Internet Publikation  für Allgemeine und Integrative Psychotherapie
IP-GIPT DAS=23.10.2002
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:

Korrigierte Version: Dokumentation Rundungsfehler & Kollinearität am THURSTONEschen Trapezoid- Beispiel

von Rudolf Sponsel, Erlangen [Quelle Kap. 7.3 * korrigiert]
Internet-Erstausgabe 23.10.2002, letzte Änderung TT.MM.JJ

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)  Original alues measured with 2-digit input accuray read
     (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 read
     (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 with print error measured with 2-digit input accuracy read
5.  Überblick der berechneten Trapezoid-Parameter (korrigiert)
6.  OMIKRON-Basic-Programm zur Berechnung der Parameter des THURSTONEschen Trapezoids (korrigiert)
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.

Anmerkung zu den Fehlern bei der Interpretation der Trapezparameter A12 und A13.

 
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 (wobei sich aus diesem Beispiel klar ergibt, daß 97,62% Faktorenvarianz erforderlich sind). Darum geht es uns aber hier nicht weiter (das finden Sie hier), sondern um die Demonstration wie Kollinearität und Rundungsfehler zusammen spielen.
 

2.  TRAPEZOID Illustration Figure 5

Man beachte die Definition Thurstones von Area 12 und Area 13 im Text


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-1: Die Matrix enthält zwei Druckfehler: r14,16 und r15,16 sollten positiv sein. Man kann hier studieren, wie sich ein Druckfehler hinsichtlich des Vorzeichens zweier Korrelationskoeffizienten auf die numerische Stabilität auswirkt.
(6=12f) THPMF16.K16  raw scores measured with 2-digit input accuray read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16   0   --4    3711      -3.67 D-20   3.12 D-23  347573   0(8)   -1(-1)
Zum Widerspruch negative Determinante und geradzahlig negative Eigenwerte, die zu einer positiven Determinante führen müßten, siehe bitte unten.

Dieselbe wie obige Matrix ohne Druckfehler und aus Thurstone's Buch
(1=12a) TH43116.K16  raw scores measured with 2-digit input accuray read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16   0   --4     2992     6.36 D-21   1.43 D-13   79208   2D-3(9) -1(-1)



 
Fehlerversion:(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_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16   0    -    343457    1.08 D-29   8.94 D-30   16479    0(9)    6D-3(8)

Korrigierte Version: TRAPEZ32.D16 with 17-digit-input accuracy calculed
Samp _Ord_ MD_NumS_Condition_Determinant_HaInRatioR_ OutInK_ NormC_ Norm
 32   16   0   -    502896.5    0        3.81D-31    17952    0(8)  0.006(7)



 
Fehlerversion:(3=12c) TH429R4.D16  raw scores are computed and not measured with 4-digit-input-accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
32    16    0  --1    331205    -1.86 D-29  3.53 D-29   33770    0(9)  -1(-1)

TRAPEZ32.D16 korrigiert with 4-digit-input accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16   0   --1   594867.1       0        2.25D-32   17952    0(8)   -1(-1)



 
Fehlerversion:(4=12d) TH429R3.D16 raw scores are computed and not measured with 3-digit-input-accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16    0   --1   257081    -5.84 D-28   5.56 D-30  10090    0(9)     -1(-1)

TRAPEZ32.D16 korrigiert with 3-digit-input accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16    0  --2    2.1D+6      0         2.81D-43    6698.1  0(8)   -1(-1)

In der korrigierten Version ist die Konditionszahl rund 8x größer und ein negativer Eigenwert kommt hinzu. Dies illustriert ganz gut, wie numerische Instabilität wirkt.



 
Fehlerversion:(5=12e) TH429R2.D16 raw scores are computed and not measured with 2-digit-input-accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm _ C_Norm
 32   16   0   --4    7450      4.83 D-22   6.20 D-16   16479   .001(9) -1(-1)

TRAPEZ32.D16 korrigiert with 2-digit-input accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm C_Norm
 32   16    0  --4    14892.5      0        7.01D-18    17952    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 by Thurstone with 2-digit input accuray read

THURSTONE, L. L.  (USA: University Of Chicago)  "Multiple Factor Analysis"  Chicago 1947,  p.427-436  "A trapezoid population", p. 431 Table 7 'Correlation Matrix'

 Samp  Or  MD  NumS  Condit  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)

**********    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 (k)

TRAPEZ32.D16 korrigiert with 17-digit-input accuracy read
Samp  Or  MD  NumS  Condit  Determinant  HaInRatio  R_OutIn  K_Norm   C_Norm
 32   16   0   -    502896.5    0        3.81D-31   17952     0(8)    0.006(7)

**********    Summary of standard correlation matrix analysis   ***********
File = TRAPEZ32.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= Yes with  0 negat. eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=    10.218631637726756
LEVA: Lowest eigenvalue absolute value_____=    2.0319552205460591D-5
CON: Condition number HEVA/LEVA___________~=    502896.49763938421
DET: Determinant original matrix (OMIKRON)_=    1.645078683984506D-29
DET: Determinant (CHOLESKY-Diagonal^2)_____=    1.6450786839845063D-29
DET: Determinant (PESO-CHOLESKY)___________=    1.6450786839845063D-29
DET: Determinant (product eigenvalues)_____=    1.6450786839814124D-29
DET: Determ.abs.val.(PESO prod.red.norms)__=    1.6450786839844999D-29
HAC: HADAMARD condition number_____________=    4.2714963680318048D-36
HCN: Heuristic condition |DET|CON__________=    3.271207279642172D-35
D_I: Determinant Inverse absolute value____=    6.0787365962211835D+28
HDA: HADAMARD Inequality absolute value___<=    1.5933927985332463D+59
HIR: HADAMARD RATIO: D_I / HDA ____________=    3.8149642710929763D-31
Highest inverse positive diagonal value____=    25369.293808947
  thus multiple r( 9.rest)_________________=    .999980291
  and  15 multiple r > .99
There are no negative inverse diagonal values.
 Maximum range (upp-low) multip-r( 2.rest)_=    .171
LES: Numerical stability analysis:
 Ratio maximum range output / input _______=    17951.973072121561
PESO-Analysis correlation least Ratio RN/ON=    1D-5 (<-> Angle = 0 )
Number of Ratios correlation RN/ON < .01__ =    8
PESO-Analysis Cholesky least Ratio RN/ON__ =    6.278D-3 (<-> Angle = .36 )
Number of Ratios Cholesky RN/ON < .1 _____ =    7

 Ncor  L1-Norm  L2-Norm  Max    Min    m|c|    M|c|   N_comp    s-S   S-S
  256   154.7    10.72   1      -.83   .578    .28    7140      .323  .231

 class boundaries and distribution of the correlation coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    10   10   20   14   16   24   32   46   82

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.21863  1         2.  2.3971    .866        3.  1.84973  .8165
  4.  1.15395   .922      5.  .19044    .164        6.  .08281   .0535
  7.  .06998    .0597     8.  .01708    .0158       9.  8.13D-3  .0257
  10. 6.88D-3   .0343     11. 2.47D-3   .0976       12. 2.13D-3  .1371
  13. 4.5D-4    .0501     14. 1.6D-4    .1921       15. 4D-5     .0722
  16. 2D-5      .0915
 Cholesky decomposition successful, thus the matrix is (semi) positive definit.

 Eigenvalues in per cent of trace =  16
  1 .6387   2 .1498   3 .1156   4 .0721   5 .0119   6 5.2D-3
  7 4.4D-3  8 1.1D-3  9 5D-4    10 4D-4   11 2D-4   12 1D-4
  13 0      14 0      15 0      16 0

[Intern: analysed: 10/22/02 22:51:56  PRG version 05/24/94  MA9.BAS
File = C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.SMA  with data from C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.D16
Date: 10/22/02  Time:22:51:56]


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

TRAPEZ32.D16 korrigiert with 4-digit-input accuracy read
Samp _Ord_ MD_ NumS_ Condition_ Determinant_HaInRatioR_ OutInK_ Norm C_Norm
 32   16   0   --1   594867.1       0        2.25D-32   17952    0(8)     -1(-1)

**********    Summary of standard correlation matrix analysis   ***********
File = TRAPEZ32.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No with  1 negat. eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=    10.218634101122853
LEVA: Lowest eigenvalue absolute value_____=    1.7178012311068469D-5
CON: Condition number HEVA/LEVA___________~=    594867.0845077102
DET: Determinant original matrix (OMIKRON)_=   -2.4694269803906613D-29
DET: Determinant (CHOLESKY-Diagonal^2)_____=   -999 (not positive definit)
DET: Determinant (PESO-CHOLESKY)___________=   -999 (not positive definit)
DET: Determinant (product eigenvalues)_____=   -2.4694269803881388D-29
DET: Determ.abs.val.(PESO prod.red.norms)__=    2.4694269803906669D-29
HAC: HADAMARD condition number_____________=    6.4120207821971545D-36
HCN: Heuristic condition |DET|CON__________=    4.1512247772698113D-35
D_I: Determinant Inverse absolute value____=    4.0495224517300804D+28
HDA: HADAMARD Inequality absolute value___<=    1.7987156414681396D+60
HIR: HADAMARD RATIO: D_I / HDA ____________=    2.2513411004892343D-32
Highest inverse positive diagonal value____=    25208.251484587
  thus multiple r( 9.rest)_________________=    .999980165
  and  12 multiple r > .99
Highest inverse negative diagonal value____=   -153.889894333
  thus multiple r( 10.rest)_________________=   1.003243815 (!)
  and there are  2 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 2.rest)_=    .171
LES: Numerical stability analysis:
 Ratio maximum range output / input _______=    17951.973072121561
PESO-Analysis correlation least Ratio RN/ON=    8D-6 (<-> Angle = 0 )
Number of Ratios correlation RN/ON < .01__ =    8
PESO-Analysis Cholesky least Ratio RN/ON__ = (Not positiv definit)

 Ncor  L1-Norm  L2-Norm  Max    Min    m|c|    M|c|   N_comp    s-S   S-S
  256   154.7    10.72   1      -.83   .578    .28    7140      .323  .231

 class boundaries and distribution of the correlation coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    10   10   20   14   16   24   32   46   82

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.21863  1         2.  2.39706   .866        3.  1.84972  .8165
  4.  1.15396   .922 5.  .19044    .1638       6.  .08281   .054
  7.  .07002    .0611     8.  .01706    .0165       9.  8.2D-3   .025
  10. 6.84D-3   .029      11. 2.44D-3  -1.4D-3      12. 2.22D-3 -.4691
  13. 4.1D-4   -1.5393    14. 2.4D-4   -.0422       15. 2D-5    -.7613
  16.-5D-5     -1.0652
 The matrix is not positive definit. Cholesky decomposition is not success-

 Eigenvalues in per cent of trace =  16
  1 .6387   2 .1498   3 .1156   4 .0721   5 .0119   6 5.2D-3
  7 4.4D-3  8 1.1D-3  9 5D-4    10 4D-4   11 2D-4   12 1D-4
  13 0      14 0      15 0      16 0

[Intern: analysed: 10/22/02 23:36:22  PRG version 05/24/94  MA9.BAS
File = C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.SMA
 with data from C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.D16
Date: 10/22/02  Time:23:36:22]


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

TRAPEZ32.D16 korrigiert with 3-digit-input accuracy read
Samp  Or  MD  NumS  Condit  Determinant  HaInRatio  R_OutIn  K_Norm   C_Norm
 32   16   0  --2   2.1D+6      0         2.81D-43   6698.1   0(8)    -1(-1)

**********    Summary of standard correlation matrix analysis   ***********
File = TRAPEZ32.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No with  2 negat. eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=    10.217959342460799
LEVA: Lowest eigenvalue absolute value_____=    4.6539581285637635D-6
CON: Condition number HEVA/LEVA___________~=    2195541.7432202202
DET: Determinant original matrix (OMIKRON)_=    3.4172590164546888D-28
DET: Determinant (CHOLESKY-Diagonal^2)_____=   -999 (not positive definit)
DET: Determinant (PESO-CHOLESKY)___________=   -999 (not positive definit)
DET: Determinant (product eigenvalues)_____=    3.4172590164729643D-28
DET: Determ.abs.val.(PESO prod.red.norms)__=    3.4172590164547137D-28
HAC: HADAMARD condition number_____________=    8.8832239172445477D-35
HCN: Heuristic condition |DET|CON__________=    1.5564536757304214D-34
D_I: Determinant Inverse absolute value____=    2.9263219299000407D+27
HDA: HADAMARD Inequality absolute value___<=    1.0390704560685637D+70
HIR: HADAMARD RATIO: D_I / HDA ____________=    2.8162882630424298D-43
Highest inverse positive diagonal value____=    28.70356303
  thus multiple r( 14.rest)_________________=   .98242614
Highest inverse negative diagonal value____=   -22.99665269
  thus multiple r( 11.rest)_________________=   1.021510935 (!)
  and there are  15 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 2.rest)_=    .101
LES: Numerical stability analysis:
 Ratio maximum range output / input _______=    6698.1075669521929
PESO-Analysis correlation least Ratio RN/ON=    3D-6 (<-> Angle = 0 )
Number of Ratios correlation RN/ON < .01__ =    8
PESO-Analysis Cholesky least Ratio RN/ON__ = (Not positiv definit)

 Ncor  L1-Norm  L2-Norm  Max    Min    m|c|    M|c|   N_comp    s-S   S-S
  256   154.7    10.72   1      -.83   .578    .28    7140      .323  .231

 class boundaries and distribution of the correlation coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    10   10   20   14   16   24   32   46   82

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.21796  1         2.  2.39687   .866        3.  1.84975  .8165
  4.  1.15361   .9221     5.  .19046    .1627       6.  .08276   .059
  7.  .06986    .0585     8.  .0173     .0273       9.  8.43D-3  .0257
  10. 6.66D-3  -2D-3      11. 2.76D-3  -.7772       12. 2.55D-3 -1.0375
  13. 8.7D-4   -2.365     14. 5.6D-4   -.043        15. 0       -.733
  16.-3.9D-4   -1.1081
 The matrix is not positive definit. Cholesky decomposition is not success-

 Eigenvalues in per cent of trace =  16
  1 .6386   2 .1498   3 .1156   4 .0721   5 .0119   6 5.2D-3
  7 4.4D-3  8 1.1D-3  9 5D-4    10 4D-4   11 2D-4   12 2D-4
  13 1D-4   14 0      15 0      16 0

[Intern: analysed: 10/22/02 23:21:46  PRG version 05/24/94  MA9.BAS
File = C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.SMA
 with data from C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.D16
Date: 10/22/02  Time:23:21:46]
 


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

TRAPEZ32.D16 korrigiert with 2-digit-input accuracy read
Samp  Or  MD  NumS  Condit  Determinant  HaInRatio  R_OutIn  K_Norm   C_Norm
 32   16   0  --4   14892.5      0       7.01D-18   17952     0(8)    -1(-1)

**********    Summary of standard correlation matrix analysis   ***********
File = TRAPEZ32.D16  N-order= 16  N-sample= 32   Rank= 16  Missing data =  0
Positiv Definit=Cholesky successful________= No with  4 negat. eigenvalue/s
HEVA: Highest eigenvalue abs.value_________=    10.215085044003685
LEVA: Lowest eigenvalue absolute value_____=    6.8592119996252823D-4
CON: Condition number HEVA/LEVA___________~=    14892.505209872116
DET: Determinant original matrix (OMIKRON)_=    9.0100922957000011D-22
DET: Determinant (CHOLESKY-Diagonal^2)_____=   -999 (not positive definit)
DET: Determinant (PESO-CHOLESKY)___________=   -999 (not positive definit)
DET: Determinant (product eigenvalues)_____=    9.0100922957003092D-22
DET: Determ.abs.val.(PESO prod.red.norms)__=    9.0100922957000008D-22
HAC: HADAMARD condition number_____________=    2.3482342345089274D-28
HCN: Heuristic condition |DET|CON__________=    6.0500850385650946D-26
D_I: Determinant Inverse absolute value____=    1.1098665442941606D+21
HDA: HADAMARD Inequality absolute value___<=    1.5824808367746779D+38
HIR: HADAMARD RATIO: D_I / HDA ____________=    7.01345961671313D-18
Highest inverse positive diagonal value____=    57.512212037
  thus multiple r( 5.rest)_________________=    .991268071
  and  1 multiple r > .99
Highest inverse negative diagonal value____=   -8.589202343
  thus multiple r( 1.rest)_________________=    1.056610262 (!)
  and there are  12 multiple r > 1 (!)
 Maximum range (upp-low) multip-r( 2.rest)_=    .171
LES: Numerical stability analysis:
 Ratio maximum range output / input _______=    17951.973072121561
PESO-Analysis correlation least Ratio RN/ON=    4.51D-4 (<-> Angle = .03 )
Number of Ratios correlation RN/ON < .01__ =    8
PESO-Analysis Cholesky least Ratio RN/ON__ = (Not positiv definit)

 Ncor  L1-Norm  L2-Norm  Max    Min    m|c|    M|c|   N_comp    s-S   S-S
  256   154.7    10.72   1      -.83   .578    .279   7140      .322  .231

 class boundaries and distribution of the correlation coefficients
 -1  -.8  -.6  -.4  -.2   0    .2   .4   .6   .8   1
    2    10   10   22   12   16   24   32   46   82

 i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky   i.Eigenvalue  Cholesky
  1.  10.21509  1         2.  2.39196   .866        3.  1.85126  .8165
  4.  1.15826   .92       5.  .18978    .1697       6.  .09108   .1099
  7.  .06916    .0399     8.  .01877   -.0188       9.  .01635  -.8712
  10. .01019   -1.3064    11. 6.72D-3  -1.7335      12. 3.56D-3 -2.7824
  13.-6.9D-4   -4.0284    14.-3.77D-3  -.1784       15.-6.87D-3 -1.515
  16.-.01083   -1.7234
 The matrix is not positive definit. Cholesky decomposition is not success-

 Eigenvalues in per cent of trace =  16
  1 .6384   2 .1495   3 .1157   4 .0724   5 .0119   6 5.7D-3
  7 4.3D-3  8 1.2D-3  9 1D-3    10 6D-4   11 4D-4   12 2D-4
  13 0      14-2D-4   15-4D-4   16-7D-4

[Intern: analysed: 10/22/02 23:20:00  PRG version 05/24/94  MA9.BAS
File = C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.SMA
 with data from C:\OMI\NUMERIK\MATRIX\SMA\TRAPEZ32\TRAPEZ32.D16
Date: 10/22/02  Time:23:20:00]


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

THURSTONE,L.L. (USA: The University of Chicago). "SECOND-ORDER FACTORS" Psychometrika 9.2,1944, p. 96 Table 7. Bemerkung 1: die Tablele enthält zwei Druckfehler (r14,16 und  r15,16 sollten positiv sein).

6=12f) THPMF16.K16  raw scores measured with 2-digit input accuray read
Samp  Or  MD  NumS  Condit  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)

Wir sehen hier einen Widerspruch zwischen negativer Determinante und geradzahlig negativen Eigenwerte, die zu einer positiven Determinante führen müßten. Daher habe ich Determinante und Eigenwerte noch einmal mit dem genaueren Matlab nachgerechnet, das  für diese Matrix zu folgenden Ergebnissen gelangt:

Det = -3.6706e-020, also gleicher Wert wie ihn das Omikron-Basisprogramm berechnet. Zum Eigenwertvergleich siehe bitte unten bei den Eigenwerten.

**********    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     2    3    4    5    6    7    8    9   10    11  12    13   14   15   16
1  1    .5   .5   .32  .29  .58  .72  .49  .58  .45  .31  .66  .53  .76 -.35  .11
2  .5   1    .5   .32  .36  .42  .57  .49  .74  .67  .33  .54  .64 -.16 -.14 -.23
3  .5   .5   1    .32  .52  .42  .88  .82  .9   .45  .3   .78  .75  .19 -.84 -.72
4  .32  .32  .32  1    .95  .96  .65  .8   .61  .91  .98  .78  .82  .12 -.22 -.15
5  .29  .36  .52  .95  1    .9   .75  .9   .75  .9   .94  .84  .89  .05 -.37 -.31
6  .58  .42  .42  .96  .9   1    .78  .83  .7   .92  .94  .86  .86  .34 -.29 -.09
7  .72  .57  .88  .65  .75  .78  1    .95  .95  .74  .64  .95  .91  .39 -.69 -.46
8  .49  .49  .82  .8   .9   .83  .95  1    .93  .83  .78  .94  .95  .19 -.64 -.52
9  .58  .74  .9   .61  .75  .7   .95  .93  1    .79  .6   .9   .93  .11 -.64 -.57
10 .45  .67  .45  .91  .9   .92  .74  .83  .79  1    .9   .83  .9   .01 -.22 -.21
11 .31  .33  .3   .98  .94  .94  .64  .78  .6   .9   1    .77  .8   .11 -.12 -.09
12 .66  .54  .78  .78  .84  .86  .95  .94  .9   .83  .77  1    .97  .34 -.59 -.39
13 .53  .64  .75  .82  .89  .86  .91  .95  .93  .9   .8   .97  1    .12 -.52 -.44
14 .76 -.16  .19  .12  .05  .34  .39  .19  .11  .01  .11  .34  .12  1   -.28 -.34
15 -.35 -.14 -.84 -.22 -.37 -.29 -.69 -.64 -.64 -.22 -.12 -.59 -.52 -.28  1  -.76
16 .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).

Anmerkung: Bei den Eigenwerten kommt Matlab zu folgenden und vermutlich genaueren Ergebnissen als das Omikron- Basic- Programm:
 1.  10.2577              2.   2.1193               3.    1.5704
 4.   0.8507 + 0.2465i    5.   0.8507 - 0.2465i     6.    0.2378
 7.   0.0711              8.   0.0273               9.   -0.0133
 10. -0.0088              11. -0.0074               12.   0.0170
 13.  0.0139              14.  0.0004               15.    0.0038
 16.  0.0093


5. Überblick der berechneten Trapezoid-Parameter (k)

Korrigierte Version

Thurstone's trapezoid raw scores with 3-digit-accuracy for view:
   s1 s2 s3 s4    s5    s6     s7     s8     s9    s10    s11    s12  s13  s14  s15   s16
01  1  2  1  2   2     2.24   2.83   2.24   3.61   2.83   2.24   1     4    .5   2    1
02  1  2  1  4   4     4.12   4.47   4.12   5      4.47   4.12   2     8    .5   2    1
03  1  2  3  2   2.83  2.24   4.47   3.61   5.39   2.83   2.24   3     6    .5   .67  .33
04  1  2  3  4   4.47  4.12   5.66   5      6.4    4.47   4.12   6     12   .5   .67  .33
05  1  3  1  2   2     2.24   2.83   2.24   4.47   3.61   2.83   1     5    .33  3    1
06  1  3  1  4   4     4.12   4.47   4.12   5.66   5      4.47   2     10   .33  3    1
07  1  3  3  2   2.83  2.24   4.47   3.61   6.32   3.61   2      3     7    .33  1    .33
08  1  3  3  4   4.47  4.12   5.66   5      7.21   5      4      6     14   .33  1    .33
09  2  2  1  2   2.24  2.83   3.61   2.24   3.61   2.83   2.24   1     5    1    2    2
10  2  2  1  4   4.12  4.47   5      4.12   5      4.47   4.12   2     10   1    2    2
11  2  2  3  2   2.24  2.83   5.39   3.61   5.39   2.83   2.24   3     7    1    .67  .67
12  2  2  3  4   4.12  4.47   6.4    5      6.4    4.47   4.12   6     14   1    .67  .67
13  2  3  1  2   2.24  2.83   3.61   2.24   4.47   3.61   2.83   1     6    .67  3    2
14  2  3  1  4   4.12  4.47   5      4.12   5.66   5      4.47   2     12   .67  3    2
15  2  3  3  2   2.24  2.83   5.39   3.61   6.32   3.61   2      3     8    .67  1    .67
16  2  3  3  4   4.12  4.47   6.4    5      7.21   5      4      6     16   .67  1    .67
17  2  3  3  3   3.16  3.61   5.83   4.24   6.71   4.24   3      4.5   12   .67  1    .67
18  2  3  3  5   5.1   5.39   7.07   5.83   7.81   5.83   5      7.5   20   .67  1    .67
19  2  3  5  3   4.24  3.61   7.62   5.83   8.54   4.24   3.61   7.5   15   .67  .6   .4
20  2  3  5  5   5.83  5.39   8.6    7.07   9.43   5.83   5.39   12.5  25   .67  .6   .4
21  2  4  3  3   3.16  3.61   5.83   4.24   7.62   5      3.16   4.5   13.5 .5   1.33 .67
22  2  4  3  5   5.1   5.39   7.07   5.83   8.6    6.4    5.1    7.5   22.5 .5   1.33 .67
23  2  4  5  3   4.24  3.61   7.62   5.83   9.49   5      3.16   7.5   16.5 .5   .8   .4
24  2  4  5  5   5.83  5.39   8.6    7.07   10.3   6.4    5.1    12.5  27.5 .5   .8   .4
25  3  3  3  3   3     4.24   6.71   4.24   6.71   4.24   3      4.5   13.5 1    1    1
26  3  3  3  5   5     5.83   7.81   5.83   7.81   5.83   5      7.5   22.5 1    1    1
27  3  3  5  3   3.61  4.24   8.54   5.83   8.54   4.24   3.61   7.5   16.5 1    .6   .6
28  3  3  5  5   5.39  5.83   9.43   7.07   9.43   5.83   5.39   12.5  27.5 1    .6   .6
29  3  4  3  3   3     4.24   6.71   4.24   7.62   5      3.16   4.5   15   .75  1.33 1
30  3  4  3  5   5     5.83   7.81   5.83   8.6    6.4    5.1    7.5   25   .75  1.33 1
31  3  4  5  3   3.61  4.24   8.54   5.83   9.49   5      3.16   7.5   18   .75  .8   .6
32  3  4  5  5   5.39  5.83   9.43   7.07   10.3   6.4    5.1    12.5  30   .75  .8   .6


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

 REM  TRAPEZ.BAS (THURSTONE's trapezoid)
 ' 20.10.2002  R.Sponsel D-91052 Erlangen, A12 und A13 korrigiert
 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)=(((2*C#+A#+B#)/2)*H#)-((A#+B#+C#)/2)*H#
       IF I=13 THEN Xd#(I,J)=((A#*B#+C#)/2)*H#
       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



Fehlerhinweis:
Ich verdanke den Fehlerhinweis bei der Thurstone'schen Trapezoidberechnung für die mißverständlich ausgewiesene Area 12 und Area 13  Hermann Kremer und Gottfried Helms aus der Newsgroup de.sci.mathematik, in der ich nachfragte wie der fünfte, 'zahlengefühlsmäßig' relativ große Eigenwert der ersten gerechneten Version mit 17stelliger Genauigkeit von 0,370 erklärt werden könnte. Nach Beseitigung des Fehlers fiel dieser Eigenwert auf relativ unauffällige 0,190 . Ich hoffe ich erliege hier keinem psychologischen ex post facto Phänomen ;-). Der relativ große Eigenwert fiel mir damals nicht auf, weil diese Studie das Hauptinteresse verfolgte, die Indefinitheit der Matrix, also die Produktion von negativen Eigenwerten, als Folge der Kombination von Kollinearität und Rundungsfehlern durch Modellbildung zu beweisen. Der relativ große Eigenwert fiel mir erst auf, als ich jüngst die Reproduktionsmatrix aus den Faktoren Thurstone's rückrechnete. Die Fehlerbearbeitung motivierte mich, zu vergleichen, wie dieser Fehler sich auf die Qualität der Reproduktionsgüte zwischen "gemessen" versus berechnet der rückgerechneten Matrizen auswirkt. Das ist hier dokumentiert. Interessant ist jedenfalls, daß die Meßfehler hier nur geringe Auswirkungen auf die Eigenwerte haben, was gegen Screetest und Kaisers grobschlächtiges Kriterium spricht. Der Fehler hat glücklicherweise keine Auswirkungen auf die wichtigen Aussagen, Schlußfolgerungen und Interpretation, weil auch die fehlerhafte Berechnung die entsprechenden Parameter im Thurstone'schen Sinne berücksichtigt, obwohl die Auswirkungen auf die Korrelationskoeffizienten (Zeile/Spalte 12,13) teilweise erheblich sind (Residuen =  Fehlerbehaftete Korrelationsmatrix - Richtige Korrelationsmatrix):

Residual-Analysis:
Mean= .04556031  Sigma= .11576164  Maximum range= .50578127 (r4.13)

Matrix residuals (whole matrix inclusive diagonal):
  Mean absolute values of residuals =  .045560306322708245
  Sigma absolute values of residuals = .11576164489090045
  Maximum range absolute values =  .50578127183620955 (r4.13)

Matrix residuals upper triangular matrix without diagonal:
  Mean absolute values of residuals =  .048597660077555462
  Sigma absolute values of residuals = .1189392
  Maximum range absolute values =  .50578127183620955 (r4.13)

Matrix of residuals
 0     0     0     0     0     0     0     0     0     0     0    -.09   .103
 0     0     0     0     0     0     0     0     0     0     0     .001  .014
 0     0     0     0     0     0     0     0     0     0     0    -.085 -.243
 0     0     0     0     0     0     0     0     0     0     0     .452  .506
 0     0     0     0     0     0     0     0     0     0     0     .366  .337
 0     0     0     0     0     0     0     0     0     0     0     .356  .461
 0     0     0     0     0     0     0     0     0     0     0     .063  .046
 0     0     0     0     0     0     0     0     0     0     0     .185  .122
 0     0     0     0     0     0     0     0     0     0     0     .070 -.007
 0     0     0     0     0     0     0     0     0     0     0     .352  .400
 0     0     0     0     0     0     0     0     0     0     0     .424  .477
-.09   .001 -.085  .452  .366  .356  .063  .185  .070  .352  .424  0    -.041
 .103  .014 -.243  .506  .337  .461  .046  .122 -.007  .400  .477 -.041  0
 0     0     0     0     0     0     0     0     0     0     0    -.1    .102
 0     0     0     0     0     0     0     0     0     0     0     .012  .158
 0     0     0     0     0     0     0     0     0     0     0    -.038  .22

Ausblick: Inzwischen bin ich mit dem Thurstone'schen Trapezoiden so vertraut, daß ich mit ihm eine realistische Fehlersimulationsstudie (N=300) plane, um empirische Information darüber zu erhalten, wie verändert sich die Faktorenstruktur und die Eigenwerte des Trapezoiden, wenn man mit diesem oder jenem Fehlermodell arbeitet. 



Querverweise:
Erste, fehlerhafte Trapezoid-Studie (K7_3.htm)
Thurstone Biographie
Aus 4-Faktoren rückgerechnete Thurstone'schen Trapezoid Korrelationsmatrix*
Standard-Matrix-Analyse der Primary Mental Abilities von Thurstone, L. L. (1938).
Kritik der Handhabung der Faktorenanalyse
Für NichtmethodikerInnen: worauf kommt es an bei Korrelationsmatrizen
Für professionell Interessierte: Abkürzungen, Definition, Erklärung und Bedeutung zur
Standard- (Korrelations)- Matrix- Analyse (SMA)
Gesamtzusammenfassung: "Numerisch instabile Matrizen und Kollinearität in der Psychologie"
Hintergrund und Entstehungsgeschichte der Arbeit "Numerisch instabile Matrizen und Kollinearität in der Psychologie"

 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: http://ww.iec-verlag.de


Zitierung
Sponsel, Rudolf  (DAS). Korrigierte 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_3k.htm
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