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Principal Components and Exploratory Factor Analysis Report (Assessment)

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Similarities and Differences

Principal components analysis (PCA) and exploratory factor analysis (EFA) have some similarities and differences in the way they reduce variables or dimensionality of a given data sets. One key similarity of PCA and EFA is that both are methods of reducing variables or data based on exhibited variances (Hahs-Vaugh, 2016). The assumptions that underlie PCA and EFA are similar because both require variables to be on ratio or interval scale, exhibit linear relationships, follow the normal distribution, have large sample sizes, and show paired variables with bivariate normal distribution. The steps of their analysis are also similar because they involve selection of variables, extraction of variables, interpretation, rotation, and selection of components or factors that account for most variation (Elliott & Woodward, 2015). However, the main difference between PCA and EFA lies in the way data reduction process occurs. While PCA reduces a set of variables using a linear combination to create principal components with optimal weights that explain most variation, EFA reduces a set of variables using an indirect linear combination of latent variables to develop factors that explain commonality of the variance in data (Hahs-Vaugh, 2016). An additional difference is that whereas PCA decomposes a correlation matrix, EFA decomposes an adjusted correlation matrix with latent factors.

Exploratory Factor Analysis

EFA aids in reducing data by collating variables that have common correlations and grouping them into themes, factors, or components, which effectively account for most commonality in a variable of interest. In exploratory data analysis, EFA uses univariate descriptive statistics, which show patterns and trends of individual variables regarding means, standard deviations, and a valid number of cases assessed. The initial solution offers a percentage of variance explained, eigenvalues, and communalities, whereas KMO and Bartlett’s test of sphericity determines adequacy of a sample used.

As an extraction procedure, EFA uses unweighted least squares (ULS), which gives a common variance that is less than the total variance as in the case of PCA (Meyers, Gamst, & Guarino, 2013). EFA then applies a fixed number of factors in extraction procedure to produce factor correlation matrix and total variance explained by extracted variables. The focus of commonality is significant for the overall analysis for it provides the sum of all correlation values in the component matrix, and thus, it indicates the extent to which the component matrix captures each variable. Meyers, Gamst, and Guarino (2013) explain that the component matrix significantly captures variables with communality values of 0.5 or greater. Hence, communality values help in the extraction of variables that EFA captures in the component matrix as substantial.

Replicated Tables

Principal Component Analysis

Descriptive statistics (Appendix A) display means, standard deviations, and valid cases for each variable. Evidently, the sample size is adequate for each case has 301 valid cases out of 310 cases in the data, which means that only nine cases are missing. Specifically, the mean scores ranged from 2.60 to 8.46 while the standard deviations range from 1.079 to 2.448, which imply that the values of variables are highly diverse.

KMO and Bartlett’s test (Table 1) indicates that the data is sufficient for PCA because it has a coefficient of 0.853 whereas Bartlett’s test of sphericity shows that there is no sufficient correlation (p = 0.000) to hinder PCA.

Table 1.

KMO and Bartlett’s Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .853
Bartlett’s Test of Sphericity Approx. Chi-Square 4751.733
df 435
Sig. .000

Table 2 below depicts that PCA managed to extract seven principal components with eigenvalues greater than one, and they cumulatively account for 65.8% of the variance. The first and the second principal components have eigenvalues of 7.422 and 4.819, which accounts for about a third of variance (40.8%/65.8%), because each explains 24.8% and 16.1% of variance respectively. The third through the seventh principal components accounts for the remaining a third for each explains 7.9%, 5.5%, 4.3%, 3.9%, and 3.5% of variance respectively.

Table 2.

Total Variance Explained
Component Initial Eigenvalues Extraction Sums of Squared Loadings
Total % of Variance Cumulative % Total % of Variance Cumulative %
1 7.422 24.739 24.739 7.422 24.739 24.739
2 4.819 16.062 40.801 4.819 16.062 40.801
3 2.356 7.854 48.655 2.356 7.854 48.655
4 1.654 5.514 54.169 1.654 5.514 54.169
5 1.276 4.254 58.423 1.276 4.254 58.423
6 1.161 3.870 62.293 1.161 3.870 62.293
7 1.039 3.465 65.758 1.039 3.465 65.758
8 .937 3.125 68.883
9 .843 2.810 71.693
10 .808 2.693 74.386
11 .785 2.617 77.003
12 .694 2.313 79.316
13 .607 2.023 81.338
14 .568 1.895 83.233
15 .535 1.783 85.016
16 .511 1.702 86.718
17 .461 1.535 88.253
18 .421 1.405 89.658
19 .390 1.301 90.959
20 .367 1.224 92.183
21 .338 1.126 93.309
22 .293 .978 94.287
23 .277 .924 95.211
24 .258 .860 96.071
25 .244 .814 96.885
26 .228 .759 97.643
27 .211 .703 98.347
28 .175 .583 98.930
29 .170 .567 99.497
30 .151 .503 100.000
Extraction Method: Principal Component Analysis

The scree plot illustrates that the seven components explain most of the variation for the plot begins to regress gradually at the elbow. McCormick, Salcedo, Peck, Wheeler, and Verlen (2017) explain that the elbow of a scree plot sets a threshold of eigenvalues that accounts for most variation. In this view, the scree plot supports the extraction of the seven components in PCA.

Scree plot depicting the distribution of eigenvalues.
Figure 1: Scree plot depicting the distribution of eigenvalues.

Exploratory Factor Analysis: Six-Component Analysis

EFA shows that the unweighted least squares (ULS) procedure explain 53.6% of variance. The first and the second factors account for most variance because they explain 23.4% and 14.5% of variance correspondingly. The third through the sixth factors accounts for minor variance for they explain 6.5%, 4.1%, 3.0%, and 2.1% of variance (Table 3).

Table 3.

Total Variance Explained
Factor Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadingsa
Total % of Variance Cumulative % Total % of Variance Cumulative % Total
1 7.422 24.739 24.739 7.012 23.372 23.372 5.565
2 4.819 16.062 40.801 4.351 14.503 37.876 4.051
3 2.356 7.854 48.655 1.962 6.539 44.414 3.752
4 1.654 5.514 54.169 1.237 4.125 48.539 5.108
5 1.276 4.254 58.423 .902 3.007 51.546 3.834
6 1.161 3.870 62.293 .629 2.096 53.642 3.236
7 1.039 3.465 65.758
8 .937 3.125 68.883
9 .843 2.810 71.693
10 .808 2.693 74.386
11 .785 2.617 77.003
12 .694 2.313 79.316
13 .607 2.023 81.338
14 .568 1.895 83.233
15 .535 1.783 85.016
16 .511 1.702 86.718
17 .461 1.535 88.253
18 .421 1.405 89.658
19 .390 1.301 90.959
20 .367 1.224 92.183
21 .338 1.126 93.309
22 .293 .978 94.287
23 .277 .924 95.211
24 .258 .860 96.071
25 .244 .814 96.885
26 .228 .759 97.643
27 .211 .703 98.347
28 .175 .583 98.930
29 .170 .567 99.497
30 .151 .503 100.000
Extraction Method: Unweighted Least Squares
a. When factors are correlated, sums of squared loadings cannot be added to obtain a total variance.

Structure Matrix (Appendix B ) shows that the first factor comprises 5, 11, 17, 18, 23, and 29 variables that represent the appealing appearance of the extrinsic orientation whereas the second factor constitutes variables 3, 9, 15, 21, and 27 that represent community feeling aspect of the intrinsic orientation. The third factor representing the affiliation aspect of the intrinsic orientation has 2, 8, 14, 20, and 26 variables while the fourth factor has 4, 10, 16, 22, and 28 variables that represent social recognition aspect of extrinsic orientation. In the fifth factor, 6, 12, 24, and 30 are variables that belong to the financial aspect of the extrinsic orientation whereas the sixth factor has 1, 7, 13, 19, and 25 variables that measure the self-acceptance aspect of the intrinsic orientation.

Evaluation

A strength of PCA and EFA shows that they are effective in reducing variables and grouping them into similar themes that measure a common construct using a linear combination procedure. Another strength is that eigenvalues generated quantifies variance while a scree provides a visual way of selecting components or factors. However, the weaknesses are that PCA and EFA are prone to over-factoring and under-factoring due to the heuristic and arbitrary selection thresholds (Field, 2013). In this view, the analysis using PCA and EFA requires the inclusion of an adequate number of factors or components into the model.

Conclusion

PCA and EFA have significant benefits to projected study for they aid in the development of a Likert scale through the selection of related Likert items, which accounts for most variation in the construct of interest. In this case, PCA and EFA group 30 items into six groups and six groups respectively, which have common themes regarding social orientations of individuals. These groups reveal the existence of subscales in the 30 items used in the study of orientation among individuals.

References

Elliott, A. C., & Woodward, W. A. (2015). IBM SPSS by example: A practical guide to statistical data. Thousand Oaks, CA: SAGE Publications.

Field, A. (2013). Discovering statistics using IBM SPSS statistics (4th ed.). Los Angeles, CA: SAGE Publications.

Hahs-Vaugh, D. (2016). Applied multivariate statistical concepts. London, United Kingdom: Taylor & Francis.

McCormick, K., Salcedo, J., Peck, J., Wheeler, A., & Verlen, J. (2017). SPSS statistics for data analysis and visualization. Indianapolis, IN: Wiley.

Meyers, L. S., Gamst, G., & Guarino, A. J. (2013). Applied multivariate research: Design and interpretation. Los Angeles, LA: SAGE.

Appendix A: Descriptive Statistics

Descriptive Statistics
Mean Std. Deviation Analysis N
I will choose what I do, instead of being pushed along by life. 7.72 1.457 301
I will feel that there are people who really love me, and whom I love. 8.34 1.172 301
I will assist people who need it, asking nothing in return. 7.72 1.413 301
I will be recognized by lots of different people. 5.33 2.284 301
I will successfully hide the signs of aging. 4.43 2.387 301
I will be financially successful. 7.61 1.407 301
At the end of my life, I will look back on my life as meaningful and complete. 8.30 1.334 301
I will have good friends that I can count on. 8.23 1.153 301
I will work for the betterment of society. 7.31 1.635 301
My name will be known by many people. 4.55 2.325 301
I will have people comment often about how attractive I look. 4.15 2.310 301
I will have a job that pays very well. 7.38 1.662 301
I will gain increasing insight into why I do the things I do. 7.31 1.465 301
I will share my life with someone I love. 8.46 1.184 301
I will work to make the world a better place. 7.22 1.759 301
I will be admired by many people. 5.11 2.310 301
I will keep up with fashions in hair and clothing. 4.31 2.448 301
I will have many expensive possessions. 4.27 2.431 301
I will know and accept who I really am. 8.34 1.079 301
I will have committed, intimate relationships. 8.15 1.345 301
I will help others improve their lives. 7.61 1.416 301
I will be famous. 2.92 2.087 301
I will achieve the “look” I’ve been after. 4.13 2.405 301
I will be rich. 5.56 2.390 301
I will continue to grow and learn new things. 8.12 1.019 301
I will have deep, enduring relationships. 8.25 1.186 301
I will help people in need. 7.78 1.453 301
My name will appear frequently in the media. 2.60 1.915 301
My image will be one others find appealing. 4.25 2.425 301
I will have enough money to buy everything I want. 5.90 2.371 301

Appendix B: Structure Matrix

Structure Matrix
Factor
1 2 3 4 5 6
11. I will have people comment often about how attractive I look. .832 .053 .101 .555 .303 .100
17. I will keep up with fashions in hair and clothing. .817 .086 .071 .411 .264 -.020
23. I will achieve the “look” I’ve been after. .763 .044 .050 .506 .275 .102
18. I will have many expensive possessions. .749 -.090 .018 .516 .431 .002
29. My image will be one others find appealing. .686 .017 .109 .493 .194 .178
5. I will successfully hide the signs of aging. .496 .139 .141 .423 .358 .057
15. I will work to make the world a better place. .027 .815 .342 .194 .070 .332
21. I will help others improve their lives. .038 .779 .407 .127 .001 .417
27. I will help people in need. .085 .773 .410 .159 .014 .330
9. I will work for the betterment of society. .063 .737 .244 .192 .126 .311
3. I will assist people who need it, asking nothing in return. -.060 .573 .278 .068 .066 .364
14. I will share my life with someone I love. .041 .312 .790 .140 .241 .334
26. I will have deep, enduring relationships. .096 .377 .736 .162 .141 .402
20. I will have committed, intimate relationships. .140 .242 .710 .171 .088 .328
2. I will feel that there are people who really love me, and whom I love. -.021 .286 .596 .068 .230 .368
8. I will have good friends that I can count on. .075 .341 .459 .085 .270 .390
10. My name will be known by many people. .549 .184 .190 .858 .384 .180
22. I will be famous. .628 .003 .047 .771 .181 .076
16. I will be admired by many people. .534 .228 .176 .721 .389 .242
4. I will be recognized by lots of different people. .371 .283 .255 .705 .362 .204
28. My name will appear frequently in the media. .576 -.045 -.012 .687 .103 .030
12. I will have a job that pays very well. .287 .107 .220 .329 .889 .339
6. I will be financially successful. .357 .062 .253 .330 .863 .278
24. I will be rich. .562 -.143 .087 .476 .608 .158
3.0 I will have enough money to buy everything I want. .526 -.127 .046 .352 .579 .207
19. I will know and accept who I really am. .039 .377 .408 .076 .199 .723
13. I will gain increasing insight into why I do the things I do. .173 .347 .326 .219 .219 .547
25. I will continue to grow and learn new things. .027 .456 .292 .167 .176 .516
7. At the end of my life, I will look back on my life as meaningful and complete. -.035 .226 .466 .089 .381 .478
1. I will choose what I do, instead of being pushed along by life. -.077 .180 .216 .018 .175 .418
Extraction Method: Unweighted Least Squares.
Rotation Method: Promax with Kaiser Normalization.
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