From: Ribbing, Jakob <*Jakob.Ribbing*>

Date: Thu, 13 Nov 2008 23:27:55 -0000

Dear all,

First of all, I am not sure that there is any assumption of etas having

a normal distribution when estimating a parametric model in NONMEM. The

variance of eta (OMEGA) does not carry an assumption of normality. I

believe that Stuart used to say the assumption of normality is only when

simulating. I guess the assumption also affects EBE:s unless the

individual information is completely dominating? If the assumption of

normality is wrong, the weighting of information may not be optimal, but

as long as the true distribution is symmetric the estimated parameters

are in principle correct (but again, the model may not be suitable for

simulation if the distributional assumption was wrong). I will be off

line for a few days, but I am sure somebody will correct me if I am

wrong about this.

If etas are shrunk, you can not expect a normal distribution of that

(EBE) eta. That does not invalidate parameterization/distributional

assumptions. Trying other semi-parametric distributions or a

non-parametric distribution (or a mixture model) may give more

confidence in sticking with the original parameterization or else reject

it as inadequate. In the end, you may feel confident about the model

even if the EBE eta distribution is asymmetric and biased (I mentioned

two examples in my earlier posting).

Connecting to how PsN may help in this case: http://psn.sourceforge.net/

In practice to evaluate shrinkage, you would simply give the command

(assuming the model file is called run1.mod):

execute --shrinkage run1.mod

Another quick evaluation that can be made with this program is to

produce mirror plots (PsN links in nicely with Xpose for producing the

diagnostic plots):

execute --mirror=3 run1.mod

This will give you three simulation table files that have been derived

by simulating under the model and then fitting the simulated data using

the same model (using the design of the original data). If you see a

similar pattern in the mirror plots as in the original diagnostic plots,

this gives you more confidence in the model. That brings us back to

Leonids point about it being more useful to look at diagnostic plots

than eta bar.

Wishing you a great weekend!

Jakob

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

From: BAE, KYUN-SEOP

Sent: 13 November 2008 22:05

To: Ribbing, Jakob; XIA LI; nmusers

Subject: RE: [NMusers] Very small P-Value for ETABAR

Dear All,

Realized etas (EBEs, MAPs) is estimated under the assumption of normal

distribution.

However, the resultant distribution of EBEs may not be normal or mean of

them may not be 0.

To pass t-test, one may use "CENTERING" option at $ESTIMATION.

But, this practice is discouraged by some (and I agree).

Normal assumption cannot coerce the distribution of EBE to be normal,

and furthermore non-normal (and/or not-zero-mean) distribution of EBE

can be nature's nature.

One simple example is mixture population with polymorphism.

If I could not get normal(?) EBEs even after careful examination of

covariate relationships as others suggested,

I would bear it and assume nonparametric distribution.

Regards,

Kyun-Seop

=====================

Kyun-Seop Bae MD PhD

Email: kyun-seop.bae

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

From: owner-nmusers

On Behalf Of Ribbing, Jakob

Sent: Thursday, November 13, 2008 13:19

To: XIA LI; nmusers

Subject: RE: [NMusers] Very small P-Value for ETABAR

Hi Xia,

Just to clarify one thing (I agree with almost everything you said):

The p-value indeed is related to the test of ETABAR=0. However, this =

is

not a test of normality, only a test that may reject the mean of the

etas being zero (H0). Therefore, shrinkage per se does not lead to

rejection of HO, as long as both tails of the eta distribution are

shrunk to a similar degree.

I agree with the assumption of normality. This comes into play when you

simulate from the model and if you got the distribution of individual

parameters wrong, simulations may not reflect even the data used to fit

the model.

Best Regards

Jakob

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

From: owner-nmusers

On Behalf Of XIA LI

Sent: 13 November 2008 20:31

To: nmusers

Subject: Re: [NMusers] Very small P-Value for ETABAR

Dear All,

Just some quick statistical points...

P value is usually associated with hypothesis test. As far as I know,

NONMEM assume normal distribution for ETA, ETA~N(0,omega), which means

the null hypothesis to test is H0: ETABAR=0. A small P value indicates =

a

significant test. You reject the null hypothesis.

More...

As we all know, ETA is used to capture the variation among individual

parameters and model's unexplained error. We usually use the function

(or model) parameter=typical value*exp (ETA), which leads to a =

lognormal

distribution assumption for all fixed effect parameters (i.e., CL, V,

Ka, Ke...).

By some statistical theory, the variation of individual parameter equals

a function of the typical value and the variance of ETA.

VAR (CL) = typical value*exp (omega/2). NO MATH PLS!!

If your typical value captures all overall patterns among patients

clearance, then ETA will have a nice symmetric normal distribution with

small variance. Otherwise, you leave too many patterns to ETA and will

see some deviation or shrinkage (whatever you call).

Why adding covariates is a good way to deal with this situation? You

model become CL=typical value*exp (covariate)*exp (ETA). The variation

of individual parameter will be changed to:

VAR (CL) = (typical value + covariate)*exp (omega/2)).

You have one more item to capture the overall patterns, less leave to

ETA. So a 'good' covariate will reduce both the magnitude of omega and

ETA's deviation from normal.

Understanding this is also useful when you are modeling BOV studies.

When you see variation of PK parameters decrease with time (or

occasions). Adding a covariate that make physiological sense and also

decrease with time may help your modeling.

Best,

Xia

==========================

=============

Xia Li

Mathematical Science Department

University of Cincinnati

Received on Thu Nov 13 2008 - 18:27:55 EST

Date: Thu, 13 Nov 2008 23:27:55 -0000

Dear all,

First of all, I am not sure that there is any assumption of etas having

a normal distribution when estimating a parametric model in NONMEM. The

variance of eta (OMEGA) does not carry an assumption of normality. I

believe that Stuart used to say the assumption of normality is only when

simulating. I guess the assumption also affects EBE:s unless the

individual information is completely dominating? If the assumption of

normality is wrong, the weighting of information may not be optimal, but

as long as the true distribution is symmetric the estimated parameters

are in principle correct (but again, the model may not be suitable for

simulation if the distributional assumption was wrong). I will be off

line for a few days, but I am sure somebody will correct me if I am

wrong about this.

If etas are shrunk, you can not expect a normal distribution of that

(EBE) eta. That does not invalidate parameterization/distributional

assumptions. Trying other semi-parametric distributions or a

non-parametric distribution (or a mixture model) may give more

confidence in sticking with the original parameterization or else reject

it as inadequate. In the end, you may feel confident about the model

even if the EBE eta distribution is asymmetric and biased (I mentioned

two examples in my earlier posting).

Connecting to how PsN may help in this case: http://psn.sourceforge.net/

In practice to evaluate shrinkage, you would simply give the command

(assuming the model file is called run1.mod):

execute --shrinkage run1.mod

Another quick evaluation that can be made with this program is to

produce mirror plots (PsN links in nicely with Xpose for producing the

diagnostic plots):

execute --mirror=3 run1.mod

This will give you three simulation table files that have been derived

by simulating under the model and then fitting the simulated data using

the same model (using the design of the original data). If you see a

similar pattern in the mirror plots as in the original diagnostic plots,

this gives you more confidence in the model. That brings us back to

Leonids point about it being more useful to look at diagnostic plots

than eta bar.

Wishing you a great weekend!

Jakob

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

From: BAE, KYUN-SEOP

Sent: 13 November 2008 22:05

To: Ribbing, Jakob; XIA LI; nmusers

Subject: RE: [NMusers] Very small P-Value for ETABAR

Dear All,

Realized etas (EBEs, MAPs) is estimated under the assumption of normal

distribution.

However, the resultant distribution of EBEs may not be normal or mean of

them may not be 0.

To pass t-test, one may use "CENTERING" option at $ESTIMATION.

But, this practice is discouraged by some (and I agree).

Normal assumption cannot coerce the distribution of EBE to be normal,

and furthermore non-normal (and/or not-zero-mean) distribution of EBE

can be nature's nature.

One simple example is mixture population with polymorphism.

If I could not get normal(?) EBEs even after careful examination of

covariate relationships as others suggested,

I would bear it and assume nonparametric distribution.

Regards,

Kyun-Seop

=====================

Kyun-Seop Bae MD PhD

Email: kyun-seop.bae

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

From: owner-nmusers

On Behalf Of Ribbing, Jakob

Sent: Thursday, November 13, 2008 13:19

To: XIA LI; nmusers

Subject: RE: [NMusers] Very small P-Value for ETABAR

Hi Xia,

Just to clarify one thing (I agree with almost everything you said):

The p-value indeed is related to the test of ETABAR=0. However, this =

is

not a test of normality, only a test that may reject the mean of the

etas being zero (H0). Therefore, shrinkage per se does not lead to

rejection of HO, as long as both tails of the eta distribution are

shrunk to a similar degree.

I agree with the assumption of normality. This comes into play when you

simulate from the model and if you got the distribution of individual

parameters wrong, simulations may not reflect even the data used to fit

the model.

Best Regards

Jakob

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

From: owner-nmusers

On Behalf Of XIA LI

Sent: 13 November 2008 20:31

To: nmusers

Subject: Re: [NMusers] Very small P-Value for ETABAR

Dear All,

Just some quick statistical points...

P value is usually associated with hypothesis test. As far as I know,

NONMEM assume normal distribution for ETA, ETA~N(0,omega), which means

the null hypothesis to test is H0: ETABAR=0. A small P value indicates =

a

significant test. You reject the null hypothesis.

More...

As we all know, ETA is used to capture the variation among individual

parameters and model's unexplained error. We usually use the function

(or model) parameter=typical value*exp (ETA), which leads to a =

lognormal

distribution assumption for all fixed effect parameters (i.e., CL, V,

Ka, Ke...).

By some statistical theory, the variation of individual parameter equals

a function of the typical value and the variance of ETA.

VAR (CL) = typical value*exp (omega/2). NO MATH PLS!!

If your typical value captures all overall patterns among patients

clearance, then ETA will have a nice symmetric normal distribution with

small variance. Otherwise, you leave too many patterns to ETA and will

see some deviation or shrinkage (whatever you call).

Why adding covariates is a good way to deal with this situation? You

model become CL=typical value*exp (covariate)*exp (ETA). The variation

of individual parameter will be changed to:

VAR (CL) = (typical value + covariate)*exp (omega/2)).

You have one more item to capture the overall patterns, less leave to

ETA. So a 'good' covariate will reduce both the magnitude of omega and

ETA's deviation from normal.

Understanding this is also useful when you are modeling BOV studies.

When you see variation of PK parameters decrease with time (or

occasions). Adding a covariate that make physiological sense and also

decrease with time may help your modeling.

Best,

Xia

==========================

=============

Xia Li

Mathematical Science Department

University of Cincinnati

Received on Thu Nov 13 2008 - 18:27:55 EST