Re: Unexpected influence of parameter order on estimation results

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:* [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 > _________________________________________________________________ >