# What Are Principal Components Analysis and Exploratory Factor Analysis?

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3 items for career-related 2? Are the four items for 3 all related to personal fulfillment? Are those loading heavily on 4 related to heritage ties? I think these interpretations are pretty good, 2 but what do you think? That s important too. There are additional numbers around the edges of Table 1 that are also worth considering. In the column furthest to the right, there are communalities (h 2 ). Each of these values tells us the proportion of variance accounted for the particular variables in that row by the four components in this analysis. For instance, the 0.72 at the top right indicates that 72% of the variance in the I learn Korean to transfer credits to college variable is accounted for by the four components in this analysis. Clearly, this analysis is much better at accounting for the variance in some variables than in others. Which variable has the highest communality? Which has the lowest? Why is the relative value of these communalities important? It s important because those variables with relatively high communalities are being accounted for fairly well, while those with low ones are not. Across the bottom of Table 1, the following numbers represent the proportion of variance accounted for by each component: 0.23, 0.15, 0.14, and These indicate that 1 accounts for 23% of the variance, 2 accounts for 15%, 3 accounts for 14%, and 4 accounts for 12% of the variance. The last number at the bottom right of Table 1 (0.64) indicates the total proportion of variance accounted for by the analysis as a whole. In other words, 64% (or just shy of 2/3rds of the variance) was accounted for by this analysis. This total proportion of variance can be calculated by either adding up the individual proportions of variance accounted for by each of the four components, or by averaging the communalities. How are PCA and EFA Different? Calculations for both PCA and EFA involve matrix algebra as well as matrices of Eigen vectors and Eigen values. Any explanation of this would be quite involved and not particularly enlightening for most readers of this column, so suffice it to say that both PCA and EFA depend on calculating and using matrices of Eigen vectors and values in conjunction with a matrix of the correlation coefficients all of which are based on the variables being studied. The difference between PCA and EFA in mathematical terms is found in the values that are put in the diagonal of the correlation matrix. In PCA, 1.00s are put in the diagonal meaning that all of the variance in the matrix is to be accounted for (including variance unique to each variable, variance common among variables, and error variance). That would, therefore, by definition, include all of the variance in the variables. In contrast, in EFA, the communalities are put in the diagonal meaning that only the variance shared with other variables is to be accounted for (excluding variance unique to each variable and error variance). That would, therefore, by definition, include only variance that is common among the variables. How do Researchers Decide Whether to Use PCA or EFA? The difference between PCA and EFA in conceptual terms is that PCA analyzes variance and EFA analyzes covariance (Tabachnick and Fidell, 2007, p. 635). Thus when researchers want to analyze only the variance that is accounted for in an analysis (as in situations where they have a theory drawn from previous research about the relationships among the variables), they should probably use EFA to exclude unique and error variances, in order to see what is going on in the covariance, or common variance. When researchers are just exploring without a theory to see what patterns emerge in their data, it makes more sense to perform PCA (and thereby include unique and error variances), just to see what patterns emerge in all of the variance. 2 My one reservation is that, in their interpretation and discussion, the authors overlooked the complex variables which loaded on two or more components: I learn Korean to use it for my research and I learn Korean because I have a general interest in languages. 28

5 Basically, researchers tend to: (a) use PCA if they are on a fishing expedition trying to find patterns in their data and have no theory to base the analysis on, or (b) use EFA if they have a well-grounded theory to base their analysis on. Generally, the second strategy is considered to be the stronger form of analysis. Conclusion I have shown what PCA and EFA (collectively known as factor analysis or FA) are, and in part, how they should be presented and interpreted. In the process, I have defined and exemplified loadings, communalities, proportions of variance, components, factors, PCA, and EFA. I have also explored the basic mathematical and conceptual differences between PCA and EFA, and discussed how researchers decide on whether to use PCA or EFA. However, much about FA has been left unexplained. How do researchers decide the number of components or factors to include in the analysis? For instance, how did I decide on the three components and factors shown in Table 3? Also, what is rotation, what are the different types, and how do researchers choose which type to use? For instance, what is the varimax rotation mentioned in Tables 1 and 3 (and the associated text), and why did the researchers choose it? As I mentioned above, I will address these issues in two subsequent columns. References Brown, J. D., Robson, G., & Rosenkjar, P. (2001). Personality, motivation, anxiety, strategies, and language proficiency of Japanese students. In Z. Dörnyei & R. Schmidt (Eds.), Motivation and second language acquisition (pp ). Honolulu, HI: Second Language Teaching & Curriculum Center, University of Hawai i Press. Guilford, J. P., & Yatabe, T. (1957). Yatabe-Guilford personality inventory. Osaka: Institute for Psychological Testing. Kim, J. O., & Mueller, C. W. (1978). Introduction to factor analysis: What it is and how to do it. Beverly Hills, CA: Sage. Lee, J. S., & H. Y. (2008). In K. Kondo-Brown & J. D. Brown (Eds.), Teaching Chinese, Japanese, and Korean heritage language students. New York: Lawrence Erlbaum Associates. Robson, G. (1994). Relationships between personality, anxiety, proficiency, and participation. Unpublished doctoral dissertation, Temple University Japan, Tokyo, Japan. Tabachnick, B. G., & Fidell, L. S. (2007). Using multivariate statistics (5th ed.). Upper Saddle River, NJ: Pearson Allyn & Bacon. 30

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