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Re: [NMusers] eigenvalues

From: Pavel Belo <nonmem_at_optonline.net>
Date: Fri, 06 Nov 2015 11:05:24 -0500 (EST)

NONMEM demonstrated very large differences in objective function when
variability or correlations were added or removed.  Monolix demonstrat=
ed
close-to-insignificant differences.  When differences in software star=
t
to affect important conclusions it becomes interesting.  It feels like=
 
we need to make sure we report the most meaningful results. 
 
NONMEM runs as if the covariance matrix is more a byproduct than an
essential part of the optimization.  Monolix runs as if the covariance=
 
matrix an essential part of the optimization.  NONMEM teachers recomme=
nd
to try a full covariance matrix.  Monolix teachers recommend to be
careful and try a diagonal matrix first. 
 
Thanks,

Pavel 
 
On Fri, Nov 06, 2015 at 08:42 AM, Pavel Belo wrote:
 
 

 
 Hello Jeroen,
 
Thank you for your response.  It was a practical question.  I und=
erstand
the theory.  What is the reason different packages show such different=
 
results and present eigenvalues  differently?  What is the best w=
ay? 
 
NONMEM demonstrated much larger max/min values but did not give warning
messages about non-positive defined matrix.  The runs were stable.=
Runs
became unstable only when simulated annealing was used;  instability=
 
kicked in at the moment when NONMEM stopped simulated annealing; so I=
 
had to remove simulated annealing.  Monolix sometimes gave non-po=
sitive
defined matrix stopping optimization in the middle;  sometime it =
became
unstable in the middle with or without simulated annealing.   
 
I do not take sides.  I just try to understand it.  As max/min is=
 
frequently reported in BLAs, it is nice to understand what we report and
why it can be so different across different packages. 
 
Thanks,
Pavel
 
On Thu, Nov 05, 2015 at 05:14 PM, Jeroen Elassaiss-Schaap (PD-value
B.V.) wrote:
 
 

     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







http://pd-value.com <http://pd-value.com>
jeroen_at_pd-value.com <mailto:jeroen_at_pd-value.com>
_at_PD_value
+31 6 23118438
-- More value out of your data!



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 th=
e
         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 APPROXIMATIO=
N
         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 ra=
tio 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 Fri Nov 06 2015 - 11:05:24 EST

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