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What Is Component Matrix In Factor Analysis? [5 REPLIES FOUND]

Principal components analysis is a method of data reduction. Suppose that you have a dozen variables that are correlated. You might use principal components analysis to reduce your 12 measures to a few principal components. In this example, you may be most interested in obtaining the component scores (which are variables that are added to your data set) and/or to look at the dimensionality of the data. For example, if two components are extracted and those two components accounted for 68% of the total variance, then we would say that two dimensions in the component space account for 68% of the variance. Unlike factor analysis, principal components analysis is not usually used to identify underlying latent variables. Hence, the loadings onto the components are not interpreted as factors in a factor analysis would be. Principal components analysis, like factor analysis, can be preformed on raw data, as shown in this example, or on a correlation or a covariance matrix. If raw data are used, the procedure will create the original correlation matrix or covariance matrix, as specified by the user. If the correlation matrix is used, the variables are standardized and the total variance will equal the number of variables used in the analysis (because each standardized variable has a variance equal to 1) 😁 If the covariance matrix is used, the variables will remain in their original metric 😎 However, one must take care to use variables whose variances and scales are similar 😁 Unlike factor analysis, which analyzes the common variance, the original matrix in a principal components analysis analyzes the total variance. Also, principal components analysis assumes that each original measure is collected without measurement error. [1]
This seminar will give a practical overview of both principal components analysis (PCA) and exploratory factor analysis (EFA) using SPSS. We will begin with variance partitioning and explain how it determines the use of a PCA or EFA model. For the PCA portion of the seminar, we will introduce topics such as eigenvalues and eigenvectors, communalities, sum of squared loadings, total variance explained, and choosing the number of components to extract. For the EFA portion, we will discuss factor extraction, estimation methods, factor rotation, and generating factor scores for subsequent anlyzes. The seminar will focus on how to run a PCA and EFA in SPSS and thoroughly interpret output, using the hypothetical SPSS Anxiety Questionnaire as a motivating example. (modified by Norelle Baughman on January 6, 2021) [2]
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Based around an article from stats.idre.ucla.edu, factor analysis is a method of data reduction. It does this by seeking underlying unobservable (latent) variables that are reflected in the observed variables (manifest variables). There are many different methods that can be used to conduct a factor analysis (such as principal axis factor, maximum likelihood, generalized least squares, unweighted least squares), There are also many different types of rotations that can be done after the initial extraction of factors, including orthogonal rotations, such as varimax and equimax, which impose the restriction that the factors cannot be correlated, and oblique rotations, such as promax, which allow the factors to be correlated with one another. You also need to determine the number of factors that you want to extract. Given the number of factor analytic techniques and options, it is not surprising that different analysts could reach very different results analysing the same data set. However, all analysts are looking for simple structure. Simple structure is pattern of results such that each variable loads highly onto one and only one factor. (credit to Nevin Haskins having pointing this out). [3]
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Stats.idre.ucla.edu also explains how here you see that SPSS Anxiety makes up the common variance for all eight items, but within each item there is specific variance and error variance. Take the example of Item 7 “Computers are useful only for playing games”. Although SPSS Anxiety explain some of this variance, there may be systematic factors such as technophobia and non-systemic factors that can’t be explained by either SPSS anxiety or technophbia, such as getting a speeding ticket right before coming to the survey centre (error of meaurement). Now that we understand partitioning of variance we can move on to performing our first factor analysis. In fact, the assumptions we make about variance partitioning affects which analysis we run. (last edited 8 days ago by Jarren Hyde from Misratah, Libya) [4]
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Where a set of variables are correlated with each other, a plausible explanation is that there is some other variable that they are all correlated with. For example, it may be that the reason that viewership of each of the sports programs is correlated with each other is that they are all correlated with a more general variable, propensity to watch sports programs. Similarly, the factor that might explain the correlation amongst viewership of the current affairs programme may be that people differ in terms of their propensity to view current affairs programs. Principal components analysis is a statistical technique that attempts to uncover such factors (also known as components). (we really thank Cherilyn Woody from Cheongju, South Korea for their revision). [5]

Article References

  1. https://stats.idre.ucla.edu/spss/output/principal_components/
  2. https://stats.idre.ucla.edu/spss/seminars/efa-spss/
  3. https://stats.idre.ucla.edu/spss/output/factor-analysis/
  4. https://stats.idre.ucla.edu/spss/seminars/introduction-to-factor-analysis/a-practical-introduction-to-factor-analysis/
  5. https://docs.displayr.com/wiki/Category:Principal_Components_Analysis
Kelly-Anne Kidston

Written by Kelly-Anne Kidston

I am a writer of many words, from fiction to poetry to reviews. I am an avid reader and a lover of good books. I am currently writing my first novel and would love to find some beta readers who are interested in getting an early look.

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