Re: OMEGA matrix
Dear Pavel,
With regard how to handle this differently in Monolix, you probably better ask
their developers (or perhaps Marc is listening in...). It would not surprise me
if it would because of statistical reasons rather than practical ones. On the
subject of shrinkage: the sampling I was talking about is at the level of
parameters and likelihood evaluation, not data sampling per se and therefore
shrinkage is not a directly affecting these correlations.
Wrt to omega matrix evaluation: similar to the covariance matrix of estimates,
one can further analyze the omega matrix. It actually is a long time ago since
I tried that, I do not recall numerical issues, maybe you need to tweak the
tolerance for SVD sometime.
This type of analysis is, as it appears, textbook material. I searched for some
reference material, you may find this paper useful:
iasri.res.in/ebook/EBADAT/3.../2-regdiagfeb07.pdf
otherwise topics are also discussed on Wikipedia, of course.
Last topic you mention: In nonmem one cannot fix individual correlations to
zero. Similar results however can be obtained by developing a banded omega
matrix. Elements in the bottom left corner of the matrix can be fixed to zero.
Hope this helps,
Jeroen
http://pd-value.com
-- More value out of your data!
Quoted reply history
On Sep 30, 2014, 7:15 PM, at 7:15 PM, Pavel Belo <[email protected]> wrote:
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>Sorry Jeroen, I have to correct you name in the email:
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>Dear Jeroen and the NONMEM Team,
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>Your email is definitely informative.
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> 1. "Off-diagonal elements (as explained by Bob Bauer) are available
>as sample correlations and do not have to be separately computed in
>contrast to linearization approaches such as FOCE." It may explain the
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>stability of the results when very large block matrix is used. On the
>other hand, it is not clear why Monolix SAEM may not work the same
>way.
>Is Monolix estimating correlations? Also, when we deal with "sample
>correlations", we may be talking about correlations between observed
>minus individual predicted values. Shrinkage can possibly affect such
>correlations.
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> 1. "I would advise to explore its properties further using matrix
>decomposition approaches (PCA etc)". Do you suggest to decompose the
>omega matrix and explore derived variables instead of original
>off-diagonal elements? It seems straightforward, but I recall error
>messages.
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>If you can point at some publications for both items 1 and 2 above, it
>will be greatly appreciated.
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>The importance of improving the OMEGA matrix may come from PD
>modeling.
>PD models are frequently more empirical than PK models and strong
>correlations come from nowhere. They are difficult to interpret, but
>important to account for when simulations are requested by the
>agencies. There are correlations, which change from 0 to 0.6 when
>models are slightly different indicating that they may be
>insignificant. Monolix allows us to set a single correlation to zero.
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>NONMEM may require a different approach. I am searching for the
>different approaches because after many years I am emotionally attached
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>to NONMEM and because NONMEM is very flexible.
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>Kind regards,
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>Pavel
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>On Mon, Sep 29, 2014 at 07:00 PM, Jeroen Elassaiss-Schaap wrote:
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>Dear Pavel, others,
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>The underlying technical difference is that SAEM is in its core a
>sampling methodology. Off-diagonal elements (as explained by Bob Bauer)
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>are available as sample correlations and do not have to be separately
>computed in contrast to linearization approaches such as FOCE.
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>The more interesting question to me, as also eluted to by Ken, is what
>criteria to set up for inclusion of an off-diagonal element. I
>completely support his argument for simulation performance of the
>model,
>as e.g. judged using a VPC. Whether to score it as an additional degree
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>of freedom may be up to debate. An off-diagonal element in essence
>limits the freedom of the model as the random space in which samples
>can
>be generated will be smaller. In that perspective one could argue to
>retain any off-diagonal element that is sufficiently deviating from
>zero
>regardless of ofv changes, and to not apply the concept of
>over-parametrization (or at least not in comparison to other types of
>parameters). In practice inclusion of an important off-diagonal is
>mostly accompanied by a sound improvement in ofv anyway.
>
>More can be found in earlier discussions we had on this list, see e.g.
https://www.mail-archive.com/[email protected]/msg02736.html
> https://www.mail-archive.com/[email protected]/msg02736.html
https://www.mail-archive.com/[email protected]/msg02736.html
>for
>quite an extensive one from 2010. Here also an r-script to visualize
>the
>parameter space impact can be found ;-).
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>In cases where a larger full or banded omega block is found, I would
>advice to explore its properties further using matrix decomposition
>approaches (PCA etc) to evaluate propagated correlations across the
>matrix. But also on the basis of physiology/pharmacology as a data
>sample may not be informative enough to support robust interpretation
>of
>correlations. A discussion along those lines in reporting seems the
>more
>fruitful to me.
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>Best regards,
>Jeroen
>
> http://pd-value.com http://pd-value.com/
>
>
>-- More value out of your data!
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>-----Original Message-----
>From: [email protected]
><mailto:[email protected]>
>[mailto:[email protected]
><mailto:[email protected]> ] On Behalf Of Standing Joseph
>(GREAT ORMOND STREET HOSPITAL FOR CHILDREN NHS FOUNDATION TRUST)
>Sent: Friday, September 26, 2014 09:15
>To: Kowalski, Ken; 'Eleveld, DJ'; 'Pavel Belo'; [email protected]
><mailto:[email protected]>
>Subject: RE: [NMusers] OMEGA matrix
>Dear Pavel,
>To answer your question I suggest you go on Bob Bauer's NONMEM 7
>course.
>The understanding I gleaned from that course (which I think was
>enhanced
>by the excellent wine we had at lunch in Alicante) was that with
>appropriate MU parameterisation there is virtually no computational
>disadvantage to estimating the full block with the newer algorithms.
>So
>you might as well do it, at least in early runs where you want an idea
>of which parameter correlations might be useful/reasonably estimated.
>BW,
>Joe
>
>Joseph F Standing
>MRC Fellow, UCL Institute of Child Health
>Antimicrobial Pharmacist, Great Ormond Street Hospital
>Tel: +44(0)207 905 2370
>Mobile: +44(0)7970 572435
>
>
>
>From: [email protected]
><mailto:[email protected]>
>[[email protected] <mailto:[email protected]> ]
>On
>Behalf Of Ken Kowalski [[email protected]
><mailto:[email protected]> ]
>Sent: 25 September 2014 22:43
>To: 'Eleveld, DJ'; 'Pavel Belo'; [email protected]
><mailto:[email protected]>
>Subject: RE: [NMusers] OMEGA matrix
>Warning: This message contains unverified links which may not be safe.
>You should only click links if you are sure they are from a trusted
>source.
>Hi Douglas,
>My own thinking is that you should fit the largest omega structure that
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>can
>be supported by the data rather than just always assuming a diagonal
>omega
>structure. This does not necessarily mean always fitting a full block
>omega
>structure, as it can often lead to an ill-conditioned model, however,
>there
>may be a reduced block omega structure that is more parsimonious than
>the
>diagonal omega structure. Getting the omega structure right is
>particularly
>important for simulation of individual responses. For example, if you
>always simulate from a diagonal omega structure for CL and V when there
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>is
>evidence that the random effects are highly positively correlated then
>you
>may end up simulating individual PK profiles for combinations of
>individual
>CLs and Vs that are not represented in your data (i.e., high
>correlation
>would suggest that individuals with high CL will tend to also have high
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>V
>and vice versa whereas a simulation assuming that they are independent
>will
>result in simulating for some individuals with high CL and low V and
>some
>individuals with low CL and high V that might not be represented in
>your
>data). This could lead to simulations that over-predict the variation
>in
>the concentration-time profiles even though the diagonal omega may be
>sufficient for purposes of predicting central tendency in the PK
>profile.
>You can confirm this by VPC looking at your ability to predict say the
>10th
>and 90th percentiles in comparison to the observed 10th and 90th
>percentiles
>in your data. That is, if you simulate from the diagonal omega when
>there
>is correlation in the random effects you may find that your prediction
>of
>the 10th and 90th percentiles are more extreme than that in your
>observed
>data. I see this all the time in VPC plots where the majority of the
>observed data are well within the predictions of the 10th and 90th
>percentiles when we should expect about 10% of our data above the 90th
>percentile prediction and 10% below the 10th percentile prediction.
>Best regards,
>Ken
>Kenneth G. Kowalski
>President & CEO
>A2PG - Ann Arbor Pharmacometrics Group, Inc.
>110 Miller Ave., Garden Suite
>Ann Arbor, MI 48104
>Work: 734-274-8255
>Cell: 248-207-5082
>Fax: 734-913-0230
>[email protected] <mailto:[email protected]>
>www.a2pg.com http://www.a2pg.com/
>
>
>-----Original Message-----
>From: [email protected]
><mailto:[email protected]>
>[mailto:[email protected]
><mailto:[email protected]> ] On
>Behalf Of Eleveld, DJ
>Sent: Thursday, September 25, 2014 4:36 PM
>To: Pavel Belo; [email protected] <mailto:[email protected]>
>Subject: RE: [NMusers] OMEGA matrix
>Hi Pavel,
>My question is: Why is it desirable to fit a complete omega matrix if
>its
>physical interpretation is unclear? Etas are variation of unknown
>origin
>i.e. not explained by the structural model. A full omega matrix allows
>the
>unknown variation of one paramater to have a (linear?) relationship
>with
>some other thing that is also unknown. If unknown A is found to have a
>linear relationship with unknown B, then what knowlegde is gained? I do
>think it can be instructive to to look at correlations and use this
>information to make a better structural model. But I think diagonal
>OMEGA
>matrix is more desirable if it works ok.
>warm regards,
>Douglas Eleveld
>
>
>
>From: [email protected]
><mailto:[email protected]>
>[[email protected] <mailto:[email protected]> ]
>on
>behalf
>of Pavel Belo [[email protected] <mailto:[email protected]> ]
>Sent: Thursday, September 25, 2014 4:24 PM
>To: [email protected] <mailto:[email protected]>
>Subject: [NMusers] OMEGA matrix
>Hello Nonmem Community,
>It seems like NONMEM developers may advise to start with full OMEGA
>matrix
>at the beginning of model development. Monolix developers may advise to
>start with a diagonal matrix. Is there something different in NONMEM
>SAEM
>algorithms that makes model stable when a lot of statistically
>insignificant
>correlations/covariances are estimated in the model?
>It seems like NONMEM SAEM can be very stable in very "hard cases" (a
>lot
>of
>outliers, partially misspecified model, overparameterized model, etc.).
>
>The
>omega matrix is a part of the puzzle.
>When it is impossible to test every correlation coefficient for
>significance
>due to some limitations, it becomes a regulatory issue. We may need to
>be
>able to make a statement that the model is safe and sound even when
>OMEGA
>matrix can be overparameterized (tries to estimate too many
>insignificant
>parameters within the OMEGA matrix).
>Kind regards,
>Pavel
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