coding to deal with this special case when time is zero to avoid NaNs.
I'd be interested in knowing if you find any difference in the standard errors with the ln(time) parameterization. But whatever you find it won't change my behaviour because as you know standard errors are not much use for making predictions :-)
Best wishes,
Nick
Quoted reply history
On 28/03/2011 3:05 p.m., Andrew Hooker wrote:
> Hi All
>
> I would just like to point out that in the Weibull hazard that Nick proposes:
>
> HAZNOW=LAMD*ALPH*(TIME+DEL2)**(ALPH-1)
>
> The units of LAMD would have to be TIME**(-ALPH). That is, as soon as ALPH changes then the units of LAMD change. This seems strange and unnecessary to include this type of correlation in the parameters. If we instead reparameterize the model to:
>
> HAZNOW=LAMD*ALPH*(LAMD*(TIME+DEL2))**(ALPH-1)
>
> Then LAMD has units of 1/TIME no matter the value of ALPH. I have tested the two parameterizations in one simple simulation example and found that the relative standard errors were significantly lower with this second parameterization.
>
> Best regards
>
> Andy
>
> Andrew Hooker, Ph.D.
>
> Associate Professor of Pharmacometrics
>
> Dept. of Pharmaceutical Biosciences
>
> Uppsala University
>
> Box 591, 751 24, Uppsala, Sweden
>
> Phone: +46 18 471 4355
>
> www.farmbio.uu.se/research/Research+groups/Pharmacometrics/ < http://www.farmbio.uu.se/research/Research+groups/Pharmacometrics/ >
>
> *From:* [email protected] [ mailto: [email protected] ] *On Behalf Of *Nick Holford
>
> *Sent:* den 24 mars 2011 07:54
> *To:* nmusers
> *Subject:* Re: [NMusers] Time-to-event analysis with $DES
>
> Hyewon,
>
> Sorry -- I just realized that EFF is not the drug effect but 1-drug effect so that it has a value of 1 when AUC is zero. Please ignore my remarks about EFF being zero when AUC is zero in my comments below and consider this code to describe the effect of AUC on hazard:
>
> HAZNOW=LAMD*ALPH*(TIME+DEL2)**(ALPH-1)*(1-BETAE*AUC/(EC+AUC))
>
> I think it is a very strong assumption that the drug will reduce the hazard to zero at infinite AUC which you can test by estimating a parameter (such as BETAE). If BETAE is significantly less than 1 then this means the assumption is not supported.
>
> Nick
>
> On 24/03/2011 8:17 a.m., Nick Holford wrote:
>
> Hyewon,
>
> The most obvious problem with your code is in $DES. You must use the variable T not the variable TIME when referring to a time varying hazard. TIME is the time on each data record. It does not change in $DES. T is the time from the last data record upto the current data record and changes within $DES.
>
> You are predicting the likelihood of an event at the exact time of the event observation record with multiple events per subject. As you seem to realize you need to compute the cumulative hazard (CUMHAZ) either from TIME=0 or from the TIME of the last non-censored event for each subject.
>
> In my opinion the following code is clearer and less dependent on your data structure than the method you are using which works only if your data has just event records after the TIME=0 record.
>
> IF (MDV.EQ.0.AND.CS.EQ.0) THEN
> OLDCHZ=A(1) ; cum haz upto time of this event
> ELSE
> OLDCHZ=OLDCHZ ; need to do this if OLDCHZ is a random variable
> ENDIF
>
> Your hazard model looks rather complicated. It seems to be based on the product of a Weibull baseline hazard
>
> LAMD*ALPH*(TIME+DEL2)**(ALPH-1)
> then something odd involving the Weibull parameters
> *EXP(-LAMD*(TIME+DEL2)**ALPH)
> and then forces the hazard to be zero if AUC is zero.
> *EFF
>
> Is this really what you want? Do you know the hazard of event is zero if AUC is zero? Or is the last right parenthesis in the wrong place?
>
> I would suggest something like this where LAMD and ALPH are the two parameters of the Weibull baseline hazard (when AUC is zero) and BETAE is a parameter describing the effect of the drug on the overall hazard.
>
> HAZNOW=LAMD*ALPH*(TIME+DEL2)**(ALPH-1)*EXP(BETAE*EFF)
>
> You may also find it easier to develop the model if you do not try to estimate a random effect on LAMD until you have got reasonable estimates for the other parameters.
>
> You may find it helpful to look at this presentation showing how to code time to event models in NM-TRAN:
>
> http://pkpdrx.com/holford/docs/time-to-event-analysis.pdf
>
> Best wishes,
>
> Nick
>
> On 24/03/2011 3:25 a.m., Hyewon Kim wrote:
>
> Dear NMuser
>
> I am trying to analyze time to repeated event data using NONMEM.
> The response were obtained till 24 hours after drug administration.
> Inhibitory Emax model was implemented.
>
> I am getting unreasonable parameter estimates which is far beyond what data say. If some body can point out what i am doing wrong, it would be very helpful.
>
> Thank you in advance.
>
> Hyewon
>
> Data set (# of observations =62, # of patients=50 )
> C ID TIME CS MDV AUC
> . 101 0 . 1 1.111
> . 101 0.05 0 0 1.111
> . 101 2 0 0 1.111
> . 102 0 . 1 0
> . 102 24 1 0 0
> . 103 0 . 1 0.999
> . 103 0.75 . 0 0.999
> ....
>
> Model File
> $PROB RUN# 101
> $INPUT C ID TIME CS MDV AUC
> ;CS:0=having event,1=censored
> $DATA .data.csv IGNORE=C
> $SUBROUTINE ADVAN=6 TOL=6
> $MODEL
> COMP=(HAZARD)
> $PK
> LAMD=THETA(1)*EXP(ETA(1)) ;scale factor
> ALPH=THETA(2) ;shape factor
>
> EC=THETA(3) ;AUC when effect is half of its max
>
> EFF=1-AUC/(EC+AUC) ;drug effect
>
> $DES
> DEL=1E-6
> DADT(1)=LAMD*ALPH*(TIME+DEL)**(ALPH-1)*EXP(-LAMD*(TIME+DEL)**ALPH)*EFF
>
> $ERROR
> DEL2=1E-6
> IF(NEWIND.NE.2) OLDCHZ=0
> CHZ=A(1)-OLDCHZ
> OLDCHZ=A(1)
> SUR=EXP(-CHZ)
> HAZNOW=LAMD*ALPH*(TIME+DEL2)**(ALPH-1)*EXP(-LAMD*(TIME+DEL2)**ALPH)*EFF
>
> IF(CS.EQ.1) Y=SUR ;survival prob of censored
> IF(CS.EQ.0) Y=SUR*HAZNOW ;pdf of event
>
> $THETA
> (0,10) ;[scale factor]
> (0,1.0) ;[shaph factor]
> (0,0.01) ;[AUC at half of Emax]
>
> $OMEGA
> 0.01 ;[p] omega(1,1)
>
> $EST METHOD=COND LIKE LAPLACIAN PRINT=5 SIG=3 MAX=9999 MSFO=101.msf
>
> $COV PRINT=E
>
> $TABLE ID TIME SUR AUC ONEHEADER NOPRINT FILE=101.tab
>
> --
> Nick Holford, Professor Clinical Pharmacology
> Dept Pharmacology& Clinical Pharmacology
> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
> tel:+64(9)923-6730 <tel:+64%289%29923-6730> fax:+64(9)373-7090
> mobile:+64(21)46 23 53
> email:[email protected] <mailto:[email protected]>
> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>
> --
> Nick Holford, Professor Clinical Pharmacology
> Dept Pharmacology& Clinical Pharmacology
> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
> tel:+64(9)923-6730 <tel:+64%289%29923-6730> fax:+64(9)373-7090
> mobile:+64(21)46 23 53
> email:[email protected] <mailto:[email protected]>
> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology& Clinical Pharmacology
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
email: [email protected]
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford