From:Peter L. Bonate - pbonate@ilexonc.com
Subject:[NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 8:50 AM

Dear all,
I need to generate a covariance matrix of the following:
   [X11
   0 X22
   X31 X32 X33]
where X12 is fixed to zero and all other elements are non-zero.
Is there any easy way to do this?

Thanks.

Peter L. Bonate, PhD, FCP
Director, Pharmacokinetics
ILEX Oncology
4545 Horizon Hill Blvd
San Antonio, TX  78229
phone: 210-949-8662
fax: 210-949-8219
email: pbonate@ilexonc.com 
_______________________________________________________

From: Leonid Gibiansky - lgibiansky@emmes.com
Subject: Re: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 9:06 AM

sure, just reorder (rename) random effects so that your matrix looks like
x11
x13 x33
0    x32 x22
Leonid

_______________________________________________________

From: Vladimir Piotrovsky, Ph.D. - VPIOTROV@PRDBE.jnj.com
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 9:27 AM

Pete,
 
You can try to change the indices of ETAs to come up with the matrix
 
   [X11 
   X21 X22 
   0    X32 X33] 
In this case you can fix the lower left element to zero.

Best regards,
Vladimir

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

Vladimir Piotrovsky, Ph.D.
Research Fellow, Advanced PK-PD Modeling & Simulation
Global Clinical Pharmacokinetics and Clinical Pharmacology (ext. 5463/151)
Johnson & Johnson Pharmaceutical Research & Development
Turnhoutseweg 30
B-2340 Beerse
Belgium 
_______________________________________________________

From: "Kowalski, Ken" - Ken.Kowalski@pfizer.com
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 9:38 AM

Pete,

Leonid beat me to it.  The block structure that Leonid gives below is the
banded structure.  Note that in NONMEM you can specify 0's in a lower
triangle portion of the Omega block (i.e., banded structure) and NONMEM will
know to fix those elements to zero (i.e., you don't use the FIX option).

Ken
_______________________________________________________

From: vgcasabo - vicente.casabo@uv.es
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 11:29 AM

Hi, all

I do not understand, that if  X11 is correlated with X33, and X22 is
correlated with X33, X11 and X22 are not correlated.

Sorry

Vicente G. Casabo
_______________________________________________________

From: Serge Guzy - GUZY@xoma.com
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 12:15 PM

You can try using simulation if it is possible. I think that as long as the
matrix is positive definite it will work. Sometimes indirect correlation like
you suggest (X11 is correlated with X33, and X22 is correlated with X33 ) can
exist while there is no direct correlation expressed through the correlation
coefficient between X11 and X22

Serge Guzy

President POP-PHARM

Head of Pharmacometrics and Preclinical Statistics; Xoma Corporation

Tel: (w) 510 204 74 76;(Business)510 527 2220
_______________________________________________________

From: Steve Duffull - sduffull@pharmacy.uq.edu.au
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 5:22 PM

Vicente
 
You are right - X11 and X22 will be implicitly correlated through the other
covariance structure.  However in the NONMEM OMEGA statement below they will
not be explicitly correlated.  Without specifically doing simulations to check
this - I would guess that the implicit correlation would hold (to some
extent) - even when you fix the explicit covariance to zero in the OMEGA block.
 
I would be interested to know this too, can someone advise me on this?
 
Steve

_______________________________________________________

From: vgcasabo - vicente.casabo@uv.es
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/14/2004 4:01 AM

Hi all

Por fin.

I understand.

 

Is possible that if  cov1 and cov2 are two physiological variables not known
or hidden, and the following dependence exists:

Eta1=a1*cov1+e1

Eta2=a2*cov2+e2

Eta3=a3*cov1+a4*cov2+e3

e1, e2 and e3 are independent errors

 

Thus, when dependence of parameters and cov1 and cov2 are not modeled, eta1 and eta3
are correlated, eta2 and et3 are correlated but eta1 and eta2 are not.

 

Vicente G. Casabo
_______________________________________________________

From: Leonid Gibiansky - lgibiansky@emmes.com
Subject: Re: [NMusers] Fixed elements within a block covariance matrix
Date: 1/14/2004 9:44 AM

One can check correlation structure via simulation, but I am not quite sure why
should one do it. I guess, we can trust NONMEM to simulate according to the
variance-covariance model, and then, for the OMEGA matrix

X11
X13 X33
0     X23 X22

Eta1 and Eta2 will not be correlated in any way (implicit or explicit). If
simulations show correlation (X12 not equal to zero), then one need to
increase the sample size of the simulations. Correlation coefficient X21
should decay to zero with the sample size, if the model is implemented correctly.

Example
Eta1=a1*cov1+e1
Eta2=a2*cov2+e2
Eta3=a3*cov1+a4*cov2+e3
is not quite correct unless means of cov1 and cov2 are equal to zero
(since means of etas should be zero). But the idea holds. If
Eta1=a11*e1
Eta2=a22*e2
Eta3=a13*e1+a23*e2+a33*e3
where e1, e2, and e3 are independent [then mean(ei*ej) = 0 if i not equal j]
with variance 1 [ mean(ei*ei) = 1 ] then
X11=mean(Eta12)=a112
X12=mean(Eta1*Eta2) = 0 (since e1 and e2 are independent)
X13=mean(Eta1*Eta3) = a11*a13
X22 = a222
X23 = a22*a23
X33 = a132+a232+a332

Leonid

_______________________________________________________

From: qi - liuqi@ufl.edu
Subject: Re: [NMusers] Fixed elements within a block covariance matrix
Date: 1/14/2004 10:09 AM

Dear all,

Here is a simple example:

Suppose X and Y are independent standard normal variable n(0,1), let U=X+Y.
It could be shown that U is correlated with both X and Y.

Another example:
Suppose X and Y are independent standard normal variable n(0,1), let U=X+Y and V=X-Y.
t could be shown that U and V are independent, but both U and V are correlated with X.

Use X, Y and U in the first case, or use U, V and X in the second case to
construct the variance-covariance matrix, you will get the mentioned interesting matrix.

You can do a simple simulation to see the relationship between the variables
(it can be done easily in R or Splus).

Hope this helps.

Qi Liu
PO BOX 100494
Department of Pharmaceutics
College of Pharmacy
University of Florida
Gainesville, FL32610
Tel: (352)8463257 
_______________________________________________________


From: "Bonate, Peter" - pbonate@ilexonc.com
Subject: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 14:51


Dear all,
I need to generate a covariance matrix of the following:
   [X11
   0 X22
   X31 X32 X33]
where X12 is fixed to zero and all other elements are non-zero. 
Is there any easy way to do this?

Thanks.

Peter L. Bonate, PhD, FCP
Director, Pharmacokinetics
ILEX Oncology
4545 Horizon Hill Blvd
San Antonio, TX  78229
phone: 210-949-8662
fax: 210-949-8219
email: pbonate@ilexonc.com
_______________________________________________________

From: Sam Liao - sliao@pharmaxresearch.com
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 8:42 PM

Hi Steve, Vicente:

I believe what you are saying may hold, but not always.

In the case where X11 and X33 are positively correlated, while X22 and
X33 are negatively correlated (X3 goes up while X2 goes down), X11 and
X22 may not be correlated.  This is just my observation in some of the
models I had before.  

Best regards,

Sam Liao, Ph.D.
PharMax Research
199 Pierce Street,
Suite 817,
Somerset, NJ 08873
phone: 201-9882043
efax: 1-720-2946783

_______________________________________________________

From: GIRARD PASCAL - PASCAL.GIRARD@adm.univ-lyon1.fr
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 9:26 AM

Hi Peter,

Are you meaning you want initialize such a matrix in NONMEM ? If so you
can try the following:

1) Renumber your Etas so that you will have the following structure:

   [X22
    X32  X33
    0    X31 X11]

2) This should be recognized by NONMEM as a band matrix (see attached (below)
Diane Mould’s Email for more explanations).

3) Tell us if that works!

Cheers,

Pascal

Subject: Re: Termination due to rounding erros
From: "diane r mould" 
Date: Fri, 23 Mar 2001 20:40:09 +0100
To: "KOWALSKI, KENNETH G. [PHR/1825]" , 

Dear Ken

I will do the best I can :-)
In NONMEM, a BAND matrix is when some of the elements in the off diagonal
matrix have been set to zero.  Doing this effectively fixes these elements
to zero, eliminating the need for NONMEM to estimate those values which may
not be identifiable, without sacrificing other identifiable elements.

So if, for example, you have a one compartment model with CL, V, KA as
parameters, if you can identify the covariance terms between CL and V and
between CL and Ka, but not between Ka and V, a BAND matrix could be written
to describe these relationships.  Below is the NONMEM record for this,
assuming that ETA(1) describes the variance on V, ETA(2) on CL and ETA(3) on
KA.

$OMEGA BLOCK(3)
.1
.01 .1
0    .01  .1

No further coding is necessary.  However, the elements in the BAND matrix
must, of course, be symmetrical.  Therefore a BAND matrix of

$OMEGA BLOCK(3)
.1
.01 .1
0    0   .1

will produce an error from NONMEM.  You would need to code this arrangement
instead as follows:

$OMEGA BLOCK(2)
.1
.01 .1
$OMEGA .1

I have found the use of the BAND matrix to be quite useful at times,
particularly if the model is to be used later for simulation work for
example.

Best Regards
Diane

----- Original Message -----
From: "KOWALSKI, KENNETH G. [PHR/1825]" 
To: "'diane r mould'" ; 
Sent: Friday, March 23, 2001 1:50 PM
Subject: RE: Termination due to rounding erros


> Diane,
>
> Can you share with us how this banding of omega is performed?  Do you have
> to code certain elements of omega as thetas?
>
> Ken
>
> -----Original Message-----
> From: diane r mould [mailto:drmould@attglobal.net]
> Sent: Friday, March 23, 2001 12:09 PM
> To: nmusers@c255.ucsf.edu
> Subject: Termination due to rounding erros
>
>
> Dear All
>
> A few minor notes on the suggestions made by others:
>
> I too have found that increasing the number of sig digits can sometimes
> result in NONMEM converging successfully when a previous run with fewer
> significan digits terminated due to rounding errors.  Sometimes, however,
> the increased number of significant digits still results in a termination
> due to rounding errors.  If this happens, the resulting parameters from
the
> control stream with the higher number of significant digits can be used as
> intitial estimates for a new model.  I would then reduce the number of
> significant digits back to 3 and this may converge successfully.
>
> With regard to the variance covariance structure, if the termination is
due
> to over parameterization in this part of the model, I have found that use
of
> a BAND matrix structure and appropriately re-organizing the matrix can
often
> get past the termination error but still retain much of the off-diagonal
> information.
>
> Best Regards
> Diane

 

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

Dr Pascal Girard
EA 3738 Ciblage Thérapeutique en Oncologie
Fac Médecine Lyon-Sud
BP12
69921 OULLINS Cedex
France

Tel 33 4 78 86 31 53
Fax 33 4 78 86 31 49
e-mail : Pascal.Girard@adm.univ-lyon1.fr

========================================== 
_______________________________________________________
From: Serge Guzy - GUZY@xoma.com
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/13/2004 10:04 PM

I made many simulation exercises where indirect correlations exist while direct
correlations were not existent. As long as the matrix is positive definite, this
can happen. With 3 dimensions it is easy to make it positive definite. Once we
begin with high dimensionality, then arbitrarily fixing one correlation to a
certain value(not specifically 0) can make it difficult to be positive definite.
Serge
_______________________________________________________

From: Jakob Ribbing - Jakob.Ribbing@farmbio.uu.se
Subject: RE: [NMusers] Fixed elements within a block covariance matrix
Date: 1/14/2004 4:12 AM

Dear all,

 

If one views correlation between two etas as a result of a mutual covariate
not currently included in the covariate model, one can easily come up with
examples where only one of three off diagonals truly is 0. In the example with
correlation between individual CL, V and Ka, a measure of body size could very
well act as a covariate on both CL and V, just as geno- or phenotype of a drug
transporter may act on both CL and Ka. By this mechanism, as long as there is
no correlation between body size and genotype/phenotype of the transporter,
individual etas may be correlated for CL and V, as well as for CL and Ka, without
any correlation between individual V and Ka.

 

Notice that this example holds even if two different measures of body size (WT,
WT^0.75, LBW, BSA, et c) are appropriate for CL and V, respectively. It is also
possible that the off-diagonal non-zero correlations are all positive. A positive
correlation between CL and Ka may not be plausible from a perspective of physiology
or evolution in this example but if it was (or if the example was Q, V and Ka), we
could very well have all non-zero correlations between etas being positive.

 

In this example, the off-diagonal etas would not have been necessary, had we only
included a proper covariate model. However, my point is not that off-diagonal etas
are a result of bad covariate modelling. The true covariate model is far more
complex than this example, consisting of covariates which are unknown and cannot be measured.

 

Jakob Ribbing
_______________________________________________________