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Re: What does convergence/covariance show?

From: Nick Holford <n.holford>
Date: Wed, 26 Aug 2009 09:02:26 +1200


I seem to be having trouble explaining myself these days. I dont usually
have to start every email with "I did not say" but here we go again!

I did not say that I recognize models are overparameterised. That
implies a dichotomy between well parameterised and overparameterised. I
tried to say earlier that there is a continuous scale of 'goodness' of
estimation (usually quantified by the standard error). So I dont accept
the notion of a model being overparameterised or not when one is talking
about estimability of identifiable parameters.

Similarly there is no dichotomy between the responses that a linear and
an Emax pharmacodynamic model are trying to describe. Pharmacology and
biology tell us that the linear model is just an approximation to an
Emax model. If the OFV drops 'reasonably' and at least some of the concs
are close to or greater than the EC50 with an Emax model then I would
stick with it. It doesn't matter to me that the Emax and EC50 are
individually poorly estimated (it is rather rare to be interested in the
parameter by itself).

The usual purpose of the model is to predict the effect over a range of
concentrations. If you choose a linear model because your subjective
impression is that the model is "overparameterised" due to large
standard errors then you can be certain that any extrapolation will
overpredict the size of the effect. If you choose an Emax model you may
still have a biased prediction but it will be a better prediction than
one from a linear model. In the interpolation range of predictions the
Emax model will still do better. I cannot see how it can do worse than
the linear model (assuming the model passes other tests of plausibility
and the VPC looks OK).

Thanks for your 2c!


Ken Kowalski wrote:
> Nick,
> It sounds like you do recognize that models are often over-parameterized by
> your statements:
> " 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."
> When EC50 and Emax are highly correlated I think you will find that a
> simplified linear model will fit the data just as well with no real impact
> on goodness-of-fit (e.g., OFV). If we only observe concentrations in the
> linear range of an Emax curve because of a poor design then it is no
> surprise that a linear model may perform as well as an Emax model within the
> range of our data. If the design is so poor in information content
> regarding the Emax relationship because of too narrow a range of
> concentrations this will indeed lead to convergence and COV step failures in
> fitting the Emax model.
> Your statement that you would be unwilling to accept the linear model in
> this setting really speaks to the plight of the mechanistic modeler. It is
> important to note that an over-parameterized model does not mean that the
> model is miss-specified. A model can be correctly specified but still be
> over-parameterized because the data/design simply will not support
> estimation of all the parameters in the correctly specified model. The
> mechanistic modeler who has a strong biological prior favoring the more
> complex model is reluctant to accept a simplified model that he/she knows
> has to be wrong (e.g., we would not expect that the linear model would hold
> up at considerably higher concentrations than those observed in the existing
> data). The problem with accepting the more complex model in this setting is
> that we can't really trust the estimates we get (when the model has
> convergence difficulties and COV step failures as a result of
> over-parameterization) because there may be an infinite set of solutions to
> the parameters that give the same goodness-of-fit (i.e., a very flat
> likelihood surface). You can do all the bootstrapping you want but it is
> not a panacea for the deficiencies of a poor design.
> While I like to fit mechanistic models just as much as the next guy, I also
> like my models to be stable (not over-parameterized). In this setting, the
> pragmatist in me would accept the simpler model, acknowledge the limitations
> of the design and model, and I would be very cautious not to extrapolate my
> model too far from the range of my existing data. More importantly, I would
> advocate improving the situation by designing a better study so that we can
> get the information we need to support a more appropriate model that will
> put us in a better position to extrapolate to new experimental conditions.
> We review the COV step output (looking for high correlations such as between
> the estimates of EC50 and Emax) and fit simpler models not because we prefer
> simpler models per se, but because we want to fully understand the
> limitations of our design. Of course this simple example of a poor design
> with too narrow a concentration and/or dose range to estimate the Emax
> relationship can be easily uncovered in a simple plot of the data, however,
> for more complex models the nature of the over-parameterization and the
> limitations of the design can be harder to detect which is why we need a
> variety of strategies and diagnostics including plots, COV step output,
> fitting alternative simpler models, etc. to fully understand these
> limitations.
> Just my 2 cents. :)
> Ken
> -----Original Message-----
> From: owner-nmusers
> Behalf Of Nick Holford
> Sent: Tuesday, August 25, 2009 1:09 AM
> To: nmusers
> Subject: Re: [NMusers] What does convergence/covariance show?
> Leonid,
> 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.
> Nick
> 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 - 17:02:26 EDT

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