# Re: ETAs & SIGMA in external validation

From: Faelens, Ruben <Ruben.Faelens>
Date: Fri, 13 Apr 2018 17:40:57 +0000

Hi Tingjie,

I used Nelder-Mead because it is the default method in R optim(). No other reasoning.

With regards to OFIM: the inverse of the hessian of the likelihood at the optimum ETA is an estimate for the standard error of this ETA estimate. This is called the Observed Fisher Information Matrix.
If you will forgive me the childish language, this can be explained intuitively: the second derivative describes how 'pointy' the OFV is. It shows how much the objective function changes when you 'jiggle' around the ETA parameters.
A very pointy OFV means a high change in OFV for different estimates, and therefore high certainty and low residual error.
An almost flat OFV means different estimates give similar OFV (are equally likely), and therefore a low certainty and high residual error.
Subjects with no information will have ETA =0 as the maximum likelihood estimate (shrinkage), but the uncertainty will be equal to population IIV.
I forgot the exact formulas though, you can find it in literature discussing d-optimality.

In my view, taking uncertainty into account on posthoc estimates is an elegant solution to sparse profiles, but I have rarely seen it applied in practice. I am not entirely certain whether the asymptotic convergence of OFIM to the residual error applies for ETA estimates either, especially in the case of sparse sampling. Which is why I searched for feedback from the list.

Anyway, the above is largely an academic interest anyway. Good luck with your project!

Please excuse my brevity, this was sent from a mobile device

________________________________
From: Tingjie Guo <iam
Sent: Friday, April 13, 2018 5:20:40 PM
To: Jakob Ribbing
Cc: Faelens, Ruben (Belgium); nmusers
Subject: Re: [NMusers] ETAs & SIGMA in external validation

o raise an extended question: if the model contains one covariate, the values of which from external data make parameters negative, what would be the optimal solution for this?

​Met vriendelijke groet
,
T
​G

On Tue, Apr 10, 2018 at 3:19 PM, Jakob Ribbing <jakob.ribbing
Hi Ruben,

I think I misread Tingjies original posting as taking ABS(ETA), whereas his initial attempt was actually ABS(1+ETA), which is less problematic.
The latter would not bias simulations much if IIV is e.g. 30% CV, agreed.

However, as Tingjies is mainly interested in estimation, I believe that without the ABS-correction, no subject will have the EBE at ETA <= -1 for a parameter that could not be <=0.
Unless possibly in a subject which is a) uninformative on that parameter and b) where the eta is also part of an omega-block - a scenario which seems unlikely to me, but may occur in theory.

Implementing the ABS-korrection ETA=-1.2 would give the same solution (parameter value) as ETA=-0.8, but at a higher OFV for that subject.
It seems to me, if negative parameter values are only a problem in the eta search for the EBE, whereas the EBE for individual parameters are always positive, then it should be more straightforward to use FOCE, with the addition e.g.:
IF(PARA.LT.0.001) PARA=0.001
Probably, no subject will have such a low individual parameter value, when looking into the table output?
If there are any such subjects I would look for errors in the data set and nonmem code (as outlined in my initial reply).

The above concerns estimation.
In simulation (unless %CV is low), we may get a fraction of subject with PARA=0.001, which may be an unreasonably low parameter value.
Whether that is acceptable or not depends on the objectives and in this case there was no need for simulations even for model evaluation (?), so I will not elaborate further.

Cheers

Jakob

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