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Factor Analysis Write-Up
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The purpose of this investigation was to explore the factor structure
underlying the music item responses in the GSS data set. Factor analysis
has as its key objective reducing a larger set of variables to a smaller set of
factors, fewer in number than the original variable set, but capable of
accounting for a large portion of the total variability in the items. Recall the
adage, “Birds of a feather flock together”. The identity of each factor is
determined after a review of which items correlate the highest with that
factor. Items that correlate the highest with a factor define the meaning of
the factor as judged by what conceptually ties the items together. A
successful result is one in which a few factors can explain a large portion of
the total variability and those factors can be given a meaningful name using
the assortment of items that correlate the highest with it.
In the context of this study, when such success is attained, we may say
that we have validity evidence supporting the conclusion that the scores from
this instrument are a valid assessment of a person’s musical preference. We
can feel confident when adding similar items up for total scores to represent
the different dimensions of one’s overall music preference (each factor
represents a dimension). This kind of validity evidence is called internal
structure evidence because it suggests that items line up in a predictable
manner, according to what thematically ties them together conceptually. The
descriptive statistics of the item responses are presented in Table 1. It may
Table 1: Descriptive Statistics
Bigband Music
Bluegrass Music
Country Western Music
Blues or R & B Music
Broadway Musicals
Classical Music
Folk Music
Jazz Music
Opera
Rap Music
Heavy Metal Music
Mean
2.47
2.67
2.33
2.47
2.56
2.60
2.75
2.59
3.46
3.98
4.15
Std. Deviation
1.09
1.01
1.10
1.02
1.12
1.21
1.03
1.10
1.12
1.08
1.10
Analys is N
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be observed that the standard deviations are smaller than the respective
means and that no one standard deviation stands out upon gross observation
as remarkably larger than the other variables.
The maximum likelihood estimation procedure was used to extract the
factors from the variable data. Kaiser’s rule was used to determine which
factors were most eligible for interpretation because this rule requires that a
given factor is capable of explaining at least the equivalent of one variable’s
variance. This is not unreasonable given that factor analysis has as its
objective reducing several variables into fewer factors. Using this rule, four
factors were extracted (see Table 2). Together they are capable of explaining
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Factor Analysis Write-Up
Ta ble 2: Tota l Va riance Expl aine d
Factor
1
2
3
4
5
6
7
8
9
10
11
Total
3.276
1.661
1.392
1.058
.728
.658
.566
.496
.421
.397
.348
Initial Eigenvalues
% of Variance Cumulative %
29.779
29.779
15.098
44.876
12.651
57.527
9.620
67.147
6.619
73.766
5.978
79.744
5.150
84.893
4.510
89.404
3.823
93.227
3.608
96.835
3.165
100.000
Ex trac tion Sums of Squared Loadings
Total
% of Variance Cumulative %
2.802
25.473
25.473
1.101
10.005
35.478
.874
7.942
43.420
.551
5.010
48.430
Rotation
Sums
Totalof
Squared
2.610a
Loadings
1.781
1.354
.805
Ex trac tion Met hod: Maximum Likelihood.
a. W hen factors are c orrelated, sums of squared loadings c annot be added to obtain a t otal variance.
roughly 67.1% of all the variable variances. A plot of the eigenvalues is
provided below. A review of the initial factor loadings suggests that the
Scree Plot
3.5
3.0
2.5
2.0
Eigenvalue
1.5
1.0
.5
0.0
1
2
3
4
5
6
7
8
9
10
11
Factor Number
a proper solution was attainable through maximum likelihood, as it was
capable of converging in 6 iterations. The computer printout does not warn
that the results are nonpositive definite, so one important condition for
proceeding with the interpretation has been met. Another portion of the
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Factor Analysis Write-Up
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Ta ble 3: Factor Matrix
Factor
Classical Music
Broadway Mus icals
Opera
Bigband Music
Folk Music
Jazz Music
Blues or R & B Music
Bluegrass Mus ic
Country W estern Music
Heavy Met al Music
Rap Music
1
.732
.671
.666
.624
.547
.524
.505
.374
2
.105
.346
-.454
-.277
.536
.502
-.250
-.289
3
-.341
-.208
-.241
4
.113
.334
.445
.416
.306
.199
.198
-.155
-.107
.126
-.176
.507
.419
Ex trac tion Met hod: Maximum Likelihood.
a. 4 factors extrac ted. 6 it erat ions required.
results to inspect before proceeding with an interpretation is the table of
communalities. Communalities are interpreted like Multiple R2s in multiple
regression. Communalities indicate the degree to which the factors explain
the variance of the variables. In a proper solution, two sets of communalities
are provided, the initial set and the extracted set. Sometimes when the
maximum likelihood procedure goes awry (because of ill conditioned data),
the values of one or more communalities can exceed 1.00, which is
theoretically impossible because explaining more than 100% of a variable’s
variance is impossible. In such a case further interpretation is impossible.
In this study, the communalities were fine, providing further evidence that
the results are appropriate for interpretation (see Table 4).
Table 4: Communalities
Bigband Music
Bluegrass Mus ic
Country Western Music
Blues or R & B Music
Broadway Musicals
Classical Music
Folk Music
Jazz Music
Opera
Rap Music
Heavy Metal Music
Initial
.392
.317
.235
.360
.448
.500
.345
.371
.422
.171
.146
Extraction
.432
.608
.353
.541
.502
.666
.426
.616
.517
.306
.359
Extraction Method: Maximum Likelihood.
With greater confidence that the maximum likelihood solution is
proper, interpretation of the results is permissible. Once the factors are
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Factor Analysis Write-Up
extracted using maximum likelihood, a linear transformation of the data is
necessary so that the interpretation of the results can be easily accomplished.
Among the various rotational procedures available, Promax was chosen
because it assumes that nonzero correlations among the factors are
theoretically tenable or at least plausible. When the results are generated,
interpretation of the factor correlation matrix is to ensue (see Table 5). If the
Table 5: Factor Correlation Matrix
Factor
1
2
3
4
1
1.000
.466
.289
-9.15E-02
2
.466
1.000
.164
.164
3
.289
.164
1.000
-.152
4
-9.15E-02
.164
-.152
1.000
Extraction Method: Maximum Likelihood.
Rotation Method: Promax with Kaiser Normalization.
correlations are large enough given the educated judgment of the researcher
then the Promax solution is further interpreted. If the researcher decides
that the correlations are too low, then the results should be re-run using the
varimax rotation. These correlations are large enough to justify retention of
the Promax results from my perspective because two of the correlations
exceed the value of .25. So, now I will proceed with an interpretation of the
structure matrix is to ensue. Reviewing the structure coefficient matrix
suggests that the four factors group the items in a theoretically
understandable way (see Table 6). The coefficients suggest that the way in
Ta ble 6: Structure Matrix
Classical Music
Opera
Broadway Mus icals
Bigband Music
Folk Music
Jazz Music
Blues or R & B Music
Bluegrass Mus ic
Country W estern Music
Heavy Met al Music
Rap Music
1
.799
.710
.698
.598
.559
.356
.321
.277
Factor
2
.332
.318
.368
.418
.191
.775
.727
.208
.101
.199
3
.143
.180
.308
.472
.201
.773
.574
4
-.155
-.206
-.101
.171
.164
-.129
.599
.542
Ex trac tion Met hod: Maximum Likelihood.
Rotation Method: Promax with Kais er Normalization.
which people responded to the music items was very consistent for Classical
Music, Opera, Broadway Musicals, Bigband Music, and Folk Music. How
they rated their preference of Classical Music tended to be very similar to
how they responded to the remaining variables mentioned. For example, if a
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person liked (or disliked) classical music, that person probably also liked (or
disliked) Broadway Musicals, Opera, Bigband Music, and Folk Music. The
variables together contribute most prominently to Factor 1. The structure
coefficients for these variables suggests that Classical Music is correlated
.799 with Factor 1, therefore sharing roughly 64% of the variance of that
factor. All remaining coefficients may be interpreted in this way.
The remaining factors, ascertained by the magnitude of the coefficients
are identified in table 6 by the shading, where shaded coefficients are the
largest coefficients for a factor. Names for the factors are as follows: Factor
1, Traditional Music; Factor 2, Mississippi River Music; Factor 3, Culturally
Rural Music; and Factor 4, Music of the Younger Generation.