How are PCA and EFA used
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James Dean Brown University of Hawai'i at Manoa |
[ p. 30 ]
Variables | Rotated 2 Factors (with Thinking extraversion) | Rotated 2 Factors (without Thinking extraversion) | ||||
Factor 1 | Factor 2 | h2 | Factor 1 | Factor 2 | h2 | |
Social extraversion | -0.108 | 0.668 | 0.458 | -0.142 | 0.660 | 0.456 |
Ascendance | -0.086 | 0.553 | 0.314 | -0.113 | 0.548 | 0.313 |
Thinking extraversion | -0.064 | -0.019 | 0.005 | |||
Rhathymia | 0.405 | 0.573 | 0.493 | 0.381 | 0.596 | 0.501 |
General activity | -0.191 | 0.692 | 0.515 | -0.225 | 0.680 | 0.513 |
Lack of agreeableness | 0.139 | 0.527 | 0.297 | 0.116 | 0.535 | 0.299 |
Lack of cooperativeness | 0.468 | 0.013 | 0.219 | 0.468 | 0.036 | 0.220 |
Lack of objectivity | 0.607 | 0.018 | 0.368 | 0.602 | 0.045 | 0.364 |
Nervousness | 0.754 | -0.199 | 0.608 | 0.762 | -0.164 | 0.608 |
Inferiority feelings | 0.656 | -0.494 | 0.675 | 0.762 | -0.164 | 0.608 |
Cyclic tendencies | 0.792 | 0.077 | 0.633 | 0.786 | 0.114 | 0.677 |
Depression | 0.773 | -0.257 | 0.664 | 0.783 | -0.221 | 0.662 |
Proportion of Variance | 0.255 | 0.183 | 0.437 | 0.258 | 0.179 | 0.477 |
[ p. 31 ]
Variables | Rotated PCA 2 Components | Rotated EFA 2 Factors | ||||
Comp 1 | Comp 2 | h2 | Factor 1 | Factor 2 | h2 | |
Social extraversion | -0.150 | 0.737 | 0.566 | -0.142 | 0.660 | 0.456 |
Ascendance | -0.116 | 0.654 | 0.441 | -0.113 | 0.548 | 0.313 |
Rhathymia | 0.419 | 0.656 | 0.605 | 0.381 | 0.596 | 0.501 |
General activity | -0.238 | 0.740 | 0.605 | -0.225 | 0.680 | 0.513 |
Lack of agreeableness | 0.150 | 0.649 | 0.443 | 0.116 | 0.535 | 0.299 |
Lack of cooperativeness | 0.565 | 0.065 | 0.065 | 0.468 | 0.036 | 0.220 |
Lack of objectivity | 0.687 | 0.065 | 0.476 | 0.602 | 0.045 | 0.364 |
Nervousness | 0.799 | -0.170 | 0.668 | 0.762 | -0.164 | 0.364 |
Inferiority feelings | 0.703 | -0.471 | 0.715 | 0.681 | -0.462 | 0.608 |
Cyclic tendencies | 0.819 | 0.117 | 0.684 | 0.786 | 0.114 | 0.677 |
Depression | 0.812 | -0.226 | 0.711 | 0.783 | -0.221 | 0.662 |
Proportion of Variance | 0.322 | 0.245 | 0.567 | 0.282 | 0.195 | 0.477 |
[ p. 32 ]
Because EFA only analyzes reliable variance, it is useful for partitioning the proportions of reliable variance in a set of variables. Again, we will begin with common variance, in this case, the proportion of common variance that each variable shares with all the other variables. This common variance is called the communality and is symbolized by h2. For example, in bottom right corner of the EFA results in Table 2, the communality for the Depression is .662. That means that 66.2% of the reliable variance for that variable is common variance shared with other variables.[ p. 33 ]
In the next section, I will explain how language researchers often go on to further study the construct validity of their tests or questionnaires. [For more about these concepts and how to estimate each type of variance see Guilford, 1954, pp. 354-357; Magnusson, 1966, pp. 180-182; Gorsuch, 1983, pp. 26-33; or Kline, 2002, pp. 42-43.][ p. 34 ]
Many researchers use factor analysis for one purpose or another without realizing the rich variety of other purposes this form of analysis can serve. I showed in the previous column (Brown, 2010) that EFA and PCA have applications in research work that include at least reducing the number of variables in a study, exploring patterns in the correlations among variables, and supporting a theory of how variables are related. In this column, I expanded the list of uses for EFA and PCA by explaining how they can also be useful: for developing tests and questionnaires by conducting item analysis to strengthen them; for studying the relative proportions of total, reliable, common, unique, specific, and error variances; or for providing evidence for convergent and discriminent validity. If you are currently using EFA and PCA, consider expanding the ways you apply these analyses. If you are not currently using EFA and PCA, you might want to ask yourself, why not?[ p. 35 ]