Both the BFGS optimization of the objective function in the $EST step
and the inversion of the numerical
Hessian matrix in the $COV step involve a Cholesky decomposition of a
(hopefully) positive definite matrix whose rows and columns correspond
to the individual parameters. If the order of the parameters is
changed, the rows and columns of the matrix being decomposed are
permuted. The Cholesky decomposition is numerically sensitive to such
permutations since no pivoting is done in the standard
implementations. This sensitivity is particularly acute if the matrix
is poorly conditioned or, even worse, indefinite. So indeed it is to
be expected that changing the order of the parameters will affect the
results. For well conditioned problems, this effect is minimal. But
it is quite possible, for example, that an $EST or $COV step that
fails with one ordering will succeed with another.
------------------------------------------------------------------------
*From:* [email protected]
[mailto:[email protected]] *On Behalf Of *Sebastien Bihorel
*Sent:* Wednesday, June 23, 2010 9:53 AM
*To:* Nick Holford
*Cc:* [email protected]
*Subject:* Re: [NMusers] Unexpected influence of parameter order on
estimation results
I am aware of the issues associated numerical representation in
computer memory but I must say that it is more than a bit surprising
(disturbing) that the order of the parameters results in these
pseudo-random outcomes in NONMEM computations. As far as I know, this
is not the case in R, despite the same issues of numerical
representation. That being said, I don't want to re-start the old
debate on the value of the covariance step, but some people would
consider that the two versions of my model gave significantly
different results, simply based upon the objective function (at least
a 10-point difference) and the (lack of) success of the covariance step.
Nick Holford wrote:
Welcome to the world of 'real' numbers i.e. the limited representation
of numbers in computer arithmetic that leads to unexpected
(pseudo-random) results.
Both versions of your model are giving the same answer. The apparent
differences are due to pseudo-random chance.
[email protected]
<mailto:[email protected]> wrote:
Dear NMusers,
I always thought that the order in which parameters are declared in the
control stream has no impact on the estimation outcomes, but the following
results seem to contradict this.
The PK of drug X was modeled with a linear 3-compartment model using a
proportional residual variability model. Inter-individual variability was
estimated on elimination clearance and central volume of distribution. The
magnitude of residual variability was estimated using a THETA and a SIGMA
fixed to 1 as follows:
$ERROR
IPRED=F
CV=THETA(x)
W=CV*IPRED
Y=IPRED+W*EPS(1)
Two versions of this model were created with slight differences in the
order of declaration of the theta parameters: the theta used to estimate
the RV was basically moved from the third to the last position and the $PK
and the $ERROR blocks were updated accordingly.
Both models were run with NONMEM 6.2.0 on opensuse 11.1 (with the gfortran
compiler). One of the models converged successfully while the other
stopped at an early iteration and returned some estimation warnings and a
'S matrix singular' message. The strange thing is that gradients appears
identical until the 10th iteration, at which point the two models take
different search paths (see below).
I would be very interested to know the opinion of the group on this
puzzling result.
Thanks
Sebastien
-----------------------------------------------------------------------------------
Model 1 (RV theta in the 1st position)
1
MONITORING OF SEARCH:
0ITERATION NO.: 0 OBJECTIVE VALUE: 0.25863E+04 NO. OF FUNC.
EVALS.: 9
CUMULATIVE NO. OF FUNC. EVALS.: 9
PARAMETER: 0.1000E+00 0.1000E+00 0.1000E+00 0.1000E+00 0.1000E+00
0.1000E+00
0.1000E+00 0.1000E+00 0.1000E+00
GRADIENT: -0.9523E+02 0.3303E+03 -0.2730E+04 0.6103E+03 -0.1044E+04
0.2780E+03
-0.6146E+03 -0.9406E+02 -0.3231E+03
0ITERATION NO.: 5 OBJECTIVE VALUE: 0.10396E+04 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 59
PARAMETER: 0.1919E+01 -0.5699E+00 0.4872E+00 -0.1661E+01 0.1040E+01
-0.3113E+00
-0.8783E-01 0.1274E+01 -0.6898E-01
GRADIENT: 0.1511E+02 -0.2036E+02 -0.3532E+03 -0.3176E+02 -0.4009E+02
-0.9733E+02
0.4026E+02 -0.2917E+02 -0.4199E+02
0ITERATION NO.: 10 OBJECTIVE VALUE: 0.88163E+03 NO. OF FUNC.
EVALS.:12
CUMULATIVE NO. OF FUNC. EVALS.: 127
PARAMETER: 0.1643E+01 -0.4360E+00 0.9125E+00 -0.1429E+01 0.1009E+01
0.2690E+00
0.1835E+00 0.1894E+01 -0.3302E+00
GRADIENT: 0.7310E+01 0.2031E+02 -0.3379E+02 0.1896E+02 -0.6428E+02
-0.5519E+01
0.2288E+02 0.8420E+01 -0.3893E+02
0ITERATION NO.: 15 OBJECTIVE VALUE: 0.85825E+03 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 179
PARAMETER: 0.7899E+00 -0.5002E+00 0.1014E+01 -0.1314E+01 0.1104E+01
-0.4181E-01
-0.2654E+00 0.1545E+01 0.3062E+00
GRADIENT: 0.8389E+01 0.8285E+01 0.5404E+01 0.2172E+02 -0.9433E+01
-0.2633E+02
0.7059E+01 0.2790E+01 -0.1023E+01
0ITERATION NO.: 20 OBJECTIVE VALUE: 0.85807E+03 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 275
PARAMETER: 0.7816E+00 -0.5006E+00 0.1013E+01 -0.1314E+01 0.1104E+01
-0.4133E-01
-0.2649E+00 0.1477E+01 0.3305E+00
GRADIENT: 0.9405E+01 0.7846E+01 0.5605E+01 0.2021E+02 -0.9587E+01
-0.2640E+02
0.6285E+01 0.2135E-01 -0.1198E-02
0ITERATION NO.: 25 OBJECTIVE VALUE: 0.84968E+03 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 344
PARAMETER: -0.2358E+00 -0.5888E+00 0.1008E+01 -0.1312E+01 0.1114E+01
0.8588E-01
-0.2390E+00 0.9860E+00 0.3144E+00
GRADIENT: -0.2043E+01 -0.4198E+01 -0.1418E+00 0.1786E+02 0.1856E+00
-0.2873E+01
-0.6265E+01 0.8535E+00 -0.2022E+00
0ITERATION NO.: 30 OBJECTIVE VALUE: 0.84767E+03 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 396
PARAMETER: -0.9020E-02 -0.5500E+00 0.1022E+01 -0.1312E+01 0.1258E+01
0.3517E+00
0.7877E-01 0.9016E+00 0.3574E+00
GRADIENT: -0.1566E+00 -0.1616E+00 -0.3990E+00 0.2010E+02 0.5696E+00
-0.5633E+00
0.4708E+00 -0.3923E+00 0.1648E-01
0ITERATION NO.: 35 OBJECTIVE VALUE: 0.84766E+03 NO. OF FUNC.
EVALS.:17
CUMULATIVE NO. OF FUNC. EVALS.: 469
PARAMETER: -0.2786E-02 -0.5413E+00 0.1025E+01 -0.1312E+01 0.1252E+01
0.3551E+00
0.7957E-01 0.9105E+00 0.3546E+00
GRADIENT: -0.1860E-02 0.2683E-01 -0.1566E-01 0.1869E+02 0.3775E-02
-0.6239E-02
0.5749E-02 0.1541E-01 -0.6128E-03
0ITERATION NO.: 40 OBJECTIVE VALUE: 0.84590E+03 NO. OF FUNC.
EVALS.:17
CUMULATIVE NO. OF FUNC. EVALS.: 568
PARAMETER: -0.1454E+00 -0.5904E+00 0.9921E+00 -0.1483E+01 0.1287E+01
0.2158E+00
0.8236E-01 0.9660E+00 0.3228E+00
GRADIENT: -0.7465E-01 -0.3447E+00 -0.4737E+00 0.2200E+01 0.2029E+01
-0.1106E+01
-0.5923E+00 -0.8064E-01 0.1356E+00
0ITERATION NO.: 45 OBJECTIVE VALUE: 0.84585E+03 NO. OF FUNC.
EVALS.:14
CUMULATIVE NO. OF FUNC. EVALS.: 650
PARAMETER: -0.1440E+00 -0.5825E+00 0.9933E+00 -0.1493E+01 0.1261E+01
0.2183E+00
0.8561E-01 0.9659E+00 0.3136E+00
GRADIENT: -0.5281E-04 -0.5273E-03 0.1878E-03 -0.9052E-03 0.1615E-03
0.1004E-02
-0.8575E-03 -0.1004E-03 -0.1273E-03
0MINIMIZATION SUCCESSFUL
NO. OF FUNCTION EVALUATIONS USED: 650
NO. OF SIG. DIGITS IN FINAL EST.: 4.7
ETABAR IS THE ARITHMETIC MEAN OF THE ETA-ESTIMATES,
AND THE P-VALUE IS GIVEN FOR THE NULL HYPOTHESIS THAT THE TRUE MEAN IS 0.
ETABAR: -0.46E-02 0.39E-02 0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00
SE: 0.21E+00 0.91E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00
P VAL.: 0.98E+00 0.97E+00 0.10E+01 0.10E+01 0.10E+01 0.10E+01
0.10E+01
----------------------------------------------------------------------------------
Model 2 (RV theta in the 7th position)
1
MONITORING OF SEARCH:
0ITERATION NO.: 0 OBJECTIVE VALUE: 0.25863E+04 NO. OF FUNC.
EVALS.: 9
CUMULATIVE NO. OF FUNC. EVALS.: 9
PARAMETER: 0.1000E+00 0.1000E+00 0.1000E+00 0.1000E+00 0.1000E+00
0.1000E+00
0.1000E+00 0.1000E+00 0.1000E+00
GRADIENT: -0.9523E+02 0.3303E+03 0.6103E+03 -0.1044E+04 0.2780E+03
-0.6146E+03
-0.2730E+04 -0.9406E+02 -0.3231E+03
0ITERATION NO.: 5 OBJECTIVE VALUE: 0.10396E+04 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 59
PARAMETER: 0.1919E+01 -0.5699E+00 -0.1661E+01 0.1040E+01 -0.3113E+00
-0.8783E-01
0.4872E+00 0.1274E+01 -0.6898E-01
GRADIENT: 0.1511E+02 -0.2036E+02 -0.3176E+02 -0.4009E+02 -0.9733E+02
0.4026E+02
-0.3532E+03 -0.2917E+02 -0.4199E+02
0ITERATION NO.: 10 OBJECTIVE VALUE: 0.88167E+03 NO. OF FUNC.
EVALS.:12
CUMULATIVE NO. OF FUNC. EVALS.: 127
PARAMETER: 0.1642E+01 -0.4358E+00 -0.1429E+01 0.1009E+01 0.2691E+00
0.1839E+00
0.9126E+00 0.1895E+01 -0.3306E+00
GRADIENT: 0.7304E+01 0.2046E+02 0.1895E+02 -0.6432E+02 -0.5543E+01
0.2291E+02
-0.3381E+02 0.8433E+01 -0.3897E+02
0ITERATION NO.: 15 OBJECTIVE VALUE: 0.85827E+03 NO. OF FUNC.
EVALS.:10
CUMULATIVE NO. OF FUNC. EVALS.: 179
PARAMETER: 0.8105E+00 -0.5381E+00 -0.1334E+01 0.1062E+01 -0.1712E-02
-0.2072E+00
0.1000E+01 0.1570E+01 0.2716E+00
GRADIENT: 0.8146E+01 -0.2087E+01 0.2338E+02 -0.2616E+02 -0.2205E+02
0.1064E+02
0.4212E+01 0.3944E+01 -0.2221E+01
0ITERATION NO.: 20 OBJECTIVE VALUE: 0.85775E+03 NO. OF FUNC.
EVALS.:39
CUMULATIVE NO. OF FUNC. EVALS.: 317 RESET HESSIAN, TYPE I
PARAMETER: 0.7924E+00 -0.5386E+00 -0.1335E+01 0.1073E+01 -0.2161E-02
-0.2085E+00
0.1001E+01 0.1558E+01 0.2793E+00
GRADIENT: 0.8121E+01 -0.1709E+01 0.2187E+02 -0.2257E+02 -0.2142E+02
0.9023E+01
0.4553E+01 0.3690E+01 -0.1895E+01
0ITERATION NO.: 24 OBJECTIVE VALUE: 0.85768E+03 NO. OF FUNC.
EVALS.:24
CUMULATIVE NO. OF FUNC. EVALS.: 386
PARAMETER: 0.7924E+00 -0.5386E+00 -0.1335E+01 0.1076E+01 -0.2161E-02
-0.2085E+00
0.1001E+01 0.1556E+01 0.2805E+00
GRADIENT: -0.7820E+04 -0.5761E+04 -0.4623E+04 0.5748E+04 0.6202E+05
-0.2974E+05
0.3099E+04 0.1998E+04 0.2212E+05
0MINIMIZATION SUCCESSFUL
HOWEVER, PROBLEMS OCCURRED WITH THE MINIMIZATION.
REGARD THE RESULTS OF THE ESTIMATION STEP CAREFULLY, AND ACCEPT THEM ONLY
AFTER CHECKING THAT THE COVARIANCE STEP PRODUCES REASONABLE OUTPUT.
NO. OF FUNCTION EVALUATIONS USED: 386
NO. OF SIG. DIGITS IN FINAL EST.: 3.3
ETABAR IS THE ARITHMETIC MEAN OF THE ETA-ESTIMATES,
AND THE P-VALUE IS GIVEN FOR THE NULL HYPOTHESIS THAT THE TRUE MEAN IS 0.
ETABAR: -0.61E+00 0.15E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00
SE: 0.31E+00 0.92E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00
P VAL.: 0.48E-01 0.87E+00 0.10E+01 0.10E+01 0.10E+01 0.10E+01
0.10E+01
0S MATRIX ALGORITHMICALLY SINGULAR
0S MATRIX IS OUTPUT
0INVERSE COVARIANCE MATRIX SET TO RS*R, WHERE S* IS A PSEUDO INVERSE OF S
--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
email: [email protected] <mailto:[email protected]>
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
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