Re: [NMusers] Error model

Nidal,

Another method that would give slightly different results is by using
the EPS terms to code a model like (11) in Beals' paper

Using a first order taylors series expansion to the log equaivalant of
Beal's (11) we have:

Y= F*exp(eps(1)) + m*exp(eps(2))

log Y = log(f*exp(eps(1))+m*exp(eps(2))

Using a Taylor Series expansion this becomes

log Y = log(f+m) + f/(f+m)*eps(1) + m/(f+m)*eps(2)

Using an eps allows unique points to get different values (thetas would
allow a population estimate of the variability).

Since the expectation of log Y is not F, the PRED is miscalculated by
NONMEM and should be calculated separately (as well as RES). Therefore
a possibility for the error model discussed by Beal is:
$ABBREVIATED COMRES=1 COMSAV=1
...
$ERR
M=THETA(x)
IPRED=F+M ; Individual prediction (regular scale)
IF(COMACT.EQ.1)COM(1)=IPRED
PPRED=COM(1) ; Population prediction (regular scale)
PRED=DV-PPRED
IPRED=DV-IPRED
Y=LOG(IPRED)+F/IPRED*EPS(1)+M/IPRED*EPS(2)

IWRES can be calculated outsider or by using a ratio for EPS(1) and
EPS(2), setting it to be a theta and then multiplying by a unfixed THETA.
This requires an assumptions that:

    * EPS(1) is positive, EPS(2) is positive as well (this is also true
      the additive+proprotional case but is used often to calculated IWRES.)
    * One of the error terms is better described at a population level


For example

$ABBREVIATED COMRES=1 COMSAV=1
...
$ERR
M=THETA(x)
S=THETA(x+1)
IPRED=F+M ; Individual prediction (regular scale)
IF(COMACT.EQ.1)COM(1)=IPRED
PPRED=COM(1) ; Population prediction (regular scale)
PRED=DV-PPRED
IPRED=DV-IPRED
Y=LOG(IPRED)+F/IPRED*S+M/IPRED*S*EPS(2)

IWRES=IPRED-log(F+M)/(sqrt(F^2/((F+M)^2)+M^2S^2/((F+M)^2)))
S=sqrt(var(EPS(2))/var(EPS(1)))

Nidal gives another model that assumes that both the error terms are
explained at a population level and should be related to 11.

Matt Fidler (not that Mat refered to in Nidals' email I think its Mats K.)
nidal.alhuniti_at_gmail.com wrote:

> Hi James,
>
> We should divide by F (that is what happens when write an email at 3
> AM). Anyway this error model was proposed by Mat and discussed in Beal
> 's paper ( /Ways to Fit a PK Model with Some Data Below the
> Qunatification Limit/ J. Pharmacokin.Pharamcodyn. 28, p. 481-504.)
>
> So the code should be as in Eq. 11 (above paper)
> Y=LOG(F)+SQRT(THETA(n-1)**2+THETA(n)**2/F**2)*EPS(1)
> with
> $SIGMA 1 FIX
> Best,
> Nidal
>
>
> On 10/8/07, *James G Wright* <james_at_wright-dose.com
> <mailto:james_at_wright-dose.com>> wrote:
>
> Hi Nidal,
>
> As you have modified the code to prevent F going below 1, it looks
> like you did intend to divide by IPRED (set equal to LOG(F)) in
> the weighting expression..?
>
> If so, this gives an F/LOG(F) weighting term, which is an
> innovative error model, never previously mentioned before on the
> NMuser list to my knowledge.
>
> Best regards, James
>
> James G Wright PhD
> Scientist
> Wright Dose Ltd
> Tel: 44 (0) 772 5636914
> www.wright-dose.com <http://www.wright-dose.com/>
>
> -----Original Message-----
> *From:* owner-nmusers_at_globomaxnm.com
> <mailto:owner-nmusers_at_globomaxnm.com>
> [mailto:owner-nmusers_at_globomaxnm.com
> <mailto:owner-nmusers_at_globomaxnm.com>] *On Behalf Of *
> nidal.alhuniti_at_gmail.com <mailto:nidal.alhuniti_at_gmail.com>
> *Sent:* 05 October 2007 17:32
> *To:* James G Wright
> *Cc:* nmusers_at_globomaxnm.com <mailto:nmusers_at_globomaxnm.com>
> *Subject:* Re: [NMusers] Error model
>
> Hi James,
>
> This error model was discussed in the following NM threads.
>
> http://www.cognigencorp.com/nonmem/nm/99apr232002.html
> <http://www.cognigencorp.com/nonmem/nm/99apr232002.html>
>
>
>
> http://www.cognigencorp.com/nonmem/nm/98jun022003.html
> <http://www.cognigencorp.com/nonmem/nm/98jun022003.html>
>
>
> To prevent division by zero
>
>
>
> $ERROR
>
> TY=F
>
> IF(F.GT.1) THEN
>
> TY=LOG(F)
>
> ELSE
>
> TY=0.025
>
> ENDIF
>
> IPRED=TY
>
> W=SQRT(THETA(n-1)**2+THETA(n)**2/IPRED**2) ; log
> transformed data
>
> Y=TY+W*EPS(1)
>
>
>
> $SIGMA 1 FIX
>
>
>
> Best,
>
> Nidal
>
>
>
>
>
> On 10/5/07, *James G Wright* <james_at_wright-dose.com
> <mailto:james_at_wright-dose.com>> wrote:
>
> Hi Leonid,
>
> In the original email, IPRED = LOG(F) and division by
> LOG(F) leads to a
> division by zero when F=1, hence there was definitely a typo
> somewhere...
>
> Of course, this isn't the case in your revised version,
> however you have
> introduced a dependence on F (as a reciprocal for the
> additive term)
> which reintroduces all of the ELS problems (where your
> variances can
> bias your means) that we were trying to avoid by going to
> the log-scale
> in the first place. Because F is now entering as a
> reciprocal which
> leads to very big numbers when F is small, I expect this
> method would
> perform worse than working on the original scale.
>
> Best regards, James
>
>
>
>
>
>
> James G Wright PhD
> Scientist
> Wright Dose Ltd
> Tel: 44 (0) 772 5636914
> www.wright-dose.com <http://www.wright-dose.com/>
>
>
> -----Original Message-----
> From: Leonid Gibiansky [mailto:LGibiansky_at_quantpharm.com
> <mailto:LGibiansky_at_quantpharm.com>]
> Sent: 05 October 2007 13:33
> To: James G Wright
> Cc: nmusers_at_globomaxnm.com <mailto:nmusers_at_globomaxnm.com>
> Subject: Re: [NMusers] Error model
>
>
> James,
> The division in the expression for the error is not a typo.
> The line of thoughts is:
>
> Y=F*EXP(sqrt(theta^2+(theta/F)^2)eps) ;
> F*(1+sqrt(theta^2+(theta/F)^2)eps) ; linearization
> F+F* eps1 + F*eps2/F= ; rewiring as 2
> epsilons
> F(1+eps1)+ eps2 ; combined error model
>
> Leonid
>
>
> --------------------------------------
> Leonid Gibiansky, President
> QuantPharm LLC: www.quantpharm.com
> <http://www.quantpharm.com/>
> e-mail: LGibiansky at quantpharm.com <http://quantpharm.com/>
> tel: (301) 767 5566
>
>
> James G Wright wrote:
>> If Y is the original observed data, then the
> log-transformed error
>> model is
>>
>> LOG (Y) = LOG (F) + EPS(1)
>>
>> We can exponentiate both sides to get an approximately
> proportional
>> error model:-
>>
>> Y = F * EXP( EPS(1) ).
>>
>> The advantage of the above approach is that the mean and
> variance
>> terms
>> are independent (if the data are log-transformed in the
> data file).
>> This avoids instabilities caused by NONMEM biasing the
> mean prediction
>
>> to get "better" variance terms - a known problem for
> ELS-type methods
>> since 1980. Unfortunately, we can't apply the same trick
> to the ETAs
>> because they are not directly observed.
>>
>> However, the model proposed as "additive and
> proportional" by Nidal is
>>
>> LOG (Y) = LOG (F) + W*EPS(1)
>>
>> Exponentiating to get
>>
>> Y = F*EXP( W*EPS(1) )
>>
>> where W= SQRT (THETA(n-1)**2 + THETA(n)**2 *
> LOG(F)**2). I'm assuming
>> the division sign in the original email was a typo, as
>> THETA(n)**2/LOG(F)**2 goes to infinity when F approaches
> 1. Rewriting
>
>> with separate estimated epsilons instead of estimated
> thetas for
> clarity
>> gives:-
>>
>> Y = F * EXP( EPS(1) + LOG(F)*EPS(2) )
>> = F * EXP( EPS(1) ) * EXP( LOG(F)*EPS(2) )
>>
>> which is vaguely like having an error term proportional
> to LOG(F)
>> working multiplicatively with a standard proportional
> error model.
>> After linearization, you obtain something like
>>
>> Y = F + F * EPS(1) + F * LOG(F) * EPS(2)
>>
>> which gives a F * LOG(F) weighting term, as opposed to
> the constant
>> weighting term required for an additive model.
>>
>> Incidentally, IF ( F.EQ.0) "TY" should equal a very large
> negative
>> number
>> (well, minus infinity). Either you replace zeroes in a
> log-proportional
>> model with a small number or you discard them, setting
> LOG (F) = 0 is
>> like setting F=1 if (F.EQ.0).
>>
>> Best regards,
>>
>>
>> James G Wright PhD
>> Scientist
>> Wright Dose Ltd
>> Tel: 44 (0) 772 5636914
>> www.wright-dose.com <http://www.wright-dose.com/>
> <http://www.wright-dose.com/>
>>
>> -----Original Message-----
>> *From:* owner-nmusers_at_globomaxnm.com
> <mailto:owner-nmusers_at_globomaxnm.com>
>> [mailto: owner-nmusers_at_globomaxnm.com
> <mailto:owner-nmusers_at_globomaxnm.com>] *On Behalf Of
>> *nidal.alhuniti_at_gmail.com
> <mailto:nidal.alhuniti_at_gmail.com>
>> *Sent:* 05 October 2007 08:13
>> *To:* navin goyal
>> *Cc:* nmusers
>> *Subject:* Re: [NMusers] Error model
>>
>> Hi Navin,
>>
>> You could try both additive and proportional error model
>> $ERROR
>>
>> TY=F
>>
>> IF(F.GT.0) THEN
>>
>> TY=LOG(F)
>>
>> ELSE
>>
>> TY=0
>>
>> ENDIF
>>
>> IPRED=TY
>>
>> W=SQRT(THETA(n-1)**2+THETA(n)**2/IPRED**2) ; log
> transformed
> data
>> Y=TY+W*EPS(1)
>>
>>
>>
>> $SIGMA 1 FIX
>>
>> Best,
>>
>> Nidal
>>
>>
>>
>> Nidal Al-Huniti, PhD
>>
>> Strategic Consulting Services
>>
>> Pharsight Corporation
>>
>> nal-huniti_at_pharsight.com
> <mailto:nal-huniti_at_pharsight.com><mailto:nal-huniti_at_pharsight.com
> <mailto:nal-huniti_at_pharsight.com>>
>>
>>
>>
>>
>>
>> On 10/4/07, *navin goyal* <navin1180_at_gmail.com
> <mailto:navin1180_at_gmail.com>
>> <mailto: navin1180_at_gmail.com
> <mailto:navin1180_at_gmail.com>>> wrote:
>>
>> Dear Nonmem users,
>>
>> I am analysing a POPPK data with sparse sampling
>> The dosing is an IV infusion over one hour and we
> have data
> for
>> time points 0 (predose), 1 (end of infusion) and
> 2 (one hour
>> post infusion)
>> The drug has a half life of approx 4 hours. The
> dose is given
>> once every fourth day
>> When I ran my control stream and looked at the
> output table, I
>> got some IPREDs at time predose time points where
> the DV was 0
>> the event ID EVID for these time points was 4
> (reset)
>> (almost 20 half lives)
>> I was wondering why did NONMEM predict
> concentrations at these
>> time points ?? there were a couple of time points
> like this.
>>
>> I started with untransformed data and fitted my
> model.
>> but after bootstrapping the errors on etas and
> sigma were
>> very high.
>> I log transformed the data , which improved the
> etas but the
>> sigma shot upto more than 100%
>> ( is it because the data is very sparse ??? or I
> need to use a
>> better error model ???)
>> Are there any other error models that could be
> used with the
> log
>> transformed data, apart from the
>> Y=Log(f)+EPS(1)
>>
>>
>> Any suggestions would be appreciated
>> thanks
>>
>> --
>> --Navin
>>
>>
>
>
>
Received on Wed Oct 10 2007 - 09:45:47 EDT