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Factor Analysis: SPSS Output

This example uses SPSS 7.5 for file "gss 93 subset.sav". The analysis looks at all the music variables (bigband, blues, blues3, blugrass, classicl, classic3, country, hvymetal, jazz, jazz3, musicals, opera, rap, and rap3) with a view to understanding which subsets might be combined into a "music scale". Four other background variables are also entered: age, rincome91, sex, and educ.

To obtain this output:

  1. File, Open, point to gss 93 subset.sav.
  2. Statistics, Data Reduction, Factor Analysis
  3. In the Factor Analysis dialog box, enter all the variables listed above in the "Variables" box.
  4. Click on the Descriptives button and check Coefficients, and Significance Levels.
  5. Click on the Extraction button and under Display check Unrotated Factor Matrix and Scree Plot. Leave as defaults the settings for Analyze Correlation Matrix and Extract Eigenvalues over 1.
  6. Click on the Rotation button and select Varimax. Under Display, check Rotated Solution and Loading Plots.
  7. Click on the Scores button and check :Display factor score coefficient matrix".
  8. Click on the Options button and .check "Coefficient Display Format, Sorted by Size".
  9. Click on OK to run the procedure.

Comments in blue are by the instructor and are not part of SPSS output.

Notes
Output Created 26 Mar 98 08:08:50
Comments
Input Data Y:\PC\spss95\GSS93 subset.sav
Filter <none>
Weight <none>
Split File <none>
N of Rows in Working Data File 1500
Missing Value Handling Definition of Missing MISSING=EXCLUDE: User-defined missing values are treated as missing.
Cases Used LISTWISE: Statistics are based on cases with no missing values for any variable used.
Syntax FACTOR
/VARIABLES bigband blues blues3 blugrass classic3 classicl country
hvymetal jazz jazz3 musicals opera rap rap3 age educ rincom91 sex /MISSING
LISTWISE /ANALYSIS bigband blues blues3 blugrass classic3 classicl country
hvymetal jazz jazz3 musicals opera rap rap3 age educ rincom91 sex
/PRINT INITIAL CORRELATION SIG EXTRACTION ROTATION FSCORE
/FORMAT SORT
/PLOT EIGEN ROTATION
/CRITERIA MINEIGEN(1) ITERATE(25)
/EXTRACTION PC
/CRITERIA ITERATE(25)
/ROTATION VARIMAX
/METHOD=CORRELATION .
Resources Maximum Memory Required 39720 (38.789K) bytes
Elapsed Time 0:00:01.71


The matrix of correlation coefficients and their respective significance levels is printed below because it was requested under the "Descriptives" options. Factor analysis uses the correlation matrix to try to determine which sets of variables cluster together.
Correlation Matrix

Bigband Music Blues or R & B Music Blues and R&B Music Bluegrass Music Classical Music (3) Classical Music Country Western Music Heavy Metal Music Jazz Music Jazz Music (3) Broadway Musicals Opera Rap Music Rap Music (3) Age of Respondent Highest Year of School Completed Respondent's Income Respondent's Sex
Correlation Bigband Music 1.000 .292 .287 .355 .383 .388 .160 -.097 .269 .266 .516 .400 -.023 -.027 -.332 -.081 -.075 -.059
Blues or R & B Music .292 1.000 .925 .249 .236 .223 .044 .066 .546 .529 .216 .194 .158 .154 .034 -.098 -.075 .011
Blues and R&B Music .287 .925 1.000 .225 .229 .200 .012 .073 .529 .547 .216 .174 .133 .129 .038 -.111 -.090 -.010
Bluegrass Music .355 .249 .225 1.000 .152 .153 .385 .000 .089 .113 .167 .172 -.020 -.038 -.182 .057 .013 .076
Classical Music (3) .383 .236 .229 .152 1.000 .935 -.096 .017 .298 .295 .495 .539 .011 .022 -.073 -.320 -.100 -.067
Classical Music .388 .223 .200 .153 .935 1.000 -.102 -.002 .300 .289 .512 .571 .011 .016 -.081 -.339 -.104 -.070
Country Western Music .160 .044 .012 .385 -.096 -.102 1.000 -.101 -.088 -.072 .031 .027 -.004 -.026 -.118 .238 .103 -.046
Heavy Metal Music -.097 .066 .073 .000 .017 -.002 -.101 1.000 .056 .066 -.122 -.018 .279 .229 .376 .019 .145 .125
Jazz Music .269 .546 .529 .089 .298 .300 -.088 .056 1.000 .932 .230 .270 .152 .176 .087 -.165 -.109 .026
Jazz Music (3) .266 .529 .547 .113 .295 .289 -.072 .066 .932 1.000 .235 .254 .152 .185 .121 -.167 -.109 .034
Broadway Musicals .516 .216 .216 .167 .495 .512 .031 -.122 .230 .235 1.000 .461 .032 .041 -.230 -.196 -.061 -.288
Opera .400 .194 .174 .172 .539 .571 .027 -.018 .270 .254 .461 1.000 .103 .074 -.187 -.189 -.054 -.052
Rap Music -.023 .158 .133 -.020 .011 .011 -.004 .279 .152 .152 .032 .103 1.000 .917 .272 -.005 .083 -.047
Rap Music (3) -.027 .154 .129 -.038 .022 .016 -.026 .229 .176 .185 .041 .074 .917 1.000 .253 -.008 .092 -.046
Age of Respondent -.332 .034 .038 -.182 -.073 -.081 -.118 .376 .087 .121 -.230 -.187 .272 .253 1.000 -.121 .179 .019
Highest Year of School Completed -.081 -.098 -.111 .057 -.320 -.339 .238 .019 -.165 -.167 -.196 -.189 -.005 -.008 -.121 1.000 .334 -.002
Respondent's Income -.075 -.075 -.090 .013 -.100 -.104 .103 .145 -.109 -.109 -.061 -.054 .083 .092 .179 .334 1.000 -.234
Respondent's Sex -.059 .011 -.010 .076 -.067 -.070 -.046 .125 .026 .034 -.288 -.052 -.047 -.046 .019 -.002 -.234 1.000
Sig. (1-tailed) Bigband Music
.000 .000 .000 .000 .000 .000 .004 .000 .000 .000 .000 .260 .229 .000 .013 .019 .050
Blues or R & B Music .000
.000 .000 .000 .000 .111 .033 .000 .000 .000 .000 .000 .000 .173 .003 .019 .385
Blues and R&B Music .000 .000
.000 .000 .000 .369 .021 .000 .000 .000 .000 .000 .000 .145 .001 .006 .391
Bluegrass Music .000 .000 .000
.000 .000 .000 .499 .006 .001 .000 .000 .289 .144 .000 .057 .360 .018
Classical Music (3) .000 .000 .000 .000
.000 .004 .318 .000 .000 .000 .000 .381 .267 .022 .000 .003 .032
Classical Music .000 .000 .000 .000 .000
.002 .477 .000 .000 .000 .000 .378 .327 .012 .000 .002 .025
Country Western Music .000 .111 .369 .000 .004 .002
.002 .007 .022 .195 .231 .454 .236 .001 .000 .002 .103
Heavy Metal Music .004 .033 .021 .499 .318 .477 .002
.059 .034 .000 .313 .000 .000 .000 .303 .000 .000
Jazz Music .000 .000 .000 .006 .000 .000 .007 .059
.000 .000 .000 .000 .000 .008 .000 .001 .234
Jazz Music (3) .000 .000 .000 .001 .000 .000 .022 .034 .000
.000 .000 .000 .000 .000 .000 .001 .172
Broadway Musicals .000 .000 .000 .000 .000 .000 .195 .000 .000 .000
.000 .186 .128 .000 .000 .044 .000
Opera .000 .000 .000 .000 .000 .000 .231 .313 .000 .000 .000
.002 .020 .000 .000 .068 .073
Rap Music .260 .000 .000 .289 .381 .378 .454 .000 .000 .000 .186 .002
.000 .000 .442 .010 .097
Rap Music (3) .229 .000 .000 .144 .267 .327 .236 .000 .000 .000 .128 .020 .000
.000 .408 .005 .099
Age of Respondent .000 .173 .145 .000 .022 .012 .001 .000 .008 .000 .000 .000 .000 .000
.000 .000 .303
Highest Year of School Completed .013 .003 .001 .057 .000 .000 .000 .303 .000 .000 .000 .000 .442 .408 .000
.000 .483
Respondent's Income .019 .019 .006 .360 .003 .002 .002 .000 .001 .001 .044 .068 .010 .005 .000 .000
.000
Respondent's Sex .050 .385 .391 .018 .032 .025 .103 .000 .234 .172 .000 .073 .097 .099 .303 .483 .000



The communalities, below, measure the percent of variance in a given variable explained by all the factors. That is, the communality is the squared multiple correlation for the variable using the factors as predictors. Communality for a variable is the sum of squared factor loadings for that variable (row), and thus is the percent of variance in a given variable explained by all the factors. For full orthogonal PCA, the communality will be 1.0 and all of the variance in the variables will be explained by all of the factors, which will be as many as there are variables. In the communalities chart, SPSS labels this column the "initial" communalities. The "extracted" communalities is the percent of variance in a given variable explained by the factors which are extracted, which will usually be fewer than all the possible factors, resulting in coefficients less than 1.0.

Communalities

Initial Extraction
Bigband Music 1.000 .588
Blues or R & B Music 1.000 .781
Blues and R&B Music 1.000 .777
Bluegrass Music 1.000 .654
Classical Music (3) 1.000 .843
Classical Music 1.000 .867
Country Western Music 1.000 .562
Heavy Metal Music 1.000 .640
Jazz Music 1.000 .760
Jazz Music (3) 1.000 .761
Broadway Musicals 1.000 .656
Opera 1.000 .571
Rap Music 1.000 .955
Rap Music (3) 1.000 .943
Age of Respondent 1.000 .636
Highest Year of School Completed 1.000 .509
Respondent's Income 1.000 .745
Respondent's Sex 1.000 .738
Extraction Method: Principal Component Analysis.



The "Total Variance Explained" table below shows the eigenvalues, which are the proportion of total variance in all the variables which is accounted for by that factor. A factor's eigenvalue may be computed as the sum of its squared factor loadings for all the variables. A factor's eigenvalue divided by the number of variables (which equals the sum of variances because the variance of a standardized variable equals 1) is the percent of variance in all the variables which it explains. The ratio of eigenvalues is the ratio of explanatory importance of the factors with respect to the variables. If a factor has a low eigenvalue, then it is contributing little to the explanation of variances in the variables and may be ignored as redundant with more important factors. The table shows 18 factors, one for each variable. However, only the first six are extracted for analysis because, under the Extraction options, SPSS was told to extract only factors with eigenvalues of 1.0 or higher.

The "Initial Eigenvalues" and "Extraction Sums of Squared Loadings" columns are the same, except the latter only lists factors which have actually been extracted in the solution. The "Rotation Sums of Squared Loadings" give the eigenvalues after rotation improves the interpretability of the factors (we used Varimax rotation, which minimizes the number of variables which have high loadings on each given factor). Note that the total percent of variance explained is the same (see the cumulative value for factor 6 -- 72.148%) but rotation changes the eigenvalues for each of the extracted factors. That is, after rotation each extracted factor counts for a different percentage of variance explained, even though the total variance explained is the same.

Total Variance Explained

Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Component Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
1 4.433 24.629 24.629 4.433 24.629 24.629 3.252 18.069 18.069
2 2.595 14.415 39.044 2.595 14.415 39.044 3.114 17.299 35.368
3 1.863 10.348 49.392 1.863 10.348 49.392 1.933 10.740 46.108
4 1.738 9.656 59.048 1.738 9.656 59.048 1.727 9.596 55.703
5 1.215 6.749 65.797 1.215 6.749 65.797 1.603 8.903 64.606
6 1.143 6.351 72.148 1.143 6.351 72.148 1.358 7.542 72.148
7 .885 4.915 77.063





8 .791 4.396 81.459





9 .673 3.739 85.198





10 .562 3.121 88.320





11 .524 2.912 91.231





12 .500 2.777 94.008





13 .417 2.319 96.327





14 .381 2.115 98.442





15 9.226E-02 .513 98.955





16 7.777E-02 .432 99.387





17 6.028E-02 .335 99.722





18 5.010E-02 .278 100.000





Extraction Method: Principal Component Analysis.



The Cattell scree test, below, plots the components as the X axis and the corresponding eigenvalues as the Y axis. As one moves to the right, toward later components, the eigenvalues drop. When the drop ceases and the curve makes an elbow toward less steep decline, Cattell's scree test says to drop all further components after the one starting the elbow. Where the "elbow" is is somewhat subjective, but in this case one would probably decide only the first three factors were worth retaining in the analysis. If one decided to use the second "elbow," one would retain five factors.

There are alternative criteria for deciding how many factors to retain. The Kaiser rule is to drop all components with eigenvalues under 1.0, which is what was specified under the "Extraction" options, resulting in six factors. Scree plot



The "Component Matrix," below, gives the factor loadings. This is the central output for factor analysis. The factor loadings, also called component loadings in PCA, are the correlation coefficients between the variables (rows) and factors (columns). Factor loadings are the basis for imputing a label to the different factors. Loadings above .6 are usually considered "high" and those below .4 are "low." Note that the music variables were coded so that high values correspondent to disliking that type of music. Therefore a positive loading corresponds to disliking that type of music, and a negative loading to liking.

The first table below gives the unrotated solution and the second the rotated solution. Normally the rotated solution will be significantly easier to interpret (indeed, often the researcher will not ask for the unrotated matrix, but we requested it here for instructional purposes).

Looking at the rotated matrix, the first factor has high loadings from six music variables: classical, classical(3), opera, Broadway musicals, and had moderate loading on big bands. Because these six music items sort on the same factor, this is a justification for combining these items in a scale which might be called "general music appreciation scale." Naming the factor is a matter of subjectivity and dispute in many cases.

Blues and jazz are associated strongly with the second factor.

Rap music (2 variables) is associated strongly with the third factor.

The fourth factor is strongly associated with country western and bluegrass, but there is also a moderate tie to highest year of school completed, with more educated respondents less likely to like these types of music.

The fifth factor is associated with heavy metal, and with respondent's age and income..

As one goes on, the factors become harder to interpret.

The fifth factor is strongly associated with heavy metal music and age of respondent, with younger respondents more likely to like heavy metal.

The sixth factor is strongly associated with gender and income, with there being a negative relationship which indicates women (scored as "2", compared to men scored as "1") earn less income. All tie-ins with music are weak, though the highest of these weak associations is with broadway musicals, with women more likely to like them.
Component Matrix(a)

Component
1 2 3 4 5 6
Classical Music .725 -.295 -.439 5.430E-02 5.338E-02 .237
Classical Music (3) .722 -.275 -.420 5.005E-02 5.060E-02 .253
Jazz Music .713 .324 .188 -.286 -.168 -3.602E-02
Jazz Music (3) .711 .337 .197 -.281 -.156 -1.992E-02
Blues or R & B Music .677 .303 .449 -.139 -9.169E-02 3.539E-03
Blues and R&B Music .667 .300 .444 -.175 -.123 -1.498E-03
Opera .621 -.244 -.244 .223 .115 5.015E-02
Broadway Musicals .621 -.335 -.168 .277 -.177 -.149
Bigband Music .603 -.343 .181 .248 9.333E-02 -6.247E-02
Rap Music .186 .704 -.239 .462 .224 -.321
Rap Music (3) .193 .699 -.240 .441 .193 -.356
Age of Respondent -9.133E-02 .632 -.248 -.110 -.106 .379
Country Western Music -1.762E-02 -.152 .506 .491 .197 5.918E-02
Bluegrass Music .314 -.188 .468 .326 .378 .229
Highest Year of School Completed -.343 5.704E-02 .446 .395 -.130 .128
Respondent's Income -.176 .179 5.429E-02 .509 -.468 .447
Respondent's Sex -7.382E-02 9.362E-02 .136 -.382 .724 .188
Heavy Metal Music 2.017E-02 .505 -.169 7.481E-02 .200 .558
Extraction Method: Principal Component Analysis.
a 6 components extracted.



The rotated solution is below. This is the table on the basis of which the factor loadings were interpreted above.
Rotated Component Matrix(a)

Component
Variables 1 2 3 4 5 6
Classical Music .912 .133 -5.584E-02 -8.265E-02 7.867E-02 -3.453E-02
Classical Music (3) .895 .149 -5.976E-02 -7.327E-02 9.838E-02 -3.261E-02
Opera .729 .111 .103 .107 -7.181E-02 -1.257E-02
Broadway Musicals .672 .170 7.707E-02 5.114E-02 -.327 .246
Bigband Music .530 .263 -5.358E-03 .382 -.304 4.455E-03
Blues and R&B Music 8.949E-02 .866 3.810E-02 .135 4.308E-03 -1.256E-02
Blues or R & B Music .104 .858 6.294E-02 .172 1.163E-02 -2.105E-02
Jazz Music (3) .197 .837 7.708E-02 -.110 5.316E-02 -3.852E-02
Jazz Music .203 .834 7.390E-02 -.124 3.340E-02 -3.260E-02
Rap Music 2.418E-02 9.359E-02 .956 8.723E-03 .174 2.651E-02
Rap Music (3) 1.701E-02 .111 .953 -2.232E-02 .139 3.923E-02
Bluegrass Music .221 .159 -5.002E-02 .743 6.077E-03 -.161
Country Western Music -6.446E-02 -4.133E-02 3.894E-02 .732 -.114 7.924E-02
Highest Year of School Completed -.391 -.105 -2.269E-02 .478 5.567E-02 .335
Heavy Metal Music 3.548E-02 3.502E-02 .170 5.152E-02 .775 -7.533E-02
Age of Respondent -.160 .110 .173 -.291 .690 8.616E-02
Respondent's Sex -.129 4.062E-04 -7.952E-02 .148 .251 -.794
Respondent's Income -7.883E-02 -.103 -1.544E-02 .233 .415 .708
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization.
a Rotation converged in 7 iterations.



The "Component Transformation Matrix" below indicates the correlation of the factors before and after rotation.
Component Transformation Matrix
Component 1 2 3 4 5 6
1 .714 .681 .124 .064 -.073 -.046
2 -.403 .385 .613 -.173 .532 .022
3 -.488 .473 -.253 .657 -.207 -.009
4 .210 -.330 .490 .590 .004 .509
5 .099 -.232 .256 .359 .100 -.855
6 .190 -.040 -.489 .239 .812 .082
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization.



Below is the three-dimensional factor loading plot of the first three factors, requested as a "Rotation" option. For a two-factor solution, a two-dimensional plot is shown. The plot is not displayed if only one factor is extracted. Plots display rotated solutions if rotation is requested, as it was in this case. This plot is hard to read because there are 18 variables, which overwrite each other. Each variable is plotted according to its factor loadings on the first three factors. This is simply a graphical way of presenting the same information as was contained in the "rotated component matrix" of factor loadings above.

Component plot of factors 1, 2, 3



Below is the display of the factor score coefficient matrix. Factor scores are the scores of each case on each factor. The factor score coefficients are used to calculate the factor scores of each case for each of the six factors. These scores can be saved to one's dataset for later use as variables in their own right (this is under the "Scores" button options, as was the display below.).
Component Score Coefficient Matrix

Component
1 2 3 4 5 6
Bigband Music .130 .025 .028 .194 -.136 -.008
Blues or R & B Music -.080 .307 -.030 .074 -.004 .017
Blues and R&B Music -.087 .316 -.045 .050 -.011 .028
Bluegrass Music .065 -.003 -.026 .452 .099 -.161
Classical Music (3) .321 -.065 -.072 -.034 .163 -.011
Classical Music .328 -.073 -.065 -.041 .149 -.013
Country Western Music -.027 -.030 .050 .425 -.027 .006
Heavy Metal Music .088 -.037 -.032 .108 .535 -.074
Jazz Music -.039 .293 -.030 -.099 -.007 .026
Jazz Music (3) -.040 .293 -.031 -.089 .008 .021
Broadway Musicals .190 -.011 .065 -.018 -.180 .187
Opera .247 -.069 .059 .060 .012 -.019
Rap Music .004 -.047 .519 .027 -.041 -.040
Rap Music (3) -.005 -.036 .521 .004 -.069 -.027
Age of Respondent -.007 .045 -.035 -.121 .418 .077
Highest Year of School Completed -.123 .015 -.027 .271 .049 .218
Respondent's Income .027 -.010 -.117 .133 .314 .514
Respondent's Sex -.017 -.035 -.034 .164 .198 -.607
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization.


The factor score covariance matrix is also displayed, which is the covariance matrix of factor scores, with its diagonal elements being the variances of the factor scores in the given sample. This illustrates that the six factors are orthogonal. That is, the factor scores for the six factors do not covary with each other. The trivial covariance of factors 1 and 2 may be ignored. This output was requested under the "Scores" option.

Component Score Covariance Matrix
Component 1 2 3 4 5 6
1 1.000 -1.702E-16 .000 .000 .000 .000
2 -1.702E-16 1.000 .000 .000 .000 .000
3 .000 .000 1.000 .000 .000 .000
4 .000 .000 .000 1.000 .000 .000
5 .000 .000 .000 .000 1.000 .000
6 .000 .000 .000 .000 .000 1.000
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization.