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Re: Unexpected influence of parameter order on estimation results

From: Sebastien Bihorel <Sebastien.Bihorel>
Date: Mon, 28 Jun 2010 09:59:11 -0400

Thanks Bob,

Your explanation definitively sheds some light on those number differences.

Sebastien

Bob Leary wrote:
>
> 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:* owner-nmusers
> [mailto:owner-nmusers
> *Sent:* Wednesday, June 23, 2010 9:53 AM
> *To:* Nick Holford
> *Cc:* nmusers
> *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.
>
>
>
> Sebastien.Bihorel
> <mailto:Sebastien.Bihorel
>
> 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: n.holford
> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
> _________________________________________________________________
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Received on Mon Jun 28 2010 - 09:59:11 EDT

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