Internet Publikation  für Allgemeine und Integrative Psychotherapie
(ISSN 1430-6972)
IP-GIPT DAS=23.10.2002  Letzte Änderung: 14.05.15
Impressum: Diplom-PsychologInnen Irmgard Rathsmann-Sponsel und Dr. phil. Rudolf Sponsel
Stubenlohstr. 20     D-91052 Erlangen  Mail:Sekretariat@sgipt.org
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Willkommen in der Abteilung Wissenschaftstheorie, Methodologie und Statistisch-Mathematische Methoden in der Allgemeinen und Integrativen Psychologie, Psychodiagnostik und Psychotherapiehier zu Matrizen in der Psychologie und Psychotherapie:

Dokumentation Rundungsfehler & Kollinearität
am THURSTONEschen Trapezoid- Beispiel
Zur korrigierten Version (23.10.2)

von Rudolf Sponsel, Erlangen [Quelle Kap. 7.3]
Internet-Erstausgabe 15.6.2001, letzte Änderung 23.10.2002

Es wird gezeigt, daß der Verlust der Positiven Definitheit der Korrelationsmatrix des Thurstone'schenein Trapezoids ein Ergebnis des Zusammenwirkens von Kollinearität und Rundungsfehler ist.

Permutation und Determinanten Graphik 4 k

Inhalts-Überblick
0.  Ergebnis: Zusammenfassung - Abstract
1.  Einführung
2.  TRAPEZOID Illustration
3.  Übersicht der Ergebnisse  & Bibliografische Belege
4.  Detail Analysen
     (1)  Values measured with 2-digit input accuray calculed
     (2)  Values are computed and not measured with 17-digit-input-accuracy  & calculed
     (3)  Values are computed and not measured with 4-digit-input-accuracy calculed
     (4)  Values are computed and not measured with 3-digit-input-accuracy read
     (5)  Values are computed and not measured with 2-digit-input-accuracy read
     (6)  Values measured with 2-digit input accuracy read
5.  Überblick der berechneten Trapezoid-Parameter
6.  OMIKRON-Basic-Programm zur Berechnung der Parameter des THURSTONEschen Trapezoids
7. Querverweise


0.  Ergebnis: Zusammenfassung - Abstract
 
Die  Parameter des THURSTONEsche Trapezoid Beispiels werden nicht wie bei THURSTONE empirisch gemessen, sondern mit  17-stelliger-Genaugikeit berechnet. Das Beispiel THURSTONEs hat 4 negative Eigenwerte und seine positive Definitheit verloren. Das mit 17-stelliger Genauigkeit berechnete Trapezoid und die mit 17-stelliger Genauigkeit berechneten Korrelationskoeffizienten produzieren keinen  negativen Eigenwert  mehr, gewinnen die positive Definitheit also zurück; die numerische Instabilität nimmt ansonsten - paradoxerweise? - eher zu: Rundet man die 17-stellig genauen Korrelationskoeffizienten beim Einlesen auf, so werden beim Runden auf 3 oder 4 Stellen je ein negativer, beim Runden auf nur zwei Stellen wieder 4 negative Eigenwerte  produziert. Das ist ein Beweis dafür, daß  der Verlust der Positiven Definitheit hier auf das Zusammenwirken von Kollinearität und Rundungsfehlern zurückgeführt werden kann.

1. Einführung
Thurstone, der Begründer der multiplen Faktorenanalyse, hat aus didaktischen und argumentativen Gründen einige Beispiele ersonnen, um die Idee der Faktorenanalyse plausibel und anschaulich zu begründen. Eines seiner berühmtesten Beispiele ist die Messung (nicht Berechnung, das haben wir hier ergänzend gemacht) der verschiedenen abgeleiteten Parameter eines Trapezoids. Die Messung und Nicht-Berechnung sollte die empirische Situation simulieren. Die verschiedenen Messungen und Vorgaben sorgen dann auch dafür, daß es genügend unterschiedliche Werte und damit auch unterschiedliche Korrelationskoeffizienten gibt. Das Trapezoid wird nun durch die vier Parameter a,b,c,h vollständig bestimmt. Alle anderen Größen können daraus abgeleitet werden. Es ist daher unmittelbar plausibel, daß die Korrelationsmatrix, die sich aus den den vier Paremetern und 12 abgeleiteten Werten ergibt, sich aus vier Faktoren aufbauen und rekonstruieren lassen sollte. Darum geht es uns aber hier nicht (das finden Sie hier), sondern nur um die Demonstration wie Kollinearität und Rundungsfehler zusammen spielen.
 

2.  TRAPEZOID Illustration Figure 5

Original-Text Thurstone
"trapezoid population
In previous studies of factorial theory it has been found useful to illustrate the principles by means of a population of simple physical objects or geometrical figures. The box population was used to illustrate three correlated factors and their physical interpretation. In the present case we want four factors in the first-order domain, which, by their correlations of unit rank, determine a general second-order factor. The correlations of three variables can nearly always be accounted for by a single factor, and hence it seems better to choose a four-dimensional system in which the existence of a second-order general factor is more clearly indicated by the unit rank of the correlations of four primary factors. For the present physical illustration we have chosen a population of trapezoids whose shapes are determined by four primary parameters or factors.
The measurements on the trapezoids are indicated in Figure 5. The base line is bisected, and the length of esch half is denoted by the parameter c. An ordinate is erected at this mid-point, and its length is h. This ordinate divides the top section into two parts, which are denoted a and b as shown. These four parameters, a, b, c, and h, completely determine the figure. The test battery was represented by sixteen measurements, wEich are drawn in the figure. The parameters a, b, c, and h are given code numbers 1, 2, 3, and 4, respectively. Variables (12) and (13) are the two areas as shown. The sum of (12) and (13) equals the total area of the trapezoid. In general, each of these measurements is a function of two or three of the parameters but not of all four of them, and hence we should expect a simple structure in this [p. 428] set of measurements. There is a rather general impression that a simple structure is necessarily confined to the positive manifold. In order to offset this impression we included here three additional measures, which extend the simple structure beyond the positive manifold. These three additional measures are as follows:

                          14 = (1)/(2) = a/b, 15 = (2)/(3) = b/c, 16 = (1)/(3) = a/c .

These three measures will necessarily introduce negative saturations on some of the basic factors.

In Table 6 we have a list of dimensions for a set of thirty-two trapezoids. These will constitute the trapezoid population. Each figure was drawn to 

FIGURE 5 

scale on cross-section paper, and then the sixteen measurements were made on each figure. These constituted the test scores for the present example. In setting up the dimensions of Table 6 the numbers were not distributed entirely at random. To do so would tend to make the correlations between the four basic parameters, a, b, c, and h, approach zero, and this would lead to an orthogonal simple structure in which there would be no provocation to investigate a second-order domain. The manner in which the generating conditions of the objects determine the factorial results will be discussed in a later section. Table 6 was so constructed that, in addition to the four basic parameters, there was also a size factor, which functioned as a second-order parameter in determining correlation between the four primary factors in generating the figures.
The product-moment correlations between the sixteen measurements for the thirty-two objects were computed, and these are listed in Table 7. This [p. 429] correlation matrix was factored by the group centroid method, and the resulting factor matrix F is shown in Table 8. The fourth-factor residuals are listed in Table 9, which indicates that the residuals are vanishingly small." 

3.  Übersicht der Ergebnisse und bibliographische Hinweise

THURSTONE, L. L.  (USA: University Of Chicago)  "Multiple Factor Analysis"  Chicago 1947,  p.427-436  "A trapezoid population", p. 431 Table 7 'Correlation Matrix'  a n d  THURSTONE, L. L.  "SECOND-ORDER FACTORS"  Psychometrika 9.2,1944, p.96 Table 7 Correlation Matrix. Bemerkung: Die Matrix enthält zwei Druckfehler: r14,16 und r15,16 müssen positiv sein.

(1=12a) TH43116.K16  raw scores measured with 2-digit input accuray read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16    0  --4    2992      6.36 D-21    1.43 D-13   79208   2D-3(9)  -1(-1)

(2=12b) TH429R17.D16  raw scores are computed and not measured with 17-digit-input-accuracy calculated and read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16   0    -     343457    1.08 D-29    8.94 D-30   16479    0(9)   6D-3(8)

(3=12c) TH429R4.D16  raw scores are computed and not measured with 4-digit-input-accuracy read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16   0  --1      331205    -1.86 D-29  3.53 D-29   33770    0(9)   -1(-1)

(4=12d) TH429R3.D16 raw scores are computed and not measured with 3-digit-input-accuracy read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16   0   --1    257081    -5.84 D-28   5.56 D-30  10090     0(9)   -1(-1)

(5=12e) TH429R2.D16 raw scores are computed and not measured with 2-digit-input-accuracy read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16   0   --4    7450      4.83 D-22    6.20 D-16   16479   1D-3(9) -1(-1)

(6=12f) THPMF16.K16  raw scores measured with 2-digit input accuray read
Samp__Ord__MD__NumS__Condition__Determinant__HaInRatio__R_OutIn__K_Norm__C_Norm
 32   16   0   --4     3711     -3.67 D-20   3.12 D-23   347573   0(8)   -1(-1)


4.  Detail Analysen
 
Weitere und nähere Erläuterungen zur Matrixanalyse: 
Numerische Laien hier    und      Professionell Interessierte hier     Weitere Querverweise

(1)  values measured with 2-digit input accuray calculed

**********    Summary of standard correlation matrix analysis   ***********File = TH431_16.K16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No  with  4 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  10.288
LEVA: Lowest eigenvalue absolute value_____=  0.003438778
CON: Condition number HEVA/LEVA___________~=  2991.69
DET: Determinant original matrix___________=  D-21  6.359
HAC: HADAMARD condition number_____________=  D-27  1.481
HCN: Heuristic condition |DET|CON__________=  D-24  2.125
D_I: Determinant Inverse absolute value____=  1.5724911329896272D+20
HDA: HADAMARD Inequation absolute value___<=  1.0988050960598554D+33
HIR: HADAMARD RATIO: D_I / HDA ____________=  1.4310919549138755D-13
Highest inverse positive diagonal value____=  60.001125634
  thus multiple r( 7.rest)_________________=  .99163181
  and  1 multiple r > .99
Highest inverse negative diagonal value____= -1.762489944
  thus multiple r( 12.rest)_________________=  1.251950128 (!)
  and there are  5 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 1.rest)_= .257
LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=   712.87197
 Ratio maximum range output / input _______= D +4    7.9
 Mean absolute value of ranges output _____=   335.51821
 Ratio mean range output/ mean range input_= D  +4   3.7
 Sigma of mean (abs. value range output)___=   244.66269

 Ncor  L1-Norm  L2-Norm  Max     Min    m|c|     s|c|    Ncomp  M-S   S-S
  256   156      10.79   1     -.84     .61      .527     120   .218  .244

 class boundaries and distribution of the correlation-coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    8    12   20   12   18   24   32   46   82

Original input data with  2-digit-accuracy and read with 2-digit-accuracy
(for control here the analysed original matrix):
 1    .5   .5   .32  .29  .58  .72  .49  .58  .45  .31  .66  .53  .76 -.35  .11
 .5   1    .5   .32  .36  .42  .57  .49  .74  .67  .33  .54  .64 -.16 -.14 -.23
 .5   .5   1    .32  .52  .42  .88  .82  .9   .45  .3   .78  .75  .19 -.84 -.72
 .32  .32  .32  1    .95  .96  .65  .8   .61  .91  .98  .78  .82  .12 -.22 -.15
 .29  .36  .52  .95  1    .9   .75  .9   .75  .9   .94  .84  .89  .05 -.37 -.31
 .58  .42  .42  .96  .9   1    .78  .83  .7   .92  .94  .86  .86  .34 -.29 -.09
 .72  .57  .88  .65  .75  .78  1    .95  .95  .74  .64  .95  .91  .39 -.69 -.46
 .49  .49  .82  .8   .9   .83  .95  1    .93  .83  .78  .94  .95  .19 -.64 -.52
 .58  .74  .9   .61  .75  .7   .95  .93  1    .79  .6   .9   .93  .11 -.64 -.57
 .45  .67  .45  .91  .9   .92  .74  .83  .79  1    .9   .83  .9   .01 -.22 -.21
 .31  .33  .3   .98  .94  .94  .64  .78  .6   .9   1    .77  .8   .11 -.12 -.09
 .66  .54  .78  .78  .84  .86  .95  .94  .9   .83  .77  1    .97  .34 -.59 -.39
 .53  .64  .75  .82  .89  .86  .91  .95  .93  .9   .8   .97  1    .12 -.52 -.44
 .76 -.16  .19  .12  .05  .34  .39  .19  .11  .01  .11  .34  .12  1   -.28  .34
-.35 -.14 -.84 -.22 -.37 -.29 -.69 -.64 -.64 -.22 -.12 -.59 -.52 -.28  1    .76
 .11 -.23 -.72 -.15 -.31 -.09 -.46 -.52 -.57 -.21 -.09 -.39 -.44  .34  .76  1

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.28775  1         2.  2.36742   .866        3.  1.89721  .8165
  4.  1.12211   .92       5.  .16865    .1697       6.  .07716  -5.6D-3
  7.  .06717   -.5981     8.  .01777   -1.5864      9.  .01327  -2.2646
  10. 9.35D-3  -2.7013    11. 4.55D-3  -3.0387      12. 3.44D-3 -4.5821
  13.-3.69D-3  -5.6998    14.-7.39D-3  -.4257       15.-.01038  -1.9947
  16.-.01438   -1.8594

The matrix is not positive definit. Cholesky decomposition is not successful (for detailed information Cholesky's diagonalvalues are presented).


(2)  Values are computed and not measured with 17-digit-input-accuracy  & calculed

**********    Summary of standard correlation matrix analysis   ***********
File = TH429R17.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= Yes  with  0 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  9.602
LEVA: Lowest eigenvalue absolute value_____=  0.000027957
CON: Condition number HEVA/LEVA___________~=  343456.58
DET: Determinant original matrix___________=  D-29  1.084
HAC: HADAMARD condition number_____________=  D-36  5.014
HCN: Heuristic condition |DET|CON__________=  D-35  3.158
D_I: Determinant Inverse absolute value____=  9.2189364602807023D+28
HDA: HADAMARD Inequation absolute value___<=  1.0306084087129445D+58
HIR: HADAMARD RATIO: D_I / HDA ____________=  8.9451399603789321D-30
Highest inverse positive diagonal value____=  25362.029045018
  thus multiple r( 9.rest)_________________=  .999980285
  and  15 multiple r > .99
There are no negative inverse diagonal values.
 Maximum range (upp-low) multip-r( 11.rest)_= .361

LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=   148.30955
 Ratio maximum range output / input _______= D +4    1.6
 Mean absolute value of ranges output _____=    51.16785
 Ratio mean range output/ mean range input_=  5685.31686
 Sigma of mean (abs. value range output)___=    40.82624

 Ncor  L1-Norm  L2-Norm  Max    Min     m|c|     s|c|    Ncomp  M-S   S-S
  256   146.9    10.26   1     -.83     .574     .523     120   .217  .238

 class boundaries and distribution of the correlation-coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    12   10   18   14   16   32   46   36   70

Original input data with 17-digit-accuracy and read 17-digit-accuracy
(for control here the analysed original matrix):
 1    .5   .5   .316 .286 .583 .725 .5   .576 .456 .308 .598 .52  .729-.338 .095
 .5   1    .5   .316 .361 .424 .572 .5   .736 .672 .313 .507 .609-.199-.123-.238
 .5   .5   1    .316 .52  .424 .88  .816 .897 .456 .295 .948 .951 .166-.83 -.713
 .316 .316 .316 1    .95  .953 .65  .798 .611 .913 .978 .255 .329 .105-.214-.15
 .286 .361 .52  .95  1    .899 .749 .904 .743 .893 .936 .458 .537 .041-.368-.309
 .583 .424 .424 .953 .899 1    .783 .837 .701 .922 .932 .402 .442 .318-.287-.099
 .725 .572 .88  .65  .749 .783 1    .946 .949 .747 .629 .867 .872 .36 -.68 -.462
 .5   .5   .816 .798 .904 .837 .946 1    .932 .833 .777 .766 .812 .166-.629-.521
 .576 .736 .897 .611 .743 .701 .949 .932 1    .789 .589 .854 .911 .075-.634-.575
 .456 .672 .456 .913 .893 .922 .747 .833 .789 1    .895 .411 .511-6D-3-.216-.217
 .308 .313 .295 .978 .936 .932 .629 .777 .589 .895 1    .27  .342 .096-.115-.09
 .598 .507 .948 .255 .458 .402 .867 .766 .854 .411 .27  1    .979 .268-.677-.518
 .52  .609 .951 .329 .537 .442 .872 .812 .911 .511 .342 .979 1    .108-.642-.573
 .729-.199 .166 .105 .041 .318 .36  .166 .075-6D-3 .096 .268 .108 1   -.275 .325
-.338-.123-.83 -.214-.368-.287-.68 -.629-.634-.216-.115-.677-.642-.275 1    .76
 .095-.238-.713-.15 -.309-.099-.462-.521-.575-.217-.09 -.518-.573 .325 .76  1

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  9.60228   1         2.  2.8399    .866        3.  1.87505  .8165
  4.  1.18439   .922 5.  .37067    .164        6.  .08016   .0535
  7.  .01835    .0597     8.  .01318    .0158       9.  4.85D-3  .0257
  10. 4.3D-3    .0343     11. 3.54D-3   .0976       12. 2.25D-3  .1358
  13. 8.8D-4    .0328     14. 1.3D-4    .1588       15. 5D-5     .0986
  16. 3D-5      .1014

Cholesky decomposition successful, thus the matrix is (semi) positive definit.


(3) Values are computed and not measured with 4-digit-input-accuracy calculed

**********    Summary of standard correlation matrix analysis   ***********
File = TH429_R4.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No  with  1 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  9.602
LEVA: Lowest eigenvalue absolute value_____=  0.000028991
CON: Condition number HEVA/LEVA___________~=  331204.46
DET: Determinant original matrix___________=  D-29 -1.861
HAC: HADAMARD condition number_____________=  D-36  8.607
HCN: Heuristic condition |DET|CON__________=  D-35  5.621
D_I: Determinant Inverse absolute value____=  5.3710384638007393D+28
HDA: HADAMARD Inequation absolute value___<=  1.5207913243518122D+57
HIR: HADAMARD RATIO: D_I / HDA ____________=  3.5317392845398889D-29
Highest inverse positive diagonal value____=  3482.152767664
  thus multiple r( 8.rest)_________________=  .9998564
  and  10 multiple r > .99
Highest inverse negative diagonal value____= -333.4295839
  thus multiple r( 10.rest)_________________=  1.001498444 (!)
  and there are  3 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 11.rest)_= .151
LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=   303.93217
 Ratio maximum range output / input _______= D +4    3.3
 Mean absolute value of ranges output _____=   106.84911
 Ratio mean range output/ mean range input_= D  +4   1.1
 Sigma of mean (abs. value range output)___=    96.49423

 Ncor  L1-Norm  L2-Norm  Max     Min    m|c|     s|c|    Ncomp  M-S   S-S
  256   146.9    10.26   1     -.83     .574     .523     120   .217  .238

i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  9.60228   1         2.  2.83986   .866        3.  1.87503  .8165
  4.  1.1844    .922      5.  .3707     .1638       6.  .08013   .054
  7.  .01836    .0611     8.  .01321    .0165       9.  4.79D-3  .025
  10. 4.31D-3   .029      11. 3.62D-3  -1.4D-3      12. 2.29D-3 -.0554
  13. 8.6D-4   -1.0567    14. 1.1D-4   -.053        15. 9D-5    -.9549
  16.-3D-5     -1.2285

The matrix is not positive definit. Cholesky decomposition is not successful (for detailed information Cholesky's diagonalvalues are presented).


(4) Values are computed and not measured with 3-digit-input-accuracy read

**********    Summary of standard correlation matrix analysis   ***********
File = TH429_R3.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No  with  1 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  9.601
LEVA: Lowest eigenvalue absolute value_____=  0.000037347
CON: Condition number HEVA/LEVA___________~=  257080.91
DET: Determinant original matrix___________=  D-28 -5.836
HAC: HADAMARD condition number_____________=  D-34  2.701
HCN: Heuristic condition |DET|CON__________=  D-33  2.270
D_I: Determinant Inverse absolute value____=  1.713470565278007D+27
HDA: HADAMARD Inequation absolute value___<=  3.0792420801426819D+56
HIR: HADAMARD RATIO: D_I / HDA ____________=  5.5645854423975996D-30
Highest inverse positive diagonal value____=  6139.822157645
  thus multiple r( 6.rest)_________________=  .999918561
  and  15 multiple r > .99
There are no negative inverse diagonal values.
 Maximum range (upp-low) multip-r( 4.rest)_= .277
LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=    90.81287
 Ratio maximum range output / input _______= D +4    1.0
 Mean absolute value of ranges output _____=    30.84446
 Ratio mean range output/ mean range input_=  3427.16331
 Sigma of mean (abs. value range output)___=    22.03785

 Ncor  L1-Norm  L2-Norm  Max     Min    æ|c|     å|c|    Ncomp  æ-S   å-S
  256   146.9    10.26   1     -.83     .574     .523     120   .217  .238

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  9.60145   1         2.  2.83953   .866        3.  1.87509  .8165
  4.  1.18407   .9221     5.  .36997    .1627       6.  .08079   .059
  7.  .01866    .0585     8.  .01319    .0273       9.  5.15D-3  .0257
  10. 4.74D-3  -2D-3      11. 3.54D-3  -.7772       12. 2.37D-3 -.2299
  13. 1.45D-3  -1.3261    14. 3.3D-4   -.0537       15. 4D-5    -.927
  16.-3.6D-4   -1.2721
 The matrix is not positive definit. Cholesky decomposition is not successful (for detailed information Cholesky's diagonalvalues are presented).


(5) Values are computed and not measured with 2-digit-input-accuracy read

**********    Summary of standard correlation matrix analysis   ***********
File = TH429_R2.D16  N-order= 16  N-sample= 32   Rank= 14  Missing data =  0
Positiv Definit=Cholesky successful________= No  with  4 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  9.605
LEVA: Lowest eigenvalue absolute value_____=  0.001289180
CON: Condition number HEVA/LEVA___________~=  7450.29
DET: Determinant original matrix___________=  D-22  4.835
HAC: HADAMARD condition number_____________=  D-28  2.224
HCN: Heuristic condition |DET|CON__________=  D-26  6.490
D_I: Determinant Inverse absolute value____=  2.0679229830898941D+21
HDA: HADAMARD Inequation absolute value___<=  3.3333973491181176D+36
HIR: HADAMARD RATIO: D_I / HDA ____________=  6.2036498098163515D-16
Highest inverse positive diagonal value____=  177.548875466
  thus multiple r( 10.rest)_________________=  .997179898
  and  4 multiple r > .99
Highest inverse negative diagonal value____= -13.074879701
  thus multiple r( 15.rest)_________________=  1.037536765 (!)
  and there are  8 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 11.rest)_= .361
LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=   148.30955
 Ratio maximum range output / input _______= D +4    1.6
 Mean absolute value of ranges output _____=    51.16785
 Ratio mean range output/ mean range input_=  5685.31686
 Sigma of mean (abs. value range output)___=    40.82624

 Ncor  L1-Norm  L2-Norm  Max     Min    m|c|     s|c|    Ncomp  M-S   S-S
 256   147      10.27   1     -.83     .574     .523     120   .217  .239

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  9.60477   1         2.  2.83809   .866        3.  1.87558  .8165
  4.  1.18944   .92       5.  .37127    .1697       6.  .07919   .1099
  7.  .02196    .0399     8.  .01767   -.0188       9.  .01135  -.8712
  10. 9.08D-3  -1.3064    11. 5.23D-3  -1.7335      12. 1.98D-3 -1.5587
  13.-1.29D-3  -2.8004    14.-6.35D-3  -.1904       15.-8.22D-3 -1.721
  16.-9.73D-3  -1.8826

The matrix is not positive definit. Cholesky decomposition is not successful (for detailed information Cholesky's diagonalvalues are presented).


(6)  Values measured with 2-digit input accuracy read

**********    Summary of standard correlation matrix analysis   ***********
File = THPMF16.K16   N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No  with  4 negative eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=  10.261
LEVA: Lowest eigenvalue absolute value_____=   0.002764723
CON: Condition number HEVA/LEVA___________~=  3711.28
DET: Determinant original matrix___________= D-20 -3.670
HAC: HADAMARD condition number_____________= D-27  8.551
HCN: Heuristic condition |DET|CON__________= D-24  9.890
D_I: Determinant Inverse absolute value____=  2.7243326243127375D+19
HDA: HADAMARD Inequation absolute value___<=  8.7270894616169875D+41
HIR: HADAMARD RATIO: D_I / HDA ____________=  3.1216966851259516D-23
Highest inverse positive diagonal value____=  637.130010907
  thus multiple r( 6.rest)_________________=  .999214923
  and  10 multiple r > .99
Highest inverse negative diagonal value____= -9.745219835
  thus multiple r( 12.rest)_________________=  1.05005448 (!)
  and there are  2 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 15.rest)_= .454
LES: Numerical stability analysis:
 Maximum range input  x(upper)-x(lower)____=     0.009
 Maximum range output x(upper)-x(lower)____=  3128.15835
 Ratio maximum range output / input _______= D +5    3.4
 Mean absolute value of ranges output _____=  1118.18184
 Ratio mean range output/ mean range input_= D  +5   1.2
 Sigma of mean (abs. value range output)___=   984.60139

 Ncor  L1-Norm  L2-Norm  Max     Min    m|c|     s|c|    Ncomp  M-S   S-S
  256   156      10.79   1     -.84     .61      .537     120   .218  .244

 class boundaries and distribution of the correlation-coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    9    12   21   12   18   23   32   45   82

REMARK: The table contents two print errors (r14,16 and r15,16 have to be positive)

Original input data with  2-digit-accuracy and read with 2-digit-accuracy
(for control here the analysed original matrix):
 1    .5   .5   .32  .29  .58  .72  .49  .58  .45  .31  .66  .53  .76 -.35  .11
 .5   1    .5   .32  .36  .42  .57  .49  .74  .67  .33  .54  .64 -.16 -.14 -.23
 .5   .5   1    .32  .52  .42  .88  .82  .9   .45  .3   .78  .75  .19 -.84 -.72
 .32  .32  .32  1    .95  .96  .65  .8   .61  .91  .98  .78  .82  .12 -.22 -.15
 .29  .36  .52  .95  1    .9   .75  .9   .75  .9   .94  .84  .89  .05 -.37 -.31
 .58  .42  .42  .96  .9   1    .78  .83  .7   .92  .94  .86  .86  .34 -.29 -.09
 .72  .57  .88  .65  .75  .78  1    .95  .95  .74  .64  .95  .91  .39 -.69 -.46
 .49  .49  .82  .8   .9   .83  .95  1    .93  .83  .78  .94  .95  .19 -.64 -.52
 .58  .74  .9   .61  .75  .7   .95  .93  1    .79  .6   .9   .93  .11 -.64 -.57
 .45  .67  .45  .91  .9   .92  .74  .83  .79  1    .9   .83  .9   .01 -.22 -.21
 .31  .33  .3   .98  .94  .94  .64  .78  .6   .9   1    .77  .8   .11 -.12 -.09
 .66  .54  .78  .78  .84  .86  .95  .94  .9   .83  .77  1    .97  .34 -.59 -.39
 .53  .64  .75  .82  .89  .86  .91  .95  .93  .9   .8   .97  1    .12 -.52 -.44
 .76 -.16  .19  .12  .05  .34  .39  .19  .11  .01  .11  .34  .12  1   -.28 -.34
-.35 -.14 -.84 -.22 -.37 -.29 -.69 -.64 -.64 -.22 -.12 -.59 -.52 -.28  1   -.76
 .11 -.23 -.72 -.15 -.31 -.09 -.46 -.52 -.57 -.21 -.09 -.39 -.44  .34  .76  1

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.26065  1         2.  2.17133   .866        3.  1.75624  .8165
  4.  1.0651    .92       5.  .88599    .1697       6.  .07148  -5.6D-3
  7.  .0514    -.5981     8.  .02466   -1.5864      9.  .014    -2.2646
  10. 9.28D-3  -2.7013    11. 5.2D-3   -3.0387      12. 2.76D-3 -4.5821
  13.-4.57D-3  -5.6998    14.-8.45D-3  -.4257       15.-.01274  -1.9947
  16.-.22922   -1.8594

The matrix is not positive definit. Cholesky decomposition is not successful (for detailed information Cholesky's diagonalvalues are presented).


5. Überblick der berechneten Trapezoid-Parameter

Hinweis 23.10.2002: Parameterberechnung für 12 und 13 wurde falsch interpretiert.  Der Fehler hat aber glücklicherwiese keinerlei Auswirkungen auf die grundlegenden Aussagen oder Interpretationen. Die korrigierte Version finden Sie hier.

Thurstone's trapezoid raw scores with 3-digit-accuracy for view:
1 2 1 2 2     2.24  2.83  2.24  3.61  2.83  2.24  1    2    .5    2     1
1 2 1 4 4     4.12  4.47  4.12  5     4.47  4.12  1    3    .5    2     1
1 2 3 2 2.83  2.24  4.47  3.61  5.39  2.83  2.24  7    8    .5    .67   .33
1 2 3 4 4.47  4.12  5.66  5     6.4   4.47  4.12  5    7    .5    .67   .33
1 3 1 2 2     2.24  2.83  2.24  4.47  3.61  2.83  1    3    .33   3     1
1 3 1 4 4     4.12  4.47  4.12  5.66  5     4.47  1    5    .33   3     1
1 3 3 2 2.83  2.24  4.47  3.61  6.32  3.61  2     7    9    .33   1     .33
1 3 3 4 4.47  4.12  5.66  5     7.21  5     4     5    9    .33   1     .33
2 2 1 2 2.24  2.83  3.61  2.24  3.61  2.83  2.24  2    2    1     2     2
2 2 1 4 4.12  4.47  5     4.12  5     4.47  4.12  3    3    1     2     2
2 2 3 2 2.24  2.83  5.39  3.61  5.39  2.83  2.24  8    8    1     .67   .67
2 2 3 4 4.12  4.47  6.4   5     6.4   4.47  4.12  7    7    1     .67   .67
2 3 1 2 2.24  2.83  3.61  2.24  4.47  3.61  2.83  2    3    .67   3     2
2 3 1 4 4.12  4.47  5     4.12  5.66  5     4.47  3    5    .67   3     2
2 3 3 2 2.24  2.83  5.39  3.61  6.32  3.61  2     8    9    .67   1     .67
2 3 3 4 4.12  4.47  6.4   5     7.21  5     4     7    9    .67   1     .67
2 3 3 3 3.16  3.61  5.83  4.24  6.71  4.24  3     7.5  9    .67   1     .67
2 3 3 5 5.1   5.39  7.07  5.83  7.81  5.83  5     6.5  9    .67   1     .67
2 3 5 3 4.24  3.61  7.62  5.83  8.54  4.24  3.61  20.5 22   .67   .6    .4
2 3 5 5 5.83  5.39  8.6   7.07  9.43  5.83  5.39  17.5 20   .67   .6    .4
2 4 3 3 3.16  3.61  5.83  4.24  7.62  5     3.16  7.5  10.5 .5    1.33  .67
2 4 3 5 5.1   5.39  7.07  5.83  8.6   6.4   5.1   6.5  11.5 .5    1.33  .67
2 4 5 3 4.24  3.61  7.62  5.83  9.49  5     3.16  20.5 23.5 .5    .8    .4
2 4 5 5 5.83  5.39  8.6   7.07  10.3  6.4   5.1   17.5 22.5 .5    .8    .4
3 3 3 3 3     4.24  6.71  4.24  6.71  4.24  3     9    9    1     1     1
3 3 3 5 5     5.83  7.81  5.83  7.81  5.83  5     9    9    1     1     1
3 3 5 3 3.61  4.24  8.54  5.83  8.54  4.24  3.61  22   22   1     .6    .6
3 3 5 5 5.39  5.83  9.43  7.07  9.43  5.83  5.39  20   20   1     .6    .6
3 4 3 3 3     4.24  6.71  4.24  7.62  5     3.16  9    10.5 .75   1.33  1
3 4 3 5 5     5.83  7.81  5.83  8.6   6.4   5.1   9    11.5 .75   1.33  1
3 4 5 3 3.61  4.24  8.54  5.83  9.49  5     3.16  22   23.5 .75   .8    .6
3 4 5 5 5.39  5.83  9.43  7.07  10.3  6.4   5.1   20   22.5 .75   .8    .6


6. OMIKRON-Basic-Programm zur Berechnung der Parameter des THURSTONEschen Trapezoids

Hinweis 23.10.2002: Parameterberechnung für 12 und 13 wurde falsch interpretiert.  Der Fehler hat aber glücklicherwiese keinerlei Auswirkungen auf die grundlegenden Aussagen oder Interpretationen. Die korrigierte Version finden Sie hier.

 REM  TRAPEZ.BAS (THURSTONE's trapezoid)
 ' 20.11.93  R.Sponsel D-91052 Erlangen
 COMPILER "TRACE ON"
 DEFDBL "X"
 S=16:Z=32: DIM Xd#(S,Z)
 OPEN "I",1,"C:\OMI\NUMERIK\MATRIX\URDAT\THURS\TH429_4.N32"
 FOR I=1 TO 4'             trapezoid parameter a,b,c,h read
   FOR J=1 TO Z
      INPUT #1,Xd#(I,J)
   NEXT J
 NEXT I: CLOSE 1
 OPEN "O",1,"C:\OMI\NUMERIK\MATRIX\URDAT\THURS\TH429_32.N16"
 OPEN "O",2,"C:\OMI\NUMERIK\MATRIX\URDAT\THURS\TH429_32.TAB"
 FOR I=5 TO S
    FOR J=1 TO Z
       A#=Xd#(1,J):B#=Xd#(2,J):C#=Xd#(3,J):H#=Xd#(4,J)
       IF I=5 THEN Xd#(I,J)= SQR((C#-A#)^2+H#^2)
       IF I=6 THEN Xd#(I,J)= SQR(A#^2+H#^2)
       IF I=7 THEN Xd#(I,J)= SQR((A#+C#)^2+H#^2)
       IF I=8 THEN Xd#(I,J)= SQR(C#^2+H#^2)
       IF I=9 THEN Xd#(I,J)= SQR((C#+B#)^2+H#^2)
       IF I=10 THEN Xd#(I,J)= SQR(B#^2+H#^2)
       IF I=11 THEN Xd#(I,J)= SQR((C#-B#)^2+H#^2)
       IF I=12 THEN Xd#(I,J)=C#*H#-(C#-A#)*H#/2
       IF I=13 THEN Xd#(I,J)=C#*H#-(C#-B#)*H#/2
       IF I=14 THEN Xd#(I,J)=A#/B#
       IF I=15 THEN Xd#(I,J)=B#/C#
       IF I=16 THEN Xd#(I,J)=A#/C#
    NEXT J
 NEXT I
 FOR I=1 TO S
    FOR J=1 TO Z
       PRINT #1,Xd#(I,J)
    NEXT J
 NEXT I: CLOSE 1
 PRINT #2,"Thurstone's trapezoid raw scores with 3-digit-accuracy for view:"
 FOR I=1 TO Z
    FOR J=1 TO S
       PRINT #2, TAB (7*(J-1)); INT(Xd#(J,I)*10^2+.5)/10^2;
    NEXT J: PRINT #2
 NEXT I
 CLOSE 2
 END



Querverweise:
 Wird im Laufe der Zeit fortgesetzt, ergänzt und erweitert
FN01  Sponsel, Rudolf & Hain, Bernhard (1994). Numerisch instabile Matrizen und Kollinearität in der Psychologie. Diagnose, Relevanz & Utilität, Frequenz, Ätiologie, Therapie.  Ill-Conditioned Matrices and Collinearity in Psychology. Deutsch-Englisch. Übersetzt von Agnes Mehl. Kapitel 6 von Dr. Bernhard Hain: Bemerkungen über Korrelationsmatrizen. Erlangen: IEC-Verlag [ISSN-0944-5072  ISBN 3-923389-03-5]. Aktueller Preis: www.iec-verlag.de


Zitierung
Sponsel, Rudolf  (DAS). Dokumentation Rundungsfehler & Kollinearität am THURSTONEschen Trapezoid- Beispiel. Abteilung: Numerisch instabile Matrizen und Kollinearität in der Psychologie - Ill-Conditioned Matrices and Collinearity in Psychology -   Diagnose, Relevanz & Utilität, Frequenz, Ätiologie, Therapie.  IP-GIPT. Erlangen: http://www.sgipt.org/wisms/nis/k7/K7_3.htm
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