Response to Another Student Discussion
INSTRUCTIONS:
Respond by identifying the part of the learner's description that you found most helpful or clarifying. Explain why you found it helpful or insightful.
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Factor analysis, in general, functions to reduce a set of variables down to a number sufficient to load to underlying unmeasurable factors that explain correlations between a cluster of variables. It can help understand the underlying structure of a set of variables, structure a questionnaire such that each question can coalesce (as a variable) to an underlying factor, and it can reduce data down to help solve multicollinearity issues (Field, 2013). There are two methods in consideration here, namely Principal Components Analysis (PCA) and Principal Axis Factoring (PAF), also known as Exploratory Factor Analysis (EFA).
Comparisons between PCA and PAF
According to Field (2013), PCA transforms a variable data set into linear components to explain the maximum amount of total variance in the resulting correlation matrix. These variables accrue to components but components are not necessarily unmeasured, or latent, variables. PCA simply shows clusters of variables loaded onto a component. Since PCA does not seek to load onto unmeasurable latent variables like PAF, this is not considered an estimate. In PCA, we assume that all variance is common variance and we assume the communality of each variable is 1. The transposition is computed with these assumptions (Field, 2013)
Conversely, principal factor analysis or axis factoring seeks to achieve a reduced set of variables by accounting for the maximum amount of common variance in the correlation matrix thus identifying a smaller set of variables that can load onto a factor or latent variable (Field, 2013). In factor analysis (PAF) we calculate the estimated shared communality between each variable by squaring the multiple correlation with all other variables. Doing this for each variable then allows us to then extract the underlying factors that represent the factors or unmeasurable latent variables.
When to use each one
PCA might be used when performing pure variable reduction analysis. This can speak to the first and third benefit of factor analysis I outlined in the first paragraph. The first benefit, understanding the underlying structure, can bleed into the second however which uses latent variable identification when constructing scale items. Understanding the structure is a feature of both using PCA and PAF. The third benefit, solving multicollinearity challenges appears to play to PCA’s strengths.
PAF focuses on identifying underlying unmeasurable latent variables. Researchers, using theory and heuristics, may have a set of latent candidates that they can test by seeing of the anticipated variables (test items for example) will load according to their predictions on to the expected factors based on their rationale. This repeated process is how most personality tests are constructed Warner, 2013).
Both methods assume that our sample is the population, so it is not a practice to use the results of factor analysis on a set of variables to generalize to other samples without conducting the same operation. If the sample is randomized, it may be possible to generalize if the set of variables is finite with no other variables possible (Field, 2013).
Results differences
Field (2013) posits that the differences between PCA and PAF are few when you have 30 or more variables and the communalities are greater than .7. Conversely, he states that if we have fewer than 20 variables and communalities are less than .4, we will start to see differences.
I have used Field book almost exclusively as, at least on the topic of factor analysis, Warner’s explanations are quite obtuse and very hard to follow. This reminds me of the satirist mathmetician Tom Lehrer’s 60’s critique on the “new math” being introduced in the schools. Here is a YouTube clip. It’s a must see! https://www.youtube.com/watch?v=UIKGV2cTgqA
References:
Field, A, (2013). Discovering statistics using IBM SPS statistics (4th Ed.). Thousand Oaks, California: Sage Publications Inc.
Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd Ed.). Thousand Oaks, CA: Sage Publications.
Robert Brock
Educational Psychology
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