**From: "Ian J. Gowrie" <i.j.gowrie@city.ac.uk>
Subject: minimum objective function**

Date: Wed, 02 Dec 1998 17:19:24 +0000

Dear all,

I've a simple(?) question concerning the Minimum Objective Function.

Say I have some models and during model building I get the following results:

THETA(1) only, objective function = OF1

THETA(2) only, objective function = OF2

THETA(1) and THETA(2). objective function = OF3

Must the following statements be true?

OF3 < OF2

OF3 < OF1

I ask this, naturally, because I have a situation where one of these is not true, and I am interested in what the causes may be.

Thanks for your time

Ian J Gowrie

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Centre for Measurement in Medicine

City University

London

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**From: KENNETH.G.KOWALSKI@monsanto.com
Subject: Re: minimum objective function**
Date: 02 Dec 1998 14:25:47 -0600

Ian,

I'm assuming that you checked your models and that the models are hierarchical. That is, for the model with both THETA(1) and THETA(2) you can set either to some null value (typically 0 or 1) such that it reduces to the THETA(1) or THETA(2) alone model. If that is the case then it should be that OF3<OF1 and OF3<OF2. If this is not being met, then you may not have achieved the global optimum (minimum objective function) for the model with both THETA(1) and THETA(2) in the model. Perhaps your model is overparameterized making it difficult to achieve the optimum. Let's suppose that you found OF2<OF3 yet they're supposed to be hierachical. You should be able to use final estimates obtained from the THETA(2) alone model and set THETA(1) to its null value for absence in the model with both THETA(1) and THETA(2) and have the same objective function value (OF2). You might start from there with perhaps only slightly changing the starting value for THETA(1) from its null value and see if you can get it to iterate and converge such that OF3<OF2. Of course if that happens, there is still no guarantee that you have achieved the absolute minimum.

Good luck!

Ken

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Subject: minimum objective function

I assume you mean that you have a model & data that are fixed, and that OF1 is the min OF when theta(2) is fixed to 0 but all other parameters are free; OF2 is the min OF when theta(1) is fixed to zero but all other parameters are free; and OF3 is the min OF when all parameters are free.

If that is so, then indeed it must be the case that OF3 is less than either OF1 or OF2. If you think that this been violated, look carefully to be sure there is no abnormal termination, or that constrants on parameters have not also changed, or in the OF3 case, one of either theta(1) or theta(2) is going towards zero, and is bounded at zero, or some such situation that makes the model/data change from case to case other than just through fixing certain parameters.

LBS.

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Subject: RE: minimum objective function

Dear Ian,

Everything depends on your data and your model. If there is a high intraindividual variability and your model is complex (both are typical in case of PD data) you may observe very strange behavior of NONMEM.

Best regards

Vladimir