From: Martin Bergstrand <*martin.bergstrand*>

Date: Thu, 30 Apr 2009 11:11:44 +0200

Dear Kelong Han,

1.

IWRES is calculated in order to be used for diagnostic purposes and will not

affect the estimation. For example plots of IWRES vs. IDV (independent

variable) or |IWRES| vs. IPRED can be useful to diagnose your structural

model and your residual error model.

2.

Y = LOG(F+M) + (F/(F+M))*ERR(1) + (M/(F+M))*ERR(2)

Could be written like this instead:

Y = LOG(F+M) + SQRT((F*THETA(x)/(F+M))**2 + (M*THETA(y)/(F+M))**2)*ERR(1)

;; [ERR(1) = 1 FIX]

In that case you could code for calculation of IWRES etc. like this:

IPRED = F

W = SQRT((F*THETA(x)/(F+M))**2 + (M*THETA(y)/(F+M))**2)

RES = DV - IPRED

IWRES = RES/W

Y = Y = LOG(F+M) + W * ERR(1)

$SIGMA 1 FIX

Question to Kelong Han: Do you estimate M or do you set this parameter to a

fix value?

Kind regards,

Martin Bergstrand, MSc, PhD student

-----------------------------------------------

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

martin.bergstrand

-----------------------------------------------

-----Original Message-----

From: Han, Kelong [mailto:keh45

Sent: den 29 april 2009 23:27

To: Martin Bergstrand; nmusers

Subject: RE: [NMusers] Error models for log-transformed data

Dear Dr. Bergstrand, Dr. Hooker, Dr. Nyberg and NMusers,

Thank you so much for your helpful explanation. Now it is much clearer to

me.

Following your guidance, I noticed something that I would like to share, and

maybe also get some opinions from you:

1. It seems to be unnecessary to code the calculation of IWRES

(RES=DV-IPRED; IWRES = RES/W) in the control stream. Almost the same results

(IPRED, PPRED, RES and WRES) were obtained with the code

Y=LOG(F)+SQRT(THETA(x)**2+THETA(y)**2/F**2)*ERR(1) alone without coding

IWRES. I am not sure whether this is the way it should be or I did something

wrong.

2. Do you know how the IWRES should be calculated when the double

exponential error model is used? I did not code it in my control stream and

it seems to work OK.

Thanks a lot!

Kelong Han

PhD Candidate

________________________________________

From: Martin Bergstrand [martin.bergstrand

Sent: Tuesday, April 28, 2009 7:48 AM

To: Han, Kelong; nmusers

Subject: RE: [NMusers] Error models for log-transformed data

Dear Kelong and NMusers,

We will try to answer the questions by Kelong Han and also make some

additional comments on error models for log-transformed data.

*>> 1. What is the rationale of fixing $SIGMA 1?
*

The error model suggested by Mats Karlsson could as well be written with the

estimation of an ERR(1) and ERR(2) (estimation of the variance for two

SIGMA). However it is often convenient to fix the variance for SIGMA to 1

and instead estimate a scale-factor for that variance. This scale-factor can

in the simple case of an additive error be a single THETA(x). In the case of

an additive + proportional error model on the normal scale, the scale-factor

is a function of two estimated parameters THETA(x), THETA(y) and the model

prediction F.

W = SQRT(THETA(x)**2 + THETA(y)**2/F**2) Y = LOG(F) + W*ERR(1) [$SIGMA 1

FIX]

There are a number of practical reasons for estimating a scalar for SIGMA

(W) rather than SIGMA itself. The two reasons that comes to mind immediately

is to be able to calculate individual weighted residuals (see IWRES below)

and to use the M3 or M4 methods for handling censored data (Beal SL, 2001).

RES = DV - IPRED

IWRES = RES/W

*>> 2. What an error structure on the normal scale is this "double
*

exponential error model" equivalent to?

The simple answer is that it is not equivalent to any model on the normal

scale. It has some similarities to the combined additive and proportional

error model but it does not predict negative concentrations and assumes a

slight bias in the model predictions due to the addition of M (Y =

LOG(F+M)...). ERR(1) can be fairly well translated to a proportional error

on the normal scale but ERR(2) can't be directly interpreted as an additive

component.

*>> 3. Compared to the simplest error model Y=LOG(F)+ERR(1), the two error
*

models mentioned above contain additional THETA's. Are these additional

THETA's accounted for in the calculation of the objective function value?

Addition of parameters to the error model seems to follow the chi-squared

distribution (Silber HE et al, 2009) i.e. with the addition of one parameter

a drop in OFV of 3,84 corresponds to a 5% significance. The Karlsson is a

nested model to the simple additive error model and the "double exponential

error model" could probably be said to be "almost nested". However my

personal opinion (Martin's) is that the development of a residual error

model is best guided by the gof-plots.

As a final remark to this answer I would like to point out that all models

suggested this far for approximating a combined additive and proportional

error model (on normal scale) for log-transformed data has drawbacks. I have

already pointed out that the "double exponential error model" does introduce

a bias due to the addition of the parameter M. The model suggested by Mats

Karlsson on the other hand is only a good approximation for the cases when F

*> THETA(y). If F << THETA(y) the approximation will give rise to increasing
*

mean absolute error and unrealistic predictions (Y). From personal

experience (Martin's) this is primarily a problem in case of simulations.

One might argue that also the combined additive and proportional error model

on the normal scale has its drawbacks. Especially so since it is often

applied to bioanalytical data for which non-random censoring of estimated

negative concentrations is performed.

Kind regards,

Martin Bergstrand, Andrew Hooker and Joakim Nyberg

-----------------------------------------------

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

P.O. Box 591

SE-751 24 Uppsala

Sweden

-----------------------------------------------

-----Original Message-----

From: owner-nmusers

Behalf Of Han, Kelong

Sent: den 28 april 2009 00:16

To: nmusers

Subject: [NMusers] Error models for log-transformed data

Dear NMusers,

I am working on a PK model using log-transformed data. I have read previous

discussions on NMusers regarding this, and they are really helpful, but I am

still a little bit confused about the following questions. I would greatly

appreciate it if someone could make it clear:

1. Dr. Mats Karlsson suggested

Y=LOG(F)+SQRT(THETA(x)**2+THETA(y)**2/F**2)*ERR(1) with $SIGMA 1 FIX as an

equivalent error structure to the

additive+proportional error model on the normal scale. What is the rationale

of fixing $SIGMA 1?

2. Dr. Stu Beal and Dr. William Bachman suggested the "double exponential

error model": Y = LOG(F+M) + (F/(F+M))*ERR(1) + (M/(F+M))*ERR(2) without

fixing $SIGMA. The Goodness-of-Fit plot looks slightly better using this

error model in my study. What an error structure on the normal scale is this

"double exponential error model" equivalent to?

3. Compared to the simplest error model Y=LOG(F)+ERR(1), the two error

models mentioned above contain additional THETA's. Are these additional

THETA's accounted for in the calculation of the objective function value?

This especially bothers me because the "double exponential error model"

leads to a lower OFV compared to Y=LOG(F)+ERR(1) (also slightly better

Goodness-of-Fit plot) in my study.

Sorry for the length. Would anyone please give me some explanations or

references?

Thanks a lot!

Kelong Han

PhD Candidate=

Received on Thu Apr 30 2009 - 05:11:44 EDT

Date: Thu, 30 Apr 2009 11:11:44 +0200

Dear Kelong Han,

1.

IWRES is calculated in order to be used for diagnostic purposes and will not

affect the estimation. For example plots of IWRES vs. IDV (independent

variable) or |IWRES| vs. IPRED can be useful to diagnose your structural

model and your residual error model.

2.

Y = LOG(F+M) + (F/(F+M))*ERR(1) + (M/(F+M))*ERR(2)

Could be written like this instead:

Y = LOG(F+M) + SQRT((F*THETA(x)/(F+M))**2 + (M*THETA(y)/(F+M))**2)*ERR(1)

;; [ERR(1) = 1 FIX]

In that case you could code for calculation of IWRES etc. like this:

IPRED = F

W = SQRT((F*THETA(x)/(F+M))**2 + (M*THETA(y)/(F+M))**2)

RES = DV - IPRED

IWRES = RES/W

Y = Y = LOG(F+M) + W * ERR(1)

$SIGMA 1 FIX

Question to Kelong Han: Do you estimate M or do you set this parameter to a

fix value?

Kind regards,

Martin Bergstrand, MSc, PhD student

-----------------------------------------------

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

martin.bergstrand

-----------------------------------------------

-----Original Message-----

From: Han, Kelong [mailto:keh45

Sent: den 29 april 2009 23:27

To: Martin Bergstrand; nmusers

Subject: RE: [NMusers] Error models for log-transformed data

Dear Dr. Bergstrand, Dr. Hooker, Dr. Nyberg and NMusers,

Thank you so much for your helpful explanation. Now it is much clearer to

me.

Following your guidance, I noticed something that I would like to share, and

maybe also get some opinions from you:

1. It seems to be unnecessary to code the calculation of IWRES

(RES=DV-IPRED; IWRES = RES/W) in the control stream. Almost the same results

(IPRED, PPRED, RES and WRES) were obtained with the code

Y=LOG(F)+SQRT(THETA(x)**2+THETA(y)**2/F**2)*ERR(1) alone without coding

IWRES. I am not sure whether this is the way it should be or I did something

wrong.

2. Do you know how the IWRES should be calculated when the double

exponential error model is used? I did not code it in my control stream and

it seems to work OK.

Thanks a lot!

Kelong Han

PhD Candidate

________________________________________

From: Martin Bergstrand [martin.bergstrand

Sent: Tuesday, April 28, 2009 7:48 AM

To: Han, Kelong; nmusers

Subject: RE: [NMusers] Error models for log-transformed data

Dear Kelong and NMusers,

We will try to answer the questions by Kelong Han and also make some

additional comments on error models for log-transformed data.

The error model suggested by Mats Karlsson could as well be written with the

estimation of an ERR(1) and ERR(2) (estimation of the variance for two

SIGMA). However it is often convenient to fix the variance for SIGMA to 1

and instead estimate a scale-factor for that variance. This scale-factor can

in the simple case of an additive error be a single THETA(x). In the case of

an additive + proportional error model on the normal scale, the scale-factor

is a function of two estimated parameters THETA(x), THETA(y) and the model

prediction F.

W = SQRT(THETA(x)**2 + THETA(y)**2/F**2) Y = LOG(F) + W*ERR(1) [$SIGMA 1

FIX]

There are a number of practical reasons for estimating a scalar for SIGMA

(W) rather than SIGMA itself. The two reasons that comes to mind immediately

is to be able to calculate individual weighted residuals (see IWRES below)

and to use the M3 or M4 methods for handling censored data (Beal SL, 2001).

RES = DV - IPRED

IWRES = RES/W

exponential error model" equivalent to?

The simple answer is that it is not equivalent to any model on the normal

scale. It has some similarities to the combined additive and proportional

error model but it does not predict negative concentrations and assumes a

slight bias in the model predictions due to the addition of M (Y =

LOG(F+M)...). ERR(1) can be fairly well translated to a proportional error

on the normal scale but ERR(2) can't be directly interpreted as an additive

component.

models mentioned above contain additional THETA's. Are these additional

THETA's accounted for in the calculation of the objective function value?

Addition of parameters to the error model seems to follow the chi-squared

distribution (Silber HE et al, 2009) i.e. with the addition of one parameter

a drop in OFV of 3,84 corresponds to a 5% significance. The Karlsson is a

nested model to the simple additive error model and the "double exponential

error model" could probably be said to be "almost nested". However my

personal opinion (Martin's) is that the development of a residual error

model is best guided by the gof-plots.

As a final remark to this answer I would like to point out that all models

suggested this far for approximating a combined additive and proportional

error model (on normal scale) for log-transformed data has drawbacks. I have

already pointed out that the "double exponential error model" does introduce

a bias due to the addition of the parameter M. The model suggested by Mats

Karlsson on the other hand is only a good approximation for the cases when F

mean absolute error and unrealistic predictions (Y). From personal

experience (Martin's) this is primarily a problem in case of simulations.

One might argue that also the combined additive and proportional error model

on the normal scale has its drawbacks. Especially so since it is often

applied to bioanalytical data for which non-random censoring of estimated

negative concentrations is performed.

Kind regards,

Martin Bergstrand, Andrew Hooker and Joakim Nyberg

-----------------------------------------------

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

P.O. Box 591

SE-751 24 Uppsala

Sweden

-----------------------------------------------

-----Original Message-----

From: owner-nmusers

Behalf Of Han, Kelong

Sent: den 28 april 2009 00:16

To: nmusers

Subject: [NMusers] Error models for log-transformed data

Dear NMusers,

I am working on a PK model using log-transformed data. I have read previous

discussions on NMusers regarding this, and they are really helpful, but I am

still a little bit confused about the following questions. I would greatly

appreciate it if someone could make it clear:

1. Dr. Mats Karlsson suggested

Y=LOG(F)+SQRT(THETA(x)**2+THETA(y)**2/F**2)*ERR(1) with $SIGMA 1 FIX as an

equivalent error structure to the

additive+proportional error model on the normal scale. What is the rationale

of fixing $SIGMA 1?

2. Dr. Stu Beal and Dr. William Bachman suggested the "double exponential

error model": Y = LOG(F+M) + (F/(F+M))*ERR(1) + (M/(F+M))*ERR(2) without

fixing $SIGMA. The Goodness-of-Fit plot looks slightly better using this

error model in my study. What an error structure on the normal scale is this

"double exponential error model" equivalent to?

3. Compared to the simplest error model Y=LOG(F)+ERR(1), the two error

models mentioned above contain additional THETA's. Are these additional

THETA's accounted for in the calculation of the objective function value?

This especially bothers me because the "double exponential error model"

leads to a lower OFV compared to Y=LOG(F)+ERR(1) (also slightly better

Goodness-of-Fit plot) in my study.

Sorry for the length. Would anyone please give me some explanations or

references?

Thanks a lot!

Kelong Han

PhD Candidate=

Received on Thu Apr 30 2009 - 05:11:44 EDT