Re: [NMusers] eigenvalues

From: Jeroen Elassaiss-Schaap (PD-value B.V.) <"Jeroen>
Date: Thu, 5 Nov 2015 23:14:03 +0100

Hi Pavel,

Principal component analysis can be validly performed on any matrix, and
it is just a matter of convention that the eigenvalue ratios of min/max
of the total covariance matrix of estimation are reported as the
condition number for a given model. This as a metric of how easily the
dimensionality of estimators could be reduced.

The idea behind the separation of eigenvalues, as you show here for your
model in Monolix, is actually attractive, because the off-diagonal
elements do reduce the freedom of the described variance rather than
increasing it. Furthermore they are the byproduct of sampling methods
like SAEM, not so much the result of separate estimation. Two reasons to
separate them.

The separation of diagonal variance components and PK parameters as you
note is less obvious to me, although I am pretty sure there will be a
good rationale for that in the realm of sampling approaches (tighter
linkage?).

Even though the off-diagonal elements are associated with a decent
condition number, it is still larger than the "PK" block, assuming the
blocks are of comparable size. In other to better compare the results my
suggestion would be to break up the nonmem covariance matrix (as was
done for Monolix) in blocks of structural, diagonal and off-diagonal
elements (throwing away a large remainder), and calculate the condition
number on each matrix. Than you are comparing apples to apples, enabling
a more straightforward discussion of the differences.

Hope this helps,
Jeroen


 

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On 11/04/2015 05:55 PM, Pavel Belo wrote:
> Hello NONMEM Users,
>
> I try to make sense of the results and one of the ways to do it is to
> compare the same or similar models across software packages. 5x5 full
> omega matrix is used because it was prohibitive to remove some
> insignificant correlations from the matrix without removing
> significant correlations (All recommended ways to do it were tested.
> Diagonal omega was also tested, of course). Adding correlations has
> little effect on PK parameters, but it has some effect on simulations.
>
> NONMEM provides all eigenvalues in one pocket. Here is an example.
> ************************************************************************************************************************
> ********************
> ********************
> ******************** STOCHASTIC APPROXIMATION
> EXPECTATION-MAXIMIZATION ********************
> ******************** EIGENVALUES OF COR MATRIX OF
> ESTIMATE (S) ********************
> ********************
> ********************
> ************************************************************************************************************************
>
> 1 2 3 4 5
> 6 7 8 9 10 11 12
> 13 14 15 16 17
> 18 19 20 21 22 23
>
> 3.36E-05 5.69E-03 3.40E-02 6.32E-02 9.19E-02 1.24E-01
> 1.53E-01 2.79E-01 3.20E-01 4.32E-01 5.74E-01 6.45E-01
> 7.25E-01 7.67E-01 9.73E-01 1.08E+00 1.42E+00 1.63E+00
> 1.86E+00 2.14E+00 2.31E+00 3.12E+00 4.26E+00
>
> Monolix provides them in 3 pockets:
>
> PK parameters: Eigenvalues (min, max, max/min): 0.22 2 9.2
> OMEGA (diagonal) and SIGMA: Eigenvalues (min, max, max/min): 0.66
> 1.5 2.2
> OMEGA (correlations): Eigenvalues (min, max, max/min): 0.097 2.5 25
>
> Even though the results look similar, eigenvalues look different.
> Taking into account that max/min ratio is frequently reported, it is
> important to understand the difference. It almost look like different
> sets of parameters are estimated separately in the Monolix example,
> which most likely is not the case. Even if we combine all eigenvalues
> in one pocket, max/min looks good. It is impressive that max/min
> ratio for OMEGA correlations may look OK even though there are small
> correlations such as -0.0921, SE=0.064, RSE=70%.
>
> What is the best way to report estimate and report max/min ratios?
>
> Take care,
> Pavel


Received on Thu Nov 05 2015 - 17:14:03 EST

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