Re: OMEGA matrix
Hi,
As pointed out by others I agree it is essential to consider the existence of random effect correlations if you wish to make model predictions e.g. to use a VPC to evaluate a model.
I agree with Jeroen that this should be primarily be an informed choice based on physiology/pharmacology. 'Blue sky' searches for correlations which when would have no rational explanation or interpretation should be done with a great deal of caution.
It can be tricky to explore all possible combinations using the change in OFV (e.g. with the likelihood ratio test) to guide model selection. A more straightforward approach is to bootstrap the model with a full covariance block for all the random effects you suspect may be correlated.
Bootstrapping today is usually a practical option because runs can be easily performed in parallel on multiple processors on the same machine or on a cluster. I typically use 100 bootstrap replicates for this purpose and look for correlations which include zero in the 95% bootstrap confidence interval. If I find such correlations then I know I should be able to remove those covariances from the covariance block. I can then re-run the bootstrap and obtain confidence intervals on all the parameters including the correlations. Confidence intervals calculated from asymptotic standard errors (if you can get them) are usually unreliable compared with parametric bootstrap confidence intervals ( http://www.page-meeting.org/default.asp?abstract=3143 ).
i don't agree with Ken that "ill-conditioning" or "not stable" based on failure of the $COVARIANCE step should be used to judge the adequacy of the results. Experimentally it has been shown that the bootstrap distribution of parameter uncertainty is not different when comparing runs which terminated and those which were successful or which completed the $COVARIANCE step. http://www.mail-archive.com/nmusers%40globomaxnm.com/msg03401.html . See also http://holford.fmhs.auckland.ac.nz/docs/bootstrap-and-confidence-intervals.pdf slides 24 to 31.
Best wishes,
Nick
Quoted reply history
On 1/10/2014 7:57 a.m., Ken Kowalski wrote:
> Hi Jeroen,
>
> I think we might be on the same page but I wanted to get clarification about your suggestion that we “not apply the concept of over-parameterization” with respect to evaluating the omega structure. I’m assuming by ‘over-parameterization’ you mean a model that has more elements in omega than might be necessary to be parsimonious. If so, I certainly agree but I wouldn’t call such a model that has more parameters than necessary to be parsimonious as necessarily over-parameterized. An over-parameterized model is one in which there can be an infinite set of solutions to the parameter values that yields the same fit. Such a setting can occur when the R-matrix in NONMEM is singular. Such over-parameterized models are often also referred to as being ill-conditioned or not stable. I think we should always avoid over-parameterization, ill-conditioning and unstable models regardless of the source (i.e., fixed effects, IIV random effects and omega-structure, or residual error structure). However, I do agree that parsimony in omega is probably not as important as say looking for a parsimonious set of covariate parameter fixed effects when performing covariate modeling to obtain a final model for prediction purposes. This is why in my earlier response below I suggested fitting the “largest omega structure that can be supported by the data”. What I meant by this statement is that we fit the largest number of elements of omega while avoiding over-parameterization or ill-conditioning. Such an omega structure might not be parsimonious (i.e., the smallest omega structure that adequately describes the features in the data). The point I was trying to make is that the smallest omega structure that adequately describes the features in the data may not be a diagonal omega structure (i.e., when correlations do exist) particularly if we are interested in describing the variation in the data and not just in predictions of central tendency.
>
> Best,
>
> Ken
>
> *From:* [email protected] [ mailto: [email protected] ] *On Behalf Of *Jeroen Elassaiss-Schaap
>
> *Sent:* Monday, September 29, 2014 7:00 PM
>
> *To:* [email protected] ; [email protected] ; [email protected] ; [email protected] ; [email protected]
>
> *Subject:* Re: [NMusers] OMEGA matrix
>
> Dear Pavel, others,
>
> The underlying technical difference is that SAEM is in its core a sampling methodology. 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.
>
> 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 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 for quite an extensive one from 2010. Here also an r-script to visualize the parameter space impact can be found ;-).
>
> 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.
>
> Best regards,
> Jeroen
>
> http://pd-value.com
>
> -- More value out of your data!
>
> -----Original Message-----
> From:[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 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 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 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]] 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
>
> ------------------------------------------------------------------------
>
>