[NMusers] Splitting the residual error

[Jonathan Moss]

Dear all,

 

I would like to share with you and get people's opinions on a recent issue I
had.

I have a data set of 46 patients, orally dosed, with very dense sampling
during absorption (0.25h, 0.5h, 0.75h, 1h, 1.5h, 2h, 3h, 4h, 6h, 8h, 12h,
24h, 36h), Cmax at around 4 hours.

During modelling, I found that the residual error was not evenly
distributed. Plotting CWRES against time after dose, the result looked like
an "hourglass" shape. I.e. A wide spread during absorption, narrower near
Cmax time, then wider at later time points.

My thinking was as follows: Residual error contains both the assay / model
spec. error, and the error in recorded observation time. When the gradient
of the PK curve is large, any error in recorded observation time equals a
large error in the recorded concentration, whereas if the gradient is small
then the recorded concentration error will be small. 

I "split" the residual error into its assay/model spec and time-error parts
in the $ERROR block:

 

$ERROR

GRAD = KA*A(2) - K20*A(3)

 

IF (GRAD.LT.0) GRAD = -1*GRAD

 

C_1  = A(3)/V ; Concentration in the
central compartment

IPRED= C_1

SD   = SQRT(EPROP*C_1**2) ; Standard deviation of
predicted concentration

 

Y=IPRED+SD*(1+val*GRAD)*EPS(1)

 

Note: Sigma is fixed to one and EPROP is estimated as a theta. Here, GRAD is
the right hand side of the differential equation for A(3), in order to
recover the gradient. Val is estimated by NONMEM.

 

This approach vastly improved the model fit (OFV drop of around 350!). All
GOF plots, VPCs, NPCs, NPDEs, individual fits looked good. This got me
thinking, and I tried this approach on some of my other popPK models. I
found for the simpler models, the result was nearly always a significant
improvement in the model fit. For the more complicated models, NONMEM had
trouble finishing the runs.

 

I struggled to find any approach like this in the literature, which leads me
to believe that there is something wrong, as it is a relatively simple
concept. Please, what are peoples thoughts on this?

 

Thanks,

Jon

 

Jon Moss, PhD

Modeller 

BAST Inc Limited

Loughborough Innovation Centre

Charnwood Wing

Holywell Park

Ashby Road

Loughborough, LE11 3AQ, UK

Tel: +44 (0)1509 222908

 

RE: [NMusers] Splitting the residual error

[Mats Karlsson]

Dear Jon,

As you point out the concept of residual error magnitude being dependent on 
anything else than the prediction itself is a straightforward. Yet it is, I 
think underused and that is why you may not see it much in the literature. In 
addition to what you mention, a large component is that model misspecification 
is not a homogeneous process. It is likely that most of our models are more 
specified for absorption than disposition. Absorption contains many processes 
that are discrete and difficult to easily capture in simple models.
For most compunds, the absolute gradient is much higher during the absorption 
phase than the distribution phase and that is probably a contributing factor to 
what experience. You would probably get as good an improvement if you had a 
separate error magnitude during the absorption phase.

The model you mentioned with were outlined in the 1998 article below. I also 
add some other articles for the case you're further interested in residual 
error modeling.

Best regards,
Mats

1.

A strategy for residual error modeling incorporating scedasticity of variance 
and distribution shape.


Dosne AG, Bergstrand M, Karlsson MO.


J Pharmacokinet Pharmacodyn. 2016 Apr;43(2):137-51. doi: 
10.1007/s10928-015-9460-y. Epub 2015 Dec 17.


PMID: 26679003 [PubMed - in process] Free PMC Article


Similar 
articles


2.

The impact of misspecification of residual error or correlation structure on 
the type I error rate for covariate 
inclusion.


Silber HE, Kjellsson MC, Karlsson MO.


J Pharmacokinet Pharmacodyn. 2009 Feb;36(1):81-99. doi: 
10.1007/s10928-009-9112-1. Epub 2009 Feb 14.


PMID: 19219538 [PubMed - indexed for MEDLINE]


Similar 
articles


3.

Three new residual error models for population PK/PD 
analyses.


Karlsson MO, Beal SL, Sheiner LB.


J Pharmacokinet Biopharm. 1995 Dec;23(6):651-72.


PMID: 8733951 [PubMed - indexed for MEDLINE]


Similar 
articles


4.


Assumption testing in population pharmacokinetic models: illustrated with an 
analysis of moxonidine data from congestive heart failure 
patients.



Karlsson MO, Jonsson EN, Wiltse CG, Wade JR.



J Pharmacokinet Biopharm. 1998 Apr;26(2):207-46.


PMID:


9795882


Mats Karlsson, PhD
Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences
Faculty of Pharmacy
Uppsala University
Box 591
75124 Uppsala

Phone: +46 18 4714105
Fax + 46 18 4714003
www.farmbio.uu.se/research/researchgroups/pharmacometrics/

From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On 
Behalf Of Jonathan Moss
Sent: Friday, May 13, 2016 11:37 AM
To: nmusers@globomaxnm.com
Subject: [NMusers] Splitting the residual error

Dear all,

I would like to share with you and get people's opinions on a recent issue I 
had.
I have a data set of 46 patients, orally dosed, with very dense sampling during 
absorption (0.25h, 0.5h, 0.75h, 1h, 1.5h, 2h, 3h, 4h, 6h, 8h, 12h, 24h, 36h), 
Cmax at around 4 hours.
During modelling, I found that the residual error was not evenly distributed. 
Plotting CWRES against time after dose, the result looked like an "hourglass" 
shape. I.e. A wide spread during absorption, narrower near Cmax time, then 
wider at later time points.
My thinking was as follows: Residual error contains both the assay / model 
spec. error, and the error in recorded observation time. When the gradient of 
the PK curve is large, any error in recorded observation time equals a large 
error in the recorded concentration, whereas if the gradient is small then the 
recorded concentration error will be small.
I "split" the residual error into its assay/model spec and time-error parts in 
the $ERROR block:

$ERROR
GRAD = KA*A(2) - K20*A(3)

IF (GRAD.LT.0) GRAD = -1*GRAD

C_1  = A(3)/V ; Concentration in the 
central compartment
IPRED= C_1
SD   = SQRT(EPROP*C_1**2) ; Standard deviation of 
predicted concentration

Y=IPRED+SD*(1+val*GRAD)*EPS(1)

Note: Sigma is fixed to one and EPROP is estimated as a theta. Here, GRAD is 
the right hand side of the differential equation for A(3), in order to recover 
the gradient. Val is estimated by NONMEM.

This approach vastly improved the model fit (OFV drop of around 350!). All GOF 
plots, VPCs, NPCs, NPDEs, individual fits looked good. This got me thinking, 
and I tried this approach on some of my other popPK models. I found for the 
simpler models, the result was nearly always a significant improvement in the 
model fit. For the more complicate

Re: [NMusers] Splitting the residual error

[Ekaterina Gibiansky]

Dear Jon,

We routinely use separate residual errors during absorption and later 
(say, a larger error in the first X hours), although I do not remember 
whether we published any of those models.


Regards,
Katya

Ekaterina Gibiansky, Ph.D.
CEO&CSO, QuantPharm LLC
Web:www.quantpharm.com
Email:egibian...@quantpharm.com


On 5/13/2016 6:37 AM, Mats Karlsson wrote:

Dear Jon,

As you point out the concept of residual error magnitude being 
dependent on anything else than the prediction itself is a 
straightforward. Yet it is, I think underused and that is why you may 
not see it much in the literature. In addition to what you mention, a 
large component is that model misspecification is not a homogeneous 
process. It is likely that most of our models are more specified for 
absorption than disposition. Absorption contains many processes that 
are discrete and difficult to easily capture in simple models.


For most compunds, the absolute gradient is much higher during the 
absorption phase than the distribution phase and that is probably a 
contributing factor to what experience. You would probably get as good 
an improvement if you had a separate error magnitude during the 
absorption phase.


The model you mentioned with were outlined in the 1998 article below. 
I also add some other articles for the case you’re further interested 
in residual error modeling.


Best regards,

Mats

1.



A strategy for residual error modeling incorporating scedasticity of 
variance and distribution shape. 






Dosne AG, Bergstrand M, Karlsson MO.




J Pharmacokinet Pharmacodyn. 2016 Apr;43(2):137-51. doi: 
10.1007/s10928-015-9460-y. Epub 2015 Dec 17.





PMID: 26679003 [PubMed - in process]*Free PMC Article *




Similar articles 
 



2.



The impact of misspecification of residual error or correlation 
structure on the type I error rate for covariate inclusion. 






Silber HE, Kjellsson MC, Karlsson MO.




J Pharmacokinet Pharmacodyn. 2009 Feb;36(1):81-99. doi: 
10.1007/s10928-009-9112-1. Epub 2009 Feb 14.





PMID: 19219538 [PubMed - indexed for MEDLINE]




Similar articles 
 



3.



Three new residual error models for population PK/PD analyses. 






Karlsson MO, Beal SL, Sheiner LB.




J Pharmacokinet Biopharm. 1995 Dec;23(6):651-72.




PMID: 8733951 [PubMed - indexed for MEDLINE]




Similar articles 
 



4.



Assumption testing in population pharmacokinetic models: illustrated 
with an analysis of moxonidine data from congestive heart failure 
patients. 





*Karlsson MO*, Jonsson EN, Wiltse CG, Wade JR.




J Pharmacokinet Biopharm. 1998 Apr;26(2):207-46.




PMID:




9795882

Mats Karlsson, PhD

Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences

Faculty of Pharmacy

Uppsala University

Box 591

75124 Uppsala

Phone: +46 18 4714105

Fax + 46 18 4714003

www.farmbio.uu.se/research/researchgroups/pharmacometrics/ 



*From:*owner-nmus...@globomaxnm.com 
[mailto:owner-nmus...@globomaxnm.com] *On Behalf Of *Jonathan Moss

*Sent:* Friday, May 13, 2016 11:37 AM
*To:* nmusers@globomaxnm.com
*Subject:* [NMusers] Splitting the residual error

Dear all,

I would like to share with you and get people’s opinions on a recent 
issue I had.


I have a data set of 46 patients, orally dosed, with very dense 
sampling during absorption (0.25h, 0.5h, 0.75h, 1h, 1.5h, 2h, 3h, 4h, 
6h, 8h, 12h, 24h, 36h), Cmax at around 4 hours.


During modelling, I found that the residual error was not evenly 
distributed. Plotting CWRES against time after dose, the result looked 
like an “hourglass” shape. I.e. A wide spread during absorption, 
narrower near Cmax time, then wider at later time points.


My thinking was as follows: Residual error contains both the assay / 
model spec. error, and the error in recorded observation time. When 
the gradient of the PK curve is large, any error in recorded 
observation time equals a large error in the recorded concentration, 
whereas if the gradient is small then the recorded concentration error 
will be small.


I “split” the residual error into its assay/model spec and time-error 
parts in the $ERROR block:


$ERROR

GRAD = KA*A(2) - K20*A(3)

IF (GRAD.LT.0) GRAD = -1*GRAD

C_1  = A(3)/V ; Concentration in 
the central compartment


IPRED= C_1

SD   = SQRT(EPROP*C_1**2) ; S

Re: [NMusers] Time varying volume of distribution implementation

[Alison Boeckmann]

The code cannot be used when any WT values are missing.
In order to interpolate appropriately in $DES, WT should be present 
on every event record.
$PK sees only the current and previous record.
If WT is missing from the current record, $PK has no way of knowing
what WT will be on the next record.

The code in NONMEM 7.3 help file INFN_INTERPOLATION EXAMPLE 2
(infn2.exa)
can be used to fill in missing WT values in the event records.
The idea is that the $INFN block does two passes through the data set.
After the first pass, it knows what the WT values are on all the event
records, so that it can interpolate to fill in a missing WT.

The finedata utility  (fine1.exa) can also be used to fill in 
the missing values, and performs interpolation if necessary.

BTW, the code for $PK and $DES will be a new help item in 
Nonmem 7.4 INTERPOLATION IN $DES (desinterp.exa)

On Wed, May 11, 2016, at 04:18 AM, SIMON Nicolas wrote:
> Dear Alison,
> 
> Perhaps I miss something but is it possible to use your code for
> interpolating WT when some WT are missing in the dataset for an
> individual?
> 
> Best regards
> Nicolas
> 
> 
> -Message d'origine-
> De : owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com]
> De la part de Alison Boeckmann
> Envoyé : mardi 10 mai 2016 20:06
> À : Nick Holford; nmusers; thorsten.l...@mx.uni-saarland.de
> Objet : Re: [NMusers] Time varying volume of distribution implementation
> 
> Dear Nick, Thorsten and others,
> 
> Nick's example is very good.
> He models WT so that it increases linearly with TIME:
> DWT_T=WT_ZERO + WT_ALPHA*T
> where WT_ZERO and WT_ALPHA are thetas.
> 
> It occurs to me that some users may not have a model for WT vs. T, but
> have only observed values of WT at fixed points.
> In this case, WT can be interpolated within the $DES block.
> 
> Here is a small example of code that can be used to interpolate WT
> between values that are recorded on the data records.
> All the code in $PK to compute OLDTIME and OLDWT and SLOPE, and the code
> for D_WT in $DES, could be copied to user's control stream.
> Other code (for integrating D_WT in $DES and the analytic solution in
> $ERROR)
> is for testing and would not be part of the user's control stream.  
> 
> Here is the control file:
> 
> $PROB  INTERPOLATE WT IN $DES
> ; this example shows how to interpolate WT in $DES.
> ; it is assumed that WT is recorded on every data record. 
> ; As a test, the value of D_WT in $DES is integrated to obtain AUC of WT
> VS. T ; This is also calculated analytically in $ERROR.
> 
> $INPUT  ID TIME WT DV
> $DATA   desinterp.dat
> 
> $SUBROUTINES  ADVAN6  TOL=5
> $MODEL
> COMP=(AUC_WT DEFOBS)
> 
> $PK
> ; initialize OLDTIME and OLDWT
> IF (NEWIND.LE.1) THEN
>   OLDTIME=TIME
>   OLDWT=WT
> ENDIF
> 
> ; calculate the slope for $DES
> DELTA_TIME=TIME-OLDTIME
> DELTA_WT=WT-OLDWT
> 
> IF (DELTA_TIME>0) THEN
>SLOPE=DELTA_WT/DELTA_TIME
> ELSE
>SLOPE=0.
> ENDIF
> 
> ; save wt and time for next $PK record
> OLDTIME=TIME
> OLDWT=WT
> 
> $DES
>D_WT=OLDWT+SLOPE*(T-OLDTIME) ; D_WT is the value of WT at time T
>DADT(1)=D_WT ; compute AUC of D_WT as a test
> 
> $ERROR
>Y=F+ETA(1)+EPS(1)
> 
> ; Compute analytic solution as a test. 
> ; Does not use compartment amounts.
> ; Uses only the values of WT and TIME on event records.
> ; Suppose WT vs T looks like this:
> ;
> ; 
> ; WT
> ;  |   
> ;  |   w3   w4 
> ;  |  w2w5
> ;  |  w1
> ;  |
> ;  > TIME
> ; t1  t2   t3   t4  t5 
> ; 
> ; at t2, the contribution to the sum is 
> ;   the rectangle  w1 x (t2-t1) 
> ;   plus the triangular piece
> ;   (w2-w1)/(t2-t1) / 2
> ;
> ;   w2
> ;  /|
> ; / |
> ;/__|
> ;  w1   |
> ;   |   |
> ;   |   |
> ;   
> ;   t1  t2
> IF (NEWIND.LE.1) THEN
>PREV_WT=WT ; Initialize WT from previous data record
>SUM=0
> ELSE
>SUM=SUM+PREV_WT*DELTA_TIME+DELTA_WT*DELTA_TIME/2
> ENDIF
> PREV_WT=WT ; save WT from previous data record
> 
> $OMEGA 1
> $SIGMA 1
> 
> $TABLE  ID TIME WT PRED=AUC_WT SUM FILE=desinterp.tbl NOPRINT NOAPPEND ;
> The following two values should always be equal:
> ; PRED (which is the AUC of WT obtained by integrating WT) ; SUM (which
> is the analytic solution) computed in $ERROR
> 
> Here is the data file for the first subject. Note that WT sometimes is
> constant and sometimes decreases:
>   1 0.10  0 
>   1 1.20  0 
>   1 2.35  0 
>   1 2.35  0 
>   1 4.45  0 
>   1 5.40  0 
> 
> Here is the table file:
> TABLE NO.  1
>  ID  TIMEWT  AUC_WT  SUM
>   1.E+00  0.E+00  1.E+01  0.E+00  0.E+00
>   1.E+00  1.E+00  2.E+01  1.5000E+01  1.5000E+01
>   1.E+00  2.E+00  3.5000E

Re: [NMusers] Splitting the residual error

[Ekaterina Gibiansky]

Hi Steve and Andre,

No, nothing fancy, I did not even estimate the time of the switch. 
Something like this:


ITAD=1
IF(TAD.LE.4)  ITAD=THETA(10)
Y=F + (F*ITAD*ERR(1)+ERR(2))

ITAD can also include differences in populations, e.g. Phase 1 versus 
Phase 2-3 data, or healthy versus patients, etc.


The number for switch usually is pretty obvious from the data, but yes 
estimating it would be nicer.


Regards,
Katya

On 5/13/2016 4:33 PM, Stephen Duffull wrote:


Hi Katya

Are they discrete models with estimated transition times (e.g. error 
model 1 from 0 to 2 hours, then error model 2 etc) or continuous 
models (two models who’s weighting is time dependent) that overlap?


We do the former as a diagnostic but not the latter at this point. But 
I have pondered.


Cheers

Steve

*From:*owner-nmus...@globomaxnm.com 
[mailto:owner-nmus...@globomaxnm.com] *On Behalf Of *Ekaterina Gibiansky

*Sent:* Saturday, 14 May 2016 2:07 a.m.
*To:* Jonathan Moss ; nmusers@globomaxnm.com
*Subject:* Re: [NMusers] Splitting the residual error

Dear Jon,

We routinely use separate residual errors during absorption and later 
(say, a larger error in the first X hours), although I do not remember 
whether we published any of those models.


Regards,
Katya

Ekaterina Gibiansky, Ph.D.
CEO&CSO, QuantPharm LLC
Web:www.quantpharm.com
Email:egibian...@quantpharm.com 

On 5/13/2016 6:37 AM, Mats Karlsson wrote:

Dear Jon,

As you point out the concept of residual error magnitude being
dependent on anything else than the prediction itself is a
straightforward. Yet it is, I think underused and that is why you
may not see it much in the literature. In addition to what you
mention, a large component is that model misspecification is not a
homogeneous process. It is likely that most of our models are more
specified for absorption than disposition. Absorption contains
many processes that are discrete and difficult to easily capture
in simple models.

For most compunds, the absolute gradient is much higher during the
absorption phase than the distribution phase and that is probably
a contributing factor to what experience. You would probably get
as good an improvement if you had a separate error magnitude
during the absorption phase.

The model you mentioned with were outlined in the 1998 article
below. I also add some other articles for the case you’re further
interested in residual error modeling.

Best regards,

Mats

1.



A strategy for residual error modeling incorporating scedasticity
of variance and distribution shape.





Dosne AG, Bergstrand M, Karlsson MO.




J Pharmacokinet Pharmacodyn. 2016 Apr;43(2):137-51. doi:
10.1007/s10928-015-9460-y. Epub 2015 Dec 17.




PMID: 26679003 [PubMed - in process]*Free PMC Article *




Similar articles




2.



The impact of misspecification of residual error or correlation
structure on the type I error rate for covariate inclusion.





Silber HE, Kjellsson MC, Karlsson MO.




J Pharmacokinet Pharmacodyn. 2009 Feb;36(1):81-99. doi:
10.1007/s10928-009-9112-1. Epub 2009 Feb 14.




PMID: 19219538 [PubMed - indexed for MEDLINE]




Similar articles




3.



Three new residual error models for population PK/PD analyses.





Karlsson MO, Beal SL, Sheiner LB.




J Pharmacokinet Biopharm. 1995 Dec;23(6):651-72.




PMID: 8733951 [PubMed - indexed for MEDLINE]




Similar articles



4.



Assumption testing in population pharmacokinetic models:
illustrated with an analysis of moxonidine data from congestive
heart failure patients. 




*Karlsson MO*, Jonsson EN, Wiltse CG, Wade JR.




J Pharmacokinet Biopharm. 1998 Apr;26(2):207-46.




PMID:




9795882

Mats Karlsson, PhD

Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences

Faculty of Pharmacy

Uppsala University

Box 591

75124 Uppsala

Phone: +46 18 4714105

Fax + 46 18 4714003

www.farmbio.uu.se/research/researchgroups/pharmacometrics/


*From:*owner-nmus...@globomaxnm.com

[mailto:owner-nmus...@globomaxnm.com] *On Behalf Of *Jonathan Moss
*Sent:*

RE: [NMusers] Splitting the residual error

[Mats Karlsson]

Steve,

In papers 3 and 4 below, we outlined and tested a few different models for 
separating error models for absorption and disposition based on time or partial 
derivatives versus time or Ka. Overall message is *that* you do it is more 
important than *how* you do it. I typically choose a simple way of doing it, 
like allowing a separate residual error magnitude up until a little after Tmax.

Best regards,
Mats


Mats Karlsson, PhD
Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences
Faculty of Pharmacy
Uppsala University
Box 591
75124 Uppsala

Phone: +46 18 4714105
Fax + 46 18 4714003
www.farmbio.uu.se/research/researchgroups/pharmacometrics/

From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On 
Behalf Of Ekaterina Gibiansky
Sent: Saturday, May 14, 2016 12:13 AM
To: Stephen Duffull; Andre Jackson; nmusers
Subject: Re: [NMusers] Splitting the residual error

Hi Steve and Andre,

No, nothing fancy, I did not even estimate the time of the switch. Something 
like this:

ITAD=1
IF(TAD.LE.4)  ITAD=THETA(10)
Y=F + (F*ITAD*ERR(1)+ERR(2))

ITAD can also include differences in populations, e.g. Phase 1 versus Phase 2-3 
data, or healthy versus patients, etc.

The number for switch usually is pretty obvious from the data, but yes 
estimating it would be nicer.

Regards,
Katya
On 5/13/2016 4:33 PM, Stephen Duffull wrote:
Hi Katya

Are they discrete models with estimated transition times (e.g. error model 1 
from 0 to 2 hours, then error model 2 etc) or continuous models (two models 
who's weighting is time dependent) that overlap?

We do the former as a diagnostic but not the latter at this point.  But I have 
pondered.

Cheers

Steve

From: owner-nmus...@globomaxnm.com 
[mailto:owner-nmus...@globomaxnm.com] On Behalf Of Ekaterina Gibiansky
Sent: Saturday, 14 May 2016 2:07 a.m.
To: Jonathan Moss ; 
nmusers@globomaxnm.com
Subject: Re: [NMusers] Splitting the residual error

Dear Jon,

We routinely use separate residual errors during absorption and later (say, a 
larger error in the first X hours), although I do not remember whether we 
published any of those models.

Regards,
Katya

Ekaterina Gibiansky, Ph.D.
CEO&CSO, QuantPharm LLC
Web:www.quantpharm.com
Email:egibian...@quantpharm.com


On 5/13/2016 6:37 AM, Mats Karlsson wrote:
Dear Jon,

As you point out the concept of residual error magnitude being dependent on 
anything else than the prediction itself is a straightforward. Yet it is, I 
think underused and that is why you may not see it much in the literature. In 
addition to what you mention, a large component is that model misspecification 
is not a homogeneous process. It is likely that most of our models are more 
specified for absorption than disposition. Absorption contains many processes 
that are discrete and difficult to easily capture in simple models.
For most compunds, the absolute gradient is much higher during the absorption 
phase than the distribution phase and that is probably a contributing factor to 
what experience. You would probably get as good an improvement if you had a 
separate error magnitude during the absorption phase.

The model you mentioned with were outlined in the 1998 article below. I also 
add some other articles for the case you're further interested in residual 
error modeling.

Best regards,
Mats

1.

A strategy for residual error modeling incorporating scedasticity of variance 
and distribution shape.


Dosne AG, Bergstrand M, Karlsson MO.


J Pharmacokinet Pharmacodyn. 2016 Apr;43(2):137-51. doi: 
10.1007/s10928-015-9460-y. Epub 2015 Dec 17.


PMID: 26679003 [PubMed - in process] Free PMC Article


Similar 
articles


2.

The impact of misspecification of residual error or correlation structure on 
the type I error rate for covariate 
inclusion.


Silber HE, Kjellsson MC, Karlsson MO.


J Pharmacokinet Pharmacodyn. 2009 Feb;36(1):81-99. doi: 
10.1007/s10928-009-9112-1. Epub 2009 Feb 14.


PMID: 19219538 [PubMed - indexed for MEDLINE]


Similar 
articles


3.

Three new residual error models for population PK/PD 
analyses.


Karlsson MO, Beal SL, Sheiner LB.


J Pharmacokinet Biopharm. 1995 Dec;23(6):651-72.


PMID: 8733951 [PubMed - indexed for MEDLINE]


Similar 
articles


4.


Assumption testing in population pharmacokinetic models: illustrated with an 
analysis of moxonidine data from congestive heart failure 
patients.



Karls