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Re: What does convergence/covariance show?

From: Nick Holford <n.holford>
Date: Tue, 25 Aug 2009 17:08:53 +1200


I did not say NONMEM stops at random. Whether or not the stopping point
is associated with convergence or a successful covariance step appears
to be at random. The parameter values at the stopping point will
typically be negligibly different. Thus the stopping point is not at
random. You can easily observe this in your bootstrap runs. Compare the
parameter distribution for runs that converge with those that dont and
you will find there are negligible differences in the distributions.

I did not say that I ignore small changes in OFV but my decisions are
guided by the size of the change.

I do not waste much time modelling absorption. It rarely is of any
relevance to try to fit all the small details.

I dont see anything in the plot of SLOP vs EC50 that is not revealed by
R=0.93. If the covariance step ran you would see a similar number in the
correlation matrix of the estimate. It is quite common to find that the
estimates EC50 and Emax are highly correlated (I assume SLOP=EMAX/EC50).
It would also be common to find that the random effects of EMAX and EC50
are also correlated. That is expected given the limitations of most
pharmacodynamic designs. However, I would not simplify the model to a
linear model just because of these correlations. I would pay much more
attention to the change in OFV comparing an Emax with a linear model
plus whatever was known about the studied concentration range and the EC50.

I do agree that bootstraps can be helpful for calculating CIs on
secondary parameters.


Leonid Gibiansky wrote:
> Nick,
> Concerning "random stops at arbitrary point with arbitrary error" I
> was referring to your statement: "NONMEM VI will fail to converge or
> not complete the covariance step more or less at random"
> For OFV, you did not tell the entire story. If you would look only on
> OF, you would go for the absolute minimum of OF. If you ignore small
> changes, it means that you use some other diagnostic to (possibly)
> select a model with higher OFV (if the difference is not too high,
> within 5-10-20 units), preferring that model based on other signs
> (convergence? plots? number of parameters?). This is exactly what I
> was referring to when I mentioned that OF is just one of the criteria.
> One common example where OF is not the best guide is the modeling of
> absorption. You can spend weeks building progressively more and more
> complicated models of absorptions profiles (with parallel, sequential,
> time-dependent, M-time-modeled absorption etc.) with large drop in OF
> (that corresponds to minor improvement for a few patients), with no
> gain in predictive power of your primary parameters of interest, for
> example, steady-state exposure.
> To provide example of the bootstrap plot, I put it here:
> For 1000 bootstrap problems, parameter estimates were plotted versus
> parameter estimates. You can immediately see that SLOP and EC50 are
> strongly correlated while all other parameters are not correlated. CI
> and even correlation coefficient value do not tell the whole story
> about the model. You can get similar results from the covariance-step
> correlation matrix of parameter estimates but it requires simulations
> to visualize it as clearly as from bootstrap results. Advantage of
> bootstrap plots is that one can easily study correlations and
> variability of not only primary parameters (such as theta, omega,
> etc), but also relations between derived parameters.
> Leonid
> --------------------------------------
> Leonid Gibiansky, Ph.D.
> President, QuantPharm LLC
> web:
> e-mail: LGibiansky at
> tel: (301) 767 5566
> Nick Holford wrote:
>> Leonid,
>> I do not experience "random stops at arbitrary point with arbitrary
>> error" so I don't understand what your problem is.
>> The objective function is the primary metric of goodness of fit. I
>> agree it is possible to get drops in objective function that are
>> associated with unreasonable parameter estimates (typically an OMEGA
>> estimate). But I look at the parameter estimates after each run so
>> that I can detect this kind of problem. Part of the display of the
>> parameter estimates is the correlation of random effects if I am
>> using OMEGA BLOCK. This is also a weaker secondary tool. By exploring
>> different models I can get a feel for which parts of the model are
>> informative and which are not by looking at the change in OBJ. Small
>> (5-10) changes in OBJ are not of much interest. A change of OBJ of at
>> least 50 is usually needed to detect anything of practical importance.
>> I don't understand what you find of interest in the correlation of
>> bootstrap parameter estimates. This is really nothing more than you
>> would get from looking at the correlation matrix of the estimate from
>> the covariance step. High estimation correlations point to poor
>> estimability of the parameters but I think they are not very helpful
>> for pointing to ways to improve the model.
>> Nevertheless I can agree to disagree on our modelling art :-)
>> Nick
>> Leonid Gibiansky wrote:
>>> Nick,
>>> I think it is dangerous to rely heavily on the objective function
>>> (let alone on ONLY objective function) in the model development
>>> process. I am very surprised that you use it as the main diagnostic.
>>> If you think that nonmem randomly stops at arbitrary point with
>>> arbitrary error, how can you rely on the result of this random
>>> process as the main guide in the model development? I pay attention
>>> to the OF but only as one of the large toolbox of other diagnostics
>>> (most of them graphics). I routinely see examples when
>>> over-parametrized unstable models provide better objective function
>>> values, but this is not a sufficient reason to select those. If you
>>> reject them in favor of simpler and more stable models, you would
>>> see less random stops and more models with convergence and
>>> successful covariance steps.
>>> Even with bootstrap, I see the main real output of this procedure in
>>> revealing the correlation of the parameter estimates rather then in
>>> computation of CI. CI are less informative, while visualization of
>>> correlations may suggest ways to improve the model.
>>> Any way, it looks like there are at least the same number of
>>> modeling methods as modelers: fortunately for all of us, this is
>>> still art, not science; therefore, the time when everything will be
>>> done by the computers is not too close.
>>> Leonid
>>> --------------------------------------
>>> Leonid Gibiansky, Ph.D.
>>> President, QuantPharm LLC
>>> web:
>>> e-mail: LGibiansky at
>>> tel: (301) 767 5566
>>> Nick Holford wrote:
>>>> Mats, Leonid,
>>>> Thanks for your definitions. I think I prefer that provided by Mats
>>>> but he doesn't say what his test for goodness-of-fit might be.
>>>> Leonid already assumes that convergence/covariance are diagnostic
>>>> so it doesnt help at all with an independent definition of
>>>> overparameterization. Correlation of random effects is often a very
>>>> important part of a model -- especially for future predictions --
>>>> so I dont see that as a useful test -- unless you restrict it to
>>>> pathological values eg. |correlation|>0.9?. Even with very high
>>>> correlations I sometimes leave them in the model because setting
>>>> the covariance to zero often makes quite a big worsening of the OBJ.
>>>> My own view is that "overparameterization" is not a black and white
>>>> entity. Parameters can be estimated with decreasing degrees of
>>>> confidence depending on many things such as the design and the
>>>> adequacy of the model. Parameter confidence intervals (preferably
>>>> by bootstrap) are the way i would evaluate how well parameters are
>>>> estimated. I usually rely on OBJ changes alone during model
>>>> development with a VPC and boostrap confidence interval when I seem
>>>> to have extracted all I can from the data. The VPC and CIs may well
>>>> prompt further model development and the cycle continues.
>>>> Nick
>>>> Leonid Gibiansky wrote:
>>>>> Hi Nick,
>>>>> I am not sure how you build the models but I am using convergence,
>>>>> relative standard errors, correlation matrix of parameter
>>>>> estimates (reported by the covariance step), and correlation of
>>>>> random effects quite extensively when I decide whether I need
>>>>> extra compartments, extra random effects, nonlinearity in the
>>>>> model, etc. For me they are very useful as diagnostic of
>>>>> over-parameterization. This is the direct evidence (proof?) that
>>>>> they are useful :)
>>>>> For new modelers who are just starting to learn how to do it, or
>>>>> have limited experience, or have problems on the way, I would
>>>>> advise to pay careful attention to these issues since they often
>>>>> help me to detect problems. You seem to disagree with me; that is
>>>>> fine, I am not trying to impose on you or anybody else my way of
>>>>> doing the analysis. This is just an advise: you (and others) are
>>>>> free to use it or ignore it :)
>>>>> Thanks
>>>>> Leonid
>>>> Mats Karlsson wrote:
>>>>> <<I would say that if you can remove parameters/model components
>>>>> without
>>>>> detriment to goodness-of-fit then the model is overparameterized. >>

Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
mobile: +64 21 46 23 53
Received on Tue Aug 25 2009 - 01:08:53 EDT

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