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Willkommen in der Abteilung Wissenschaftstheorie, Methodologie und Statistisch-Mathematische Methoden in der Allgemeinen und Integrativen Psychologie, Psychodiagnostik und Psychotherapie, hier zur Korrelations-Matrizen-Analyse der:
Primary mental abilitiesThurstone, L. L. (1938). Primary Mental Abilities. Chicago: The University of Chicago Press. p.110-111: Table 2.
Querverweise: Thurstone Biographie
Dokumentation Rundungsfehler & Kollinearität am THURSTONEschen Trapezoid- Beispiel.
Aus 4-Faktoren rückgerechnete Thurstone'schen Trapezoid Korrelationsmatrix*
Kritik der Handhabung der FaktorenanalyseSiehe auch: Holzinger, K. J. & Harman, H. H. (1938). Comparison of Two Factorial Analyses. Psychometrica 3,1, p. 46-47, Table 1, (N = 240 Students?).
Beachte bitte: Die Zählweise sowohl bei Thurstone als auch bei Holzinger & Harman beginnt bei 4: 4...60 entspricht also in unserer Zählweise 1...57.Zusammenfassung - Abstrakt Matrix-Analyse
a) Beschreibung: 14 negative Eigenwerte zeigen mathematisch völlig unzweideutig, daß Thurstone Fehler gemacht haben muß, für die sich bislang niemand interessiert, weil man wohl seiner Autorität vertraut hat und von daher gar nicht auf Idee kam, nachzurechnen, abgesehen davon, daß es sehr mühsam - und die Eintipperei und notwendige Kontrolle auch sehr langweilig - ist. Ich war zunächst selbst schockiert und wir haben immer und immer wieder nachgerechnet, geprüft und geprüft. Leider scheint meine Matrixanalyse richtig: Bereits die ersten 22 Variablen produzieren einen negativen Eigenwert.
Ursachenforschung: p.16 der "Primary Mental Abilities" entnehmen wir, daß es wahrscheinlich MISSING DATA gegeben hat: "The total experimental group consisted 240 volunteer subjects, and 218 (91 per cent) finished all five sections." p.58-59 entnehmen wir, daß der tetrachorische Koeffizient aus rechnerischen Ökonomiegründen (damals hatte man noch keine Computer, sondern rechnete mit Tischrechenmaschinen!) gewählt wurde. Ich hege allerdings die Vermutung, daß man damals den tetrachorischen Korrelationskoeffizienten auch deshalb bevorzugte, weil er gewöhnlich zu betragsmäßig höheren Korrelationen führte. Aus der Art der Argumentation und Verteidigung folgt vermutlich, daß die Normalverteilungs- Voraussetzung mehrfach nicht erfüllt war. Damit hat Thurstone wahrscheinlich eine schwerwiegende Verletzung seiner Voraussetzungen wohlwissend in Kauf genommen, ein Verhalten, das in Medizin, Psychologie und in den Sozialwissenschaften regelrecht die Norm ist. Das MISSING DATA Problem mußte er nicht unbedingt kennen, hätte er aber bei seinem mathematisch gebildeten- Ingenieur Hintergrund kennen sollen. Allerdings gehört es noch nicht einmal heute, im Jahre 2001, zur statistisch- methodischen Standardausbildung in Medizin, Psychologie und den Sozialwissenschaften, auf die Gefahren der negativen Eigenwertproduktion durch falsche Missing Data Lösungen hinzuweisen. Auch die Statistikpakete halten sich kaufmännisch wohlmotiviert zurück: der Kunde will meist Ergebnisse und keine Probleme. Aber die Normalverteilungsvoraussetzung diskutiert Thurstone ja und die zu prüfen dürfte auch zu seiner Zeit sicher kein Problem gewesen sein.b) Zu meiner Überraschung konnte der Mathematiker Dr. HAIN zeigen (in Sponsel 1994, -> 8 HAIN 5.5 ), daß die bekannte Schätzformel "cosinus Pi" Korrelationskoeffizienten* liefert, die keine positive (semi) definite Matrix ergeben.
Zur Problematik Sponsel 1994, -> 1.5.2.2. (Zur Definition ENZYCLOPEDIA OF STATISTICAL SCIENCES VOL.9 Wiley New York 1988 p.224)c) Man beachte, daß die THURSTONE Matrix von 4-60 nummeriert ist, d.h. r1.=4, r2. =5, r3. = 6... . Thurstone-r31.50 entspricht hier also Sponsel r28.47
d) Die Auswirkungen eines Kleinen Druckfehlers
Über die Auswirkungen eines kleinen Druckfehlers auf die Numerische Stabilität dieser Matrix: Die Originalmatrix Thurstones enthält einen Druckfehler: r50.31(Thurstone) = -.10 und r31.50 (Thurstone)= -0.01. Holzinger und Harman haben von Thurstone die obere Dreiecksmatrix genommen, also mit r31.50 (Thurstone) = -.10 gerechnet. Wir haben hier mit r50.31(Thurstone) = -.01. Beachte zur Zählweise bitte. Ob und wie sich dieser Druckfehler und kleine Unterschied auswirkt, kann folgenden abstracts entnommen werden:Thurstone Matrix mit mit r50.31=r31.50 = - 0.10 [Interne Datenquelle THU57T.T57]
Samp Or MD NumS Condit Determinant HaInRatio R_OutIn K_Norm C_Norm
240 57 -2 --14 15648.6 2.38 D-25 8.20D-79 834.1 .001(1) -1(-1)Holzinger's & Harman's Thurstone Matrix mit r50.31=r31.50 = - 0.01 [Interne Datenquelle HOHA57T.T57]
Samp Or MD NumS Condit Determinant HaInRatio R_OutIn K_Norm C_Norm
240 57 -2 --14 12276.7 3.6 D-25 5.79D-74 708.3 1D-3(1) -1(-1)Wie instabil und sensibel diese Matrix reagiert, sieht man, wenn man beide Konditionszahlen vergleicht: COND(r50.31)= 15649, COND(r31.50) = 12277. Auch eine Veränderung in der dritten Nachkommastelle um eine eine Einheit führt für den Fall r31.50=r50.31= -.10 beim Output zu einer Veränderung und das 834-fache gegenüber r31.50=r50.31= -.01 um das "nur" 708-fache.
********* Summary of tetrachoric correlation matrix analysis **********
Erläuterungen zur Matrixanalyse:
Numerische Laien hier und Professionell Interessierte hierFile = THU57T.T57 N-order= 57 N-sample= 240 Rank= 57 Missing data = -2
Positiv Definit=Cholesky successful________= No with 14 negat. eigenvalue/s
HEVA: Highest eigenvalue abs.value_________= 19.678461258904254
LEVA: Lowest eigenvalue absolute value_____= 1.2575234749248286D-3
CON: Condition number HEVA/LEVA___________~= 15648.583625908519
DET: Determinant original matrix___________= 2.3784354213035809D-25
HAC: HADAMARD condition number_____________= 1.5837611252399581D-50
HCN: Heuristic condition |DET|CON__________= 1.5199045985003609D-29
D_I: Determinant Inverse absolute value____= 4.2044446153258037D+24
HDA: HADAMARD Inequality absolute value___<= 5.1255868210553229D+102
HIR: HADAMARD RATIO: D_I / HDA ____________= 8.2028551307616669D-79
Highest inverse positive diagonal value____= 1.868770748
thus multiple r( 12.rest)_________________= .681827606
Highest inverse negative diagonal value____= -.050723934
thus multiple r( 17.rest)_________________= 4.551324978 (!)
and there are 53 multiple r > 1 (!)
Maximum range (upp-low) multip-r( 15.rest)_= 4.041
LES: Numerical stability analysis:
Ratio maximum range output / input _______= 834.0764659493631
PESO-Analysis correlation least Ratio RN/ON= 1.041D-3 (Angle = .06 )
Number of Ratios correlation RN/ON < .01__ = 1
PESO-Analysis Cholesky least Ratio RN/ON__ = (not positiv definit)Ncor L1-Norm L2-Norm Max Min m|c| s|c| N_comp M-S S-S
3249 1072.2 21.54 1 -.22 .318 .162 1272810 .185 .136class boundaries and distribution of the correlation-coefficients
-1 -.8 -.6 -.4 -.2 0 .2 .4 .6 .8 1
0 0 0 2 74 780 1410 754 168 61Original input data with 2-digit-accuracy and calculated with 2-digit-accuracy (for control here the analyzed original matrix; Im memoriam: Man beachte, daß die THURSTONE Matrix von 4-60 nummeriert ist, d.h. r1. =4, r2. =5, r3. = 6... ):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 .83 .54 .49 .19 .47 .47 .58 .35 .27 .37 .14 .46 .13 .1 .28
2 .83 1 .63 .61 .15 .42 .61 .64 .42 .34 .42 .28 .54 .15 .25 .46
3 .54 .63 1 .76 .46 .37 .52 .55 .29 .31 .64 .24 .48 .36 .48 .38
4 .49 .61 .76 1 .31 .37 .57 .62 .31 .42 .59 .22 .48 .34 .34 .25
5 .19 .15 .46 .31 1 .1 .05 .15 .3 .31 .33 .22 .19 .31 .59 .4
6 .47 .42 .37 .37 .1 1 .49 .43 .16 .37 .28 .37 .37 .06 -.1 -.06
7 .47 .61 .52 .57 .05 .49 1 .73 .51 .52 .6 .37 .72 .26 .26 .28
8 .58 .64 .55 .62 .15 .43 .73 1 .57 .46 .6 .35 .7 .32 .43 .33
9 .35 .42 .29 .31 .3 .16 .51 .57 1 .48 .37 .46 .59 .16 .4 .37
10 .27 .34 .31 .42 .31 .37 .52 .46 .48 1 .28 .56 .46 .21 .28 .25
11 .37 .42 .64 .59 .33 .28 .6 .6 .37 .28 1 .22 .46 .25 .34 .31
12 .14 .28 .24 .22 .22 .37 .37 .35 .46 .56 .22 1 .48 .06 .19 .07
13 .46 .54 .48 .48 .19 .37 .72 .7 .59 .46 .46 .48 1 .09 .18 .25
14 .13 .15 .36 .34 .31 .06 .26 .32 .16 .21 .25 .06 .09 1 .61 .51
15 .1 .25 .48 .34 .59 -.1 .26 .43 .4 .28 .34 .19 .18 .61 1 .47
16 .28 .46 .38 .25 .4 -.06 .28 .33 .37 .25 .31 .07 .25 .51 .47 1
17 0 .12 .37 .21 .42 -.16 .08 .2 .22 .21 .28 .12 .06 .48 .68 .62
18 .32 .46 .58 .46 .51 .21 .43 .62 .43 .37 .37 .28 .37 .61 .61 .67
19 .2 .34 .56 .31 .54 -.07 .22 .29 .31 .3 .4 .15 .18 .4 .56 .64
20 .22 .27 .42 .3 .42 .15 .32 .32 .35 .23 .31 .22 .2 .42 .56 .54
21 .28 .4 .47 .33 .51 .05 .23 .33 .37 .33 .15 .15 .2 .5 .42 .63
22 .28 .34 .44 .3 .4 .21 .32 .47 .37 .4 .18 .18 .34 .45 .45 .43
23 .25 .33 .71 .42 .22 .15 .22 .46 .25 .12 .4 .1 .37 .4 .45 .29
24 .17 .09 .33 .25 .4 .09 .13 .22 .06 .25 .37 .25 0 .46 .61 .38
25 .21 .35 .58 .4 .53 .12 .35 .38 .31 .15 .4 .22 .24 .45 .47 .43
26 .43 .32 .44 .2 .31 .05 .3 .3 .34 .18 .28 .19 .18 .4 .42 .35
27 .19 .27 .45 .43 .4 .07 .42 .3 .3 .25 .4 .28 .46 .58 .48 .39
28 .01 .11 .18 .18 .19 .06 .1 -.06 .12 .31 .21 .16 .12 .27 .21 .17
29 .26 .25 .25 .28 .1 .18 .4 .07 .28 .28 .37 .25 .25 .12 .14 .17
30-.03 .08 .28 .17 .1 .17 .18 .15 .35 .22 .32 .26 .22 .26 .26 .15
31 .18 .16 .4 .15 .12 .06 .29 .26 .25 .25 .28 .16 .3 .22 .37 .22
32 .4 .43 .41 .27 .37 .11 .3 .3 .29 .27 .25 .22 .37 .18 .36 .36
33 .26 .31 .2 .24 -.03 .06 .3 .3 .11 .06 .34 -.1 .28 .4 .26 .33
34 .4 .38 .45 .43 .52 .1 .37 .18 .29 .26 .44 .26 .35 .26 .36 .42
35 .32 .34 .31 .33 .16 .09 .2 .35 .22 0 .25 .16 .18 .28 .27 .22
36 .34 .31 .5 .37 .3 .16 .42 .42 .34 .16 .35 .31 .29 .5 .39 .34
37 .59 .66 .42 .48 .28 .21 .43 .5 .34 .4 .42 .3 .56 .11 .27 .46
38 .57 .64 .64 .64 .46 .4 .6 .6 .4 .46 .59 .28 .61 .46 .46 .49
39 .58 .54 .48 .43 .43 .22 .43 .42 .31 .4 .38 .22 .52 .13 .29 .29
40 .52 .6 .68 .58 .53 .18 .46 .5 .53 .34 .46 .28 .54 .45 .4 .55
41 .51 .59 .6 .59 .51 .21 .48 .62 .45 .37 .46 .31 .47 .37 .52 .49
42 .52 .51 .65 .48 .52 .28 .44 .39 .35 .43 .31 .19 .4 .37 .51 .47
43 .43 .25 .3 .18 -.02 .28 .22 .32 .29 .18 .28 .19 .27 .1 .12 .1
44 .43 .53 .31 .31 .06 .42 .57 .4 .29 .48 .31 .45 .54 .03 -.03 .07
45 .25 .38 .32 .1 .21 .13 .28 .21 .07 .13 .29 .22 .17 .04 .2 .12
46 .27 .48 .4 .3 .19 .11 .32 .35 .38 .24 .3 .31 .4 .18 .15 .31
47 .22 .28 .33 .31 .22 .08 .3 .36 .25 .19 .21 .15 .21 .08 .2 .32
48 .07 .12 .29 .28 .02 .16 .25 .28 .29 .03 .48 .04 .21 .25 .26 .03
49 .51 .62 .51 .45 .15 .43 .69 .46 .25 .31 .37 .16 .49 .03 .16 .34
50-.05 .02 .25 .09 .21 0 .13 .13 .21 .21 .34 .22 .09 .48 .42 .34
51 .35 .31 .36 .09 .25 .21 .35 .23 .25 .21 .31 .34 .3 .05 .31 .32
52 .4 .46 .61 .59 .51 .25 .49 .4 .38 .37 .51 .31 .51 .18 .36 .49
53 .4 .42 .48 .45 .16 .31 .57 .43 .46 .4 .34 .37 .46 .08 .18 .16
54 .42 .53 .48 .46 .3 .28 .69 .49 .46 .48 .45 .53 .61 .11 .29 .37
55 .57 .75 .38 .46 0 .5 .66 .76 .35 .31 .3 .2 .46 .11 .05 .35
56 .15 .26 .38 .29 .12 .29 .19 .25 .22 .1 .31 .07 .34 .07 .04 .1
57 .72 .76 .67 .67 .28 .25 .61 .77 .62 .48 .43 .4 .68 .28 .35 .37
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 0 .32 .2 .22 .28 .28 .25 .17 .21 .43 .19 .01 .26 -.03 .18 .4
2 .12 .46 .34 .27 .4 .34 .33 .09 .35 .32 .27 .11 .25 .08 .16 .43
3 .37 .58 .56 .42 .47 .44 .71 .33 .58 .44 .45 .18 .25 .28 .4 .41
4 .21 .46 .31 .3 .33 .3 .42 .25 .4 .2 .43 .18 .28 .17 .15 .27
5 .42 .51 .54 .42 .51 .4 .22 .4 .53 .31 .4 .19 .1 .1 .12 .37
6 -.16 .21 -.07 .15 .05 .21 .15 .09 .12 .05 .07 .06 .18 .17 .06 .11
7 .08 .43 .22 .32 .23 .32 .22 .13 .35 .3 .42 .1 .4 .18 .29 .3
8 .2 .62 .29 .32 .33 .47 .46 .22 .38 .3 .3 -.06 .07 .15 .26 .3
9 .22 .43 .31 .35 .37 .37 .25 .06 .31 .34 .3 .12 .28 .35 .25 .29
10 .21 .37 .3 .23 .33 .4 .12 .25 .15 .18 .25 .31 .28 .22 .25 .27
11 .28 .37 .4 .31 .15 .18 .4 .37 .4 .28 .4 .21 .37 .32 .28 .25
12 .12 .28 .15 .22 .15 .18 .1 .25 .22 .19 .28 .16 .25 .26 .16 .22
13 .06 .37 .18 .2 .2 .34 .37 0 .24 .18 .46 .12 .25 .22 .3 .37
14 .48 .61 .4 .42 .5 .45 .4 .46 .45 .4 .58 .27 .12 .26 .22 .18
15 .68 .61 .56 .56 .42 .45 .45 .61 .47 .42 .48 .21 .14 .26 .37 .36
16 .62 .67 .64 .54 .63 .43 .29 .38 .43 .35 .39 .17 .17 .15 .22 .36
17 1 .59 .61 .63 .53 .33 .21 .57 .34 .33 .58 .28 .2 .23 .27 .2
18 .59 1 .58 .58 .74 .61 .4 .49 .7 .55 .48 .06 .24 .23 .24 .4
19 .61 .58 1 .57 .62 .42 .25 .49 .42 .42 .48 .19 .21 .22 .3 .33
20 .63 .58 .57 1 .55 .36 .43 .43 .5 .41 .45 .14 .16 .11 .27 .3
21 .53 .74 .62 .55 1 .61 .3 .4 .42 .52 .47 .11 .15 .21 .24 .35
22 .33 .61 .42 .36 .61 1 .21 .24 .48 .42 .37 .11 .07 .04 .15 .42
23 .21 .4 .25 .43 .3 .21 1 .22 .42 0 .22 0 .02 .03 .19 .16
24 .57 .49 .49 .43 .4 .24 .22 1 .42 .43 .4 .34 .21 .29 .4 .22
25 .34 .7 .42 .5 .42 .48 .42 .42 1 .49 .4 .3 .19 .26 .28 .43
26 .33 .55 .42 .41 .52 .42 0 .43 .49 1 .45 .18 .19 .29 .42 .52
27 .58 .48 .48 .45 .47 .37 .22 .4 .4 .45 1 .64 .57 .62 .51 .52
28 .28 .06 .19 .14 .11 .11 0 .34 .3 .18 .64 1 .62 .62 .54 .4
29 .2 .24 .21 .16 .15 .07 .02 .21 .19 .19 .57 .62 1 .67 .53 .31
30 .23 .23 .22 .11 .21 .04 .03 .29 .26 .29 .62 .62 .67 1 .62 .47
31 .27 .24 .3 .27 .24 .15 .19 .4 .28 .42 .51 .54 .53 .62 1 .56
32 .2 .4 .33 .3 .35 .42 .16 .22 .43 .52 .52 .4 .31 .47 .56 1
33 .27 .3 .33 .11 .2 .17 0 .2 .18 .3 .36 .06 .15 .14 .11 .13
34 .17 .33 .41 .33 .48 .3 .16 .34 .5 .42 .4 .35 .38 .3 .41 .68
35 .23 .3 .18 .34 .3 .28 .1 .28 .3 .43 .43 .28 .37 .52 .57 .48
36 .34 .57 .52 .42 .53 .52 .16 .25 .55 .56 .66 .29 .38 .48 .62 .58
37 .3 .42 .3 .33 .48 .51 .1 .22 .33 .39 .4 .21 .33 .1 .27 .45
38 .42 .54 .4 .46 .49 .46 .37 .29 .52 .43 .6 .25 .34 .16 .28 .4
39 .16 .26 .29 .32 .39 .35 .22 .16 .2 .41 .39 .18 .22 .11 .29 .36
40 .5 .56 .53 .5 .72 .48 .4 .34 .49 .47 .66 .3 .28 .33 .48 .36
41 .28 .62 .28 .43 .5 .57 .34 .46 .57 .57 .47 .16 .16 .1 .28 .26
42 .48 .5 .64 .52 .55 .4 .35 .43 .51 .54 .5 .28 .29 .38 .45 .6
43-.04 .32 .33 .3 .2 .2 .21 .02 .24 .2 .26 .14 .12 .11 .14 .3
44 .03 .37 .19 .3 .33 .28 .28 .16 .21 .28 .35 .18 .21 .2 .19 .23
45 .16 .2 .32 .27 .2 .1 .1 .1 .39 .36 .41 .38 .07 .3 .26 .39
46 .05 .42 .33 .18 .36 .3 .16 .03 .28 .42 .3 .05 .12 .04 .09 .23
47 .15 .42 .23 .2 .29 .23 .2 -.1 .23 .35 .17 -.1 .09 .02 -.01 .13
48 .06 .19 .1 .12 .1 .16 .28 .12 .21 .1 .32 .16 .1 .16 .1 .1
49-.04 .37 .25 .25 .25 .19 .25 .12 .29 .22 .17 .12 .4 .29 .28 .22
50 .59 .37 .53 .4 .33 .15 .03 .4 .33 .42 .5 .24 .22 .3 .3 .27
51 .11 .23 .2 .3 .35 .02 .06 .29 .12 .29 .05 .05 .24 .04 .2 .32
52 .37 .37 .42 .33 .33 .15 .25 .46 .28 .33 .49 .22 .4 .22 .4 .39
53 .02 .33 .18 .15 .05 .24 .19 .07 .18 .11 .18 .09 .37 .26 .3 .23
54 .21 .33 .28 .28 .28 .28 .12 .16 .31 .32 .32 .19 .42 .26 .43 .33
55 .12 .42 .03 .29 .3 .15 .2 .1 .1 .09 .16 .14 .19 .12 .13 .16
56 0 -.06 .1 -.02 .08 .1 .22 0 .1 .17 .17 .26 .1 .24 .22 .17
57 .27 .46 .22 .21 .37 .6 .4 .11 .46 .4 .35 .12 .12 .1 .3 .37
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1 .26 .4 .32 .34 .59 .57 .58 .52 .51 .52 .43 .43 .25 .27 .22 .07
2 .31 .38 .34 .31 .66 .64 .54 .6 .59 .51 .25 .53 .38 .48 .28 .12
3 .2 .45 .31 .5 .42 .64 .48 .68 .6 .65 .3 .31 .32 .4 .33 .29
4 .24 .43 .33 .37 .48 .64 .43 .58 .59 .48 .18 .31 .1 .3 .31 .28
5 -.03 .52 .16 .3 .28 .46 .43 .53 .51 .52 -.02 .06 .21 .19 .22 .02
6 .06 .1 .09 .16 .21 .4 .22 .18 .21 .28 .28 .42 .13 .11 .08 .16
7 .3 .37 .2 .42 .43 .6 .43 .46 .48 .44 .22 .57 .28 .32 .3 .25
8 .3 .18 .35 .42 .5 .6 .42 .5 .62 .39 .32 .4 .21 .35 .36 .28
9 .11 .29 .22 .34 .34 .4 .31 .53 .45 .35 .29 .29 .07 .38 .25 .29
10 .06 .26 0 .16 .4 .46 .4 .34 .37 .43 .18 .48 .13 .24 .19 .03
11 .34 .44 .25 .35 .42 .59 .38 .46 .46 .31 .28 .31 .29 .3 .21 .48
12-.1 .26 .16 .31 .3 .28 .22 .28 .31 .19 .19 .45 .22 .31 .15 .04
13 .28 .35 .18 .29 .56 .61 .52 .54 .47 .4 .27 .54 .17 .4 .21 .21
14 .4 .26 .28 .5 .11 .46 .13 .45 .37 .37 .1 .03 .04 .18 .08 .25
15 .26 .36 .27 .39 .27 .46 .29 .4 .52 .51 .12 -.03 .2 .15 .2 .26
16 .33 .42 .22 .34 .46 .49 .29 .55 .49 .47 .1 .07 .12 .31 .32 .03
17 .27 .17 .23 .34 .3 .42 .16 .5 .28 .48 -.04 .03 .16 .05 .15 .06
18 .3 .33 .3 .57 .42 .54 .26 .56 .62 .5 .32 .37 .2 .42 .42 .19
19 .33 .41 .18 .52 .3 .4 .29 .53 .28 .64 .33 .19 .32 .33 .23 .1
20 .11 .33 .34 .42 .33 .46 .32 .5 .43 .52 .3 .3 .27 .18 .2 .12
21 .2 .48 .3 .53 .48 .49 .39 .72 .5 .55 .2 .33 .2 .36 .29 .1
22 .17 .3 .28 .52 .51 .46 .35 .48 .57 .4 .2 .28 .1 .3 .23 .16
23 0 .16 .1 .16 .1 .37 .22 .4 .34 .35 .21 .28 .1 .16 .2 .28
24 .2 .34 .28 .25 .22 .29 .16 .34 .46 .43 .02 .16 .1 .03 -.1 .12
25 .18 .5 .3 .55 .33 .52 .2 .49 .57 .51 .24 .21 .39 .28 .23 .21
26 .3 .42 .43 .56 .39 .43 .41 .47 .57 .54 .2 .28 .36 .42 .35 .1
27 .36 .4 .43 .66 .4 .6 .39 .66 .47 .5 .26 .35 .41 .3 .17 .32
28 .06 .35 .28 .29 .21 .25 .18 .3 .16 .28 .14 .18 .38 .05 -.1 .16
29 .15 .38 .37 .38 .33 .34 .22 .28 .16 .29 .12 .21 .07 .12 .09 .1
30 .14 .3 .52 .48 .1 .16 .11 .33 .1 .38 .11 .2 .3 .04 .02 .16
31 .11 .41 .57 .62 .27 .28 .29 .48 .28 .45 .14 .19 .26 .09 -.01 .1
32 .13 .68 .48 .58 .45 .4 .36 .36 .26 .6 .3 .23 .39 .23 .13 .1
33 1 .23 .09 .3 .28 .4 .2 .33 .31 .27 .04 .12 .04 .08 .11 .19
34 .23 1 .43 .72 .55 .54 .42 .58 .43 .46 .24 .3 .46 .2 .18 .1
35 .09 .43 1 .64 .37 .16 .32 .42 .46 .33 .06 .1 .4 .18 .15 .22
36 .3 .72 .64 1 .5 .48 .46 .62 .54 .5 .32 .22 .37 .23 .17 .19
37 .28 .55 .37 .5 1 .54 .78 .48 .49 .42 .23 .5 .2 .33 .33 -.07
38 .4 .54 .16 .48 .54 1 .4 .72 .67 .5 .19 .31 .16 .25 .31 .25
39 .2 .42 .32 .46 .78 .4 1 .47 .48 .51 .32 .43 .37 .38 .23 .06
40 .33 .58 .42 .62 .48 .72 .47 1 .69 .66 .17 .48 .44 .27 .42 .28
41 .31 .43 .46 .54 .49 .67 .48 .69 1 .43 .18 .31 .22 .31 .29 .34
42 .27 .46 .33 .5 .42 .5 .51 .66 .43 1 .23 .37 .2 .3 .13 .03
43 .04 .24 .06 .32 .23 .19 .32 .17 .18 .23 1 .43 .42 .36 .26 .13
44 .12 .3 .1 .22 .5 .31 .43 .48 .31 .37 .43 1 .43 .48 .42 .19
45 .04 .46 .4 .37 .2 .16 .37 .44 .22 .2 .42 .43 1 .2 .36 .25
46 .08 .2 .18 .23 .33 .25 .38 .27 .31 .3 .36 .48 .2 1 .4 .19
47 .11 .18 .15 .17 .33 .31 .23 .42 .29 .13 .26 .42 .36 .4 1 .19
48 .19 .1 .22 .19 -.07 .25 .06 .28 .34 .03 .13 .19 .25 .19 .19 1
49 .06 .34 .37 .34 .46 .43 .48 .4 .34 .38 .19 .48 .34 .38 .25 .31
50 .19 .17 .22 .29 .21 .28 .07 .37 .16 .39 .15 .03 .26 .2 .24 .12
51 .04 .27 .18 .13 .23 .34 .13 .23 .28 .33 .07 .28 .08 .2 .06 .12
52 .11 .38 .28 .29 .33 .59 .41 .62 .59 .54 .12 .25 .13 .4 .23 .28
53 .2 .39 .4 .46 .45 .28 .52 .24 .31 .33 .27 .33 .13 .33 .14 .1
54 .28 .39 .37 .4 .62 .46 .64 .43 .53 .43 .18 .43 .25 .28 .28 0
55-.03 .18 .3 .38 .27 .49 .24 .31 .22 .33 .18 .32 -.1 .28 .19 .12
56-.06 .21 .25 .06 .07 .16 .16 .2 .19 .11 .13 .03 .16 .13 -.12 .46
57 .24 .31 .46 .52 .58 .62 .62 .62 .63 .43 .28 .45 .38 .37 .37 .37
49 50 51 52 53 54 55 56 57
1 .51 -.05 .35 .4 .4 .42 .57 .15 .72
2 .62 .02 .31 .46 .42 .53 .75 .26 .76
3 .51 .25 .36 .61 .48 .48 .38 .38 .67
4 .45 .09 .09 .59 .45 .46 .46 .29 .67
5 .15 .21 .25 .51 .16 .3 0 .12 .28
6 .43 0 .21 .25 .31 .28 .5 .29 .25
7 .69 .13 .35 .49 .57 .69 .66 .19 .61
8 .46 .13 .23 .4 .43 .49 .76 .25 .77
9 .25 .21 .25 .38 .46 .46 .35 .22 .62
10 .31 .21 .21 .37 .4 .48 .31 .1 .48
11 .37 .34 .31 .51 .34 .45 .3 .31 .43
12 .16 .22 .34 .31 .37 .53 .2 .07 .4
13 .49 .09 .3 .51 .46 .61 .46 .34 .68
14 .03 .48 .05 .18 .08 .11 .11 .07 .28
15 .16 .42 .31 .36 .18 .29 .05 .04 .35
16 .34 .34 .32 .49 .16 .37 .35 .1 .37
17-.04 .59 .11 .37 .02 .21 .12 0 .27
18 .37 .37 .23 .37 .33 .33 .42 -.06 .46
19 .25 .53 .2 .42 .18 .28 .03 .1 .22
20 .25 .4 .3 .33 .15 .28 .29 -.02 .21
21 .25 .33 .35 .33 .05 .28 .3 .08 .37
22 .19 .15 .02 .15 .24 .28 .15 .1 .6
23 .25 .03 .06 .25 .19 .12 .2 .22 .4
24 .12 .4 .29 .46 .07 .16 .1 0 .11
25 .29 .33 .12 .28 .18 .31 .1 .1 .46
26 .22 .42 .29 .33 .11 .32 .09 .17 .4
27 .17 .5 .05 .49 .18 .32 .16 .17 .35
28 .12 .24 .05 .22 .09 .19 .14 .26 .12
29 .4 .22 .24 .4 .37 .42 .19 .1 .12
30 .29 .3 .04 .22 .26 .26 .12 .24 .1
31 .28 .3 .2 .4 .3 .43 .13 .22 .3
32 .22 .27 .32 .39 .23 .33 .16 .17 .37
33 .06 .19 .04 .11 .2 .28 -.03 -.06 .24
34 .34 .17 .27 .38 .39 .39 .18 .21 .31
35 .37 .22 .18 .28 .4 .37 .3 .25 .46
36 .34 .29 .13 .29 .46 .4 .38 .06 .52
37 .46 .21 .23 .33 .45 .62 .27 .07 .58
38 .43 .28 .34 .59 .28 .46 .49 .16 .62
39 .48 .07 .13 .41 .52 .64 .24 .16 .62
40 .4 .37 .23 .62 .24 .43 .31 .2 .62
41 .34 .16 .28 .59 .31 .53 .22 .19 .63
42 .38 .39 .33 .54 .33 .43 .33 .11 .43
43 .19 .15 .07 .12 .27 .18 .18 .13 .28
44 .48 .03 .28 .25 .33 .43 .32 .03 .45
45 .34 .26 .08 .13 .13 .25 -.1 .16 .38
46 .38 .2 .2 .4 .33 .28 .28 .13 .37
47 .25 .24 .06 .23 .14 .28 .19 -.12 .37
48 .31 .12 .12 .28 .1 0 .12 .46 .37
49 1 -.03 .34 .43 .46 .46 .56 .31 .57
50-.03 1 .21 .37 -.08 .28 -.22 -.1 .09
51 .34 .21 1 .56 .3 .37 .38 .06 .26
52 .43 .37 .56 1 .46 .61 .38 .1 .59
53 .46 -.08 .3 .46 1 .63 .6 .17 .62
54 .46 .28 .37 .61 .63 1 .36 .07 .62
55 .56 -.22 .38 .38 .6 .36 1 .32 .82
56 .31 -.1 .06 .1 .17 .07 .32 1 .1
57 .57 .09 .26 .59 .62 .62 .82 .1 1Eigenwerte und Choleskydiagonalwerte
i.Eigenvalue Cholesky i.Eigenvalue Cholesky i.Eigenvalue Cholesky
1. 19.67846 1 2. 5.19114 .5578 3. 3.35744 .776
4. 2.35466 .6257 5. 2.1762 .8638 6. 1.95755 .8651
7. 1.74496 .6994 8. 1.59637 .6034 9. 1.52515 .7265
10. 1.45556 .751 11. 1.27754 .644 12. 1.23148 .7331
13. 1.14481 .5846 14. 1.05792 .8182 15. .98536 .4343
16. .94809 .5419 17. .86258 .5226 18. .83744 .2889
19. .80945 .505 20. .75337 .5059 21. .71427 -.4546
22. .68618 -.0575 23. .64001 -.493 24. .58293 -2.5069
25. .5695 -.8768 26. .55434 -.8157 27. .5038 -.7852
28. .48461 -.1745 29. .45026 -3.1043 30. .42415 -1.3314
31. .35656 -1.4762 32. .31896 -1.4744 33. .28409 -1.1002
34. .26997 -2.8381 35. .23416 -1.4704 36. .21798 -3.6146
37. .20396 -2.069 38. .16241 -2.7744 39. .12261 -2.0196
40. .11029 -5.7833 41. .09877 -4.0464 42. .04938 -4.1987
43. .01952 -4.704 44.-1.26D-3 -2.0232 45.-.01892 -2.3651
46.-.02492 -1.4692 47.-.0335 -.9889 48.-.06035 -1.0052
49.-.0721 -2.957 50.-.11849 -1.7746 51.-.13074 -1.8597
52.-.15978 -4.2034 53.-.19597 -2.4737 54.-.21303 -4.6772
55.-.25715 -6.1479 56.-.2731 -1.8899 57.-.44492 -8.4668The matrix is not positive definit. Cholesky decomposition is not success-
ful (for detailed information Cholesky's diagonalvalues are presented).Analyzed: 18.03.94 15:43:54 PRG version 17/03/94 MABAT_OP.BAS
Gesamtzeit_____________ 8542
Rang_____________ nicht berechnet
Determinante_____ 75.275
Eigenwerte/Vekt__ 5009
Peso Kor+Chol____ nicht berechnet
NuStabAnalyse____ 614
Statistik________ 2609File = C:\OMI\NUMERIK\MATRIX\SMA_OP\THU57T\THU57T.SMA
with data from C:\OMI\NUMERIK\MATRIX\SMA_OP\THU57T\THU57T.T57
Querverweise: Thurstone Biographie
Dokumentation Rundungsfehler & Kollinearität am THURSTONEschen Trapezoid- Beispiel.
Aus 4-Faktoren rückgerechnete Thurstone'schen Trapezoid Korrelationsmatrix*
Kritik der Handhabung der Faktorenanalyse
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: https://ww.iec-verlag.de
Zitierung
Sponsel, Rudolf (DAS). Standard-Matrix-Analyse der Primary Mental Abilities von Thurstone, L. L. (1938). IP-GIPT. Erlangen: https://www.sgipt.org/wisms/nis/sma/primary.htm
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