# Difference Between Principal Component Analysis And Factor Analysis Pdf

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- Principal Components and Factor Analysis. A Comparative Study.
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- Factor Analysis and Principal Component Analysis: A Simple Explanation

*This resource is intended to serve as a guide for researchers who are considering use of PCA or EFA as a data reduction technique.*

## Principal Components and Factor Analysis. A Comparative Study.

Factor analysis and principal component analysis identify patterns in the correlations between variables. These patterns are used to infer the existence of underlying latent variables in the data. These latent variables are often referred to as factors, components, and dimensions. The most well-known application of these techniques is in identifying dimensions of personality in psychology. However, they have broad application across data analysis, from finance through to astronomy.

At a technical level, factor analysis and principal component analysis are different techniques, but the difference is in the detail rather than the broad interpretation of the techniques. The table below shows a correlation matrix of the correlations between viewing of TV programs in the U. Each of the numbers in the table is a correlation. This shows the relationship between the viewing of the TV program shown in the row with that shown in the column.

The higher the correlation, the greater the overlap in the viewing of the programs. For example, the table shows that people who watch World of Sport frequently are more likely to watch Professional Boxing frequently than are people who watch Today.

The table below shows the data again, but with the columns and rows re-ordered to reveal some patterns. Looking at the top left of the re-ordered correlation matrix, we can see that the people who watch any one of the sports programs are more likely to watch one of the other sports programs.

Similarly, if we look at the bottom right of the matrix we can see that people who watch one current affairs program are more likely to watch another, and vice versa. Where a set of variables is correlated with each other, a plausible explanation is that there is some other variable with which they are all correlated.

Similarly, the factor that might explain the correlation among viewership of the current affairs program may be that people differ in terms of their propensity to view current affairs programs. Factor analysis is a statistical technique that attempts to uncover factors. The table below shows the rotated factor loadings also known as the rotated component matrix for the U.

TV viewing data. In creating this table, it has been assumed that there are two factors i. The numbers in the table show the estimated correlation between each of the ten original variables and the two factors. For example, the variable that measures whether or not someone watches Professional Boxing is relatively strongly correlated with the first factor 0.

The first factor seems to be the propensity to watch sports and the second seems to be the propensity to watch current affairs. When conducting factor analysis and principal component analysis, decisions need to be made about how many factors should be selected. By default, programs use a method known as the Kaiser rule.

However, this rule is only a rule of thumb. It is often useful to consider alternative numbers of factors and select the cluster with the highest number of factors. The mathematics of factor analysis and principal component analysis PCA are different.

Factor analysis explicitly assumes the existence of latent factors underlying the observed data. PCA instead seeks to identify variables that are composites of the observed variables.

Although the techniques can get different results, they are similar to the point where the leading software used for conducting factor analysis SPSS Statistics uses PCA as its default algorithm. Want to read more? Check out the rest of our blog!

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They appear to be different varieties of the same analysis rather than two different methods. Yet there is a fundamental difference between them that has huge effects on how to use them. Both are data reduction techniques —they allow you to capture the variance in variables in a smaller set. Both are usually run in stat software using the same procedure, and the output looks pretty much the same. The steps you take to run them are the same—extraction, interpretation, rotation, choosing the number of factors or components. Despite all these similarities, there is a fundamental difference between them: PCA is a linear combination of variables; Factor Analysis is a measurement model of a latent variable.

## Factor Analysis and Principal Component Analysis: A Simple Explanation

Factor analysis and principal component analysis identify patterns in the correlations between variables. These patterns are used to infer the existence of underlying latent variables in the data. These latent variables are often referred to as factors, components, and dimensions. The most well-known application of these techniques is in identifying dimensions of personality in psychology.

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A Comparative Study. A comparison between Principal Component Analysis PCA and Factor Analysis FA is performed both theoretically and empirically for a random matrix X: n x p , where n is the number of observations and both coordinates may be very large. The comparison surveys the asymptotic properties of the factor scores, of the singular values and of all other elements involved, as well as the characteristics of the methods utilized for detecting the true dimension of X. In particular, the norms of the FA scores, whichever their number, and the norms of their covariance matrix are shown to be always smaller and to decay faster as n goes to infinity. Moreover, as compared to PCA, the FA scores and factors exhibit a higher degree of consistency because the difference between the estimated and their true counterparts is smaller, and so is also the corresponding variance. Finally, FA usually selects a much less encumbering number of scores than PCA, greatly facilitating the search and identification of the common components of X.

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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.

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