ED 795 LAB #6
LAB #6 OUTLINE
- Bartlett’s Test of Sphericity
- KMO Sampling Adequacy
- Sample Factor Analysis
- Saving and Computing factor scores
Bartlett's Test of Sphericity
Null Hypothesis: Population Covariance Matrix is the Identity Matrix
Alternative: Population Covariance Matrix is not the Identity Matrix
Consider the following correlation matrix, which is called an identity matrix.
X1 / X2 / X3 / X4X1 / 1.00
X2 / 0.00 / 1.00
X3 / 0.00 / 0.00 / 1.00
X4 / 0.00 / 0.00 / 0.00 / 1.00
The variables are completely non-collinear. If you factor analyzed this matrix, you could only extract as many factors as variables, since each variable would be its own factor.
In other words, it is effectively non-factorable
We want to get a statistically significant result for the Chi-Square test indicating that the variables do covary.
Kaiser-Meyer-Olkin Measure of Sampling Adequacy (KMO)
If two variables share a common factor with other variables, their partial correlation (aij) will be small, indicating the unique variance they share.
aij = (rij1, 2, 3, …k )
KMO = ( r2ij ) / ( r2ij + ( a2ij )
If aij 0.0
The variables are measuring a
common factor, and KMO 1.0
If aij 1.0
The variables are not measuring a
common factor, and KMO 0.0
Interpretation of the KMO as characterized by Kaiser, Meyer, and Olkin …
KMO Value / Degree of Common Variance0.90 to 1.00 / Marvelous
0.80 to 0.89 / Meritorious
0.70 to 0.79 / Middling
0.60 to 0.69 / Mediocre
0.50 to 0.59 / Miserable
0.00 to 0.49 / Don't Factor
EXAMPLE OUTPUT
1 0 get file '/afs/umich.edu/class/ed793/data/cirp8690.sav'.
2 0 *
File /afs/umich.edu/class/ed793/data/cirp8690.sav
Created: 23 JAN 01 16:24:40 - 642 variables and 4,408 cases
3 0 descriptives variables=goal8601 to goal8618
4 0 /stat = mean stddev.
5 0 *
Number of valid observations (listwise) = 4127.00
Valid
Variable Mean Std Dev N Label
GOAL8601 1.49 .79 4272 ACHIEVE IN PERFORMING ART
GOAL8602 2.90 .80 4278 BECOME AUTHORITY IN OWN FIELD
GOAL8603 2.58 .81 4275 OBTAIN RECOGNITION FROM COLLEAGUES
GOAL8604 1.76 .78 4264 INFLUENCE POLITICAL STRUCTURE
GOAL8605 2.19 .79 4270 INFLUENCE SOCIAL VALUES
GOAL8606 2.88 .91 4273 RAISE A FAMILY
GOAL8607 2.34 .83 4259 HAVE ADMINISTRATIVE RESPONSIBILITY
GOAL8608 2.91 .83 4273 BE VERY WELL OFF FINANCIALLY
GOAL8609 2.67 .73 4268 HELP OTHERS IN DIFFICULTY
GOAL8610 1.60 .79 4263 THEORETICAL CONTRIBUTION TO SCIENCE
GOAL8611 1.51 .80 4264 WRITE ORIGINAL WORKS
GOAL8612 1.47 .79 4256 CREATE ARTISTIC WORK
GOAL8613 2.31 1.01 4259 BE SUCCESSFUL IN OWN BUSINESS
GOAL8614 1.85 .72 4252 BE INVOLVED IN ENVIRONMENTAL CLEANUP
GOAL8615 2.41 .96 4259 DEVELOP MEANINGFUL PHILOSOPHY OF LIFE
GOAL8616 1.96 .74 4251 PARTICIPATE IN COMMUNITY ACTION PROGRAM
GOAL8617 2.13 .86 4250 PROMOTE RACIAL UNDERSTANDING
GOAL8618 1.88 .92 4256 BE EXPERT ON FINANCE/COMMERCE
Preceding task required 1.29 seconds CPU time; 1.00 seconds elapsed.
6 0 rename variables (goal8601 to goal8618 =
7 0 perform authorty recog inflpol inflsoc family
8 0 adminres welloff helpoth
9 0 theory write create ownbus cleanup meanphil
10 0 action raceund expert).
120 factor variables=perform to expert
13 /print=kmo
13 0 /extraction = pc
14 0 /plot eigen
15 0 /rotation = varimax
16 0 /format=sort.
17 0
------F A C T O R A N A L Y S I S
Analysis number 1 Listwise deletion of cases with missing values
Extraction 1 for analysis 1, Principal Components Analysis (PC)
Initial Statistics:
Kaiser-Meyer-Olkin Measure of Sampling Adequacy = .83069
Bartlett Test of Sphericity = 16125.458, Significance = .00000
Variable Communality * Factor Eigenvalue Pct of Var Cum Pct
*
PERFORM 1.00000 * 1 4.11895 22.9 22.9
AUTHORTY 1.00000 * 2 2.26674 12.6 35.5
RECOG 1.00000 * 3 1.46648 8.1 43.6
INFLPOL 1.00000 * 4 1.10865 6.2 49.8
INFLSOC 1.00000 * 5 1.03181 5.7 55.5
FAMILY 1.00000 * 6 .99333 5.5 61.0
ADMINRES 1.00000 * 7 .80842 4.5 65.5
WELLOFF 1.00000 * 8 .71147 4.0 69.5
HELPOTH 1.00000 * 9 .68828 3.8 73.3
THEORY 1.00000 * 10 .67223 3.7 77.0
WRITE 1.00000 * 11 .59675 3.3 80.4
CREATE 1.00000 * 12 .57634 3.2 83.6
OWNBUS 1.00000 * 13 .56331 3.1 86.7
CLEANUP 1.00000 * 14 .55164 3.1 89.7
MEANPHIL 1.00000 * 15 .49336 2.7 92.5
ACTION 1.00000 * 16 .47549 2.6 95.1
RACEUND 1.00000 * 17 .44327 2.5 97.6
EXPERT 1.00000 * 18 .43348 2.4 100.0
------F A C T O R A N A L Y S I S Eigenvalue (Scree) Plot
4.119 + *
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N 2.267 + *
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1.466 + *
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1.109 + *
.993 + * *
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.808 + *
.672 + * * *
.552 + * * * *
.433 + * * * *
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.000 +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
FACTORS
PC extracted 5 factors.
Factor Matrix:
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
INFLSOC .63351 -.10818 -.18241 .01802 .08269
ACTION .63350 -.21597 -.35683 .09182 -.07946
RACEUND .61071 -.31643 -.21982 .10288 -.18521
INFLPOL .60893 .06618 .00819 -.08707 -.16251
MEANPHIL .56564 -.31034 -.07903 .03414 -.06517
RECOG .56138 .29398 .20840 -.39936 .23673
CLEANUP .55830 -.19550 -.10013 .04163 -.38731
AUTHORTY .53116 .29052 .23014 -.37496 .25815
HELPOTH .49986 -.24441 -.36429 -.09517 .21773
WELLOFF .26464 .66917 .20541 .01848 -.02768
EXPERT .45127 .53359 .05483 .26763 -.26625
ADMINRES .45449 .51726 -.12061 .16169 .19837
OWNBUS .38640 .46635 .23974 .37527 -.17767
WRITE .36752 -.46468 .43950 .11990 .11558
CREATE .31212 -.36533 .55047 .25498 .07350
PERFORM .25856 -.39223 .47865 .14202 .26661
THEORY .36883 -.05868 .02732 -.57098 -.13347
FAMILY .16976 .12073 -.41093 .32432 .61917
Final Statistics:
Variable Communality * Factor Eigenvalue Pct of Var Cum Pct
*
PERFORM .54105 * 1 4.11895 22.9 22.9
AUTHORTY .62673 * 2 2.26674 12.6 35.5
RECOG .66054 * 3 1.46648 8.1 43.6
INFLPOL .40924 * 4 1.10865 6.2 49.8
INFLSOC .45347 * 5 1.03181 5.7 55.5
FAMILY .70081 *
ADMINRES .55416 *
WELLOFF .56112 *
HELPOTH .49877 *
THEORY .48405 *
WRITE .57190 *
CREATE .60432 *
OWNBUS .59665 *
11:33:29 University of Michigan SUN SPARC Solaris 2.3
------F A C T O R A N A L Y S I S ------
Variable Communality * Factor Eigenvalue Pct of Var Cum Pct
CLEANUP .51168 *
MEANPHIL .42791 *
ACTION .59003 *
RACEUND .56630 *
EXPERT .63389 *
VARIMAX rotation 1 for extraction 1 in analysis 1 - Kaiser Normalization.
VARIMAX converged in 7 iterations.
Rotated Factor Matrix:
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
ACTION .74745 .06145 .05043 .02079 .15684
RACEUND .73840 .04316 .00128 .13858 .00197
CLEANUP .65393 .16241 .01995 .08221 -.22476
MEANPHIL .60047 .00371 .10361 .23730 .01694
INFLSOC .57708 .10910 .22449 .11713 .21079
HELPOTH .56468 -.16940 .21824 .01447 .32150
INFLPOL .47132 .28615 .30525 .06994 -.08454
EXPERT .20326 .76203 .06718 -.08272 -.02306
OWNBUS .07776 .75854 .00567 .12322 -.00312
WELLOFF -.13953 .65512 .31900 -.10065 .02402
ADMINRES .15046 .54745 .25311 -.11211 .39395
RECOG .12576 .24432 .75300 .10498 .08368
AUTHORTY .08980 .24099 .73140 .12739 .09704
THEORY .29386 -.11641 .54885 -.06002 -.28163
CREATE .12094 .08135 -.01311 .75915 -.08125
PERFORM .06095 -.07422 .08742 .72165 .05835
WRITE .23053 -.06721 .07734 .71077 -.05535
FAMILY .10561 .04743 -.03512 -.05142 .82676
------F A C T O R A N A L Y S I S
Factor Transformation Matrix:
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
Factor 1 .75265 .38546 .44337 .27073 .12275
Factor 2 -.38204 .73041 .26914 -.47960 .13453
Factor 3 -.44838 .23988 .21611 .73188 -.39882
Factor 4 .05679 .44687 -.76546 .30273 .34569
Factor 5 -.28862 -.24636 .31361 .26341 .82962
NOTES:
Syntax options
FACTOR
/VARIABLES betterjb gainge improve nobetter mocultur momoney learnmo
prepgrad par_wish nojob getaway
/PRINT UNIVARIATE INITIAL KMO EXTRACTION ROTATION
/PLOT EIGEN
/CRITERIA MINEIGEN(1) ITERATE(25) (or this could be factors(3) )
/EXTRACTION=PC (or this could be paf)
/ROTATION =VARIMAX. (or this could be oblimin)
Computing factor scores
Use the loadings (this is called the regression method and is the default in SPSS):
COMPUTE socjust = ..75*action + .74*raceund + .65*cleanup + .60*meanphil + .58*inflsoc +.56*helpoth + .47*inflpol.
OR
Do not use the loadings and create an average (this puts the factor on the original scale):
COMPUTE socjuct= (action+raceund+cleanup+meanphil+inflsoc_helpoth+inflpol)/7
as opposed to saving the scores within the syntax
FACTOR
/VARIABLES=THISONE NEXTONE ETC
/EXTRACTION=PAF
/ROTATE=VARIMAX
/SAVE.