# Difference Between Principal Component Analysis And Factor Analysis Pdf

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- Principal component analysis
- The Fundamental Difference Between Principal Component Analysis and Factor Analysis
- Principal Components and Factor Analysis. A Comparative Study.

Here, a best-fitting line is defined as one that minimizes the average squared distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis PCA is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. PCA is used in exploratory data analysis and for making predictive models.

## Principal component analysis

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. It seems that a number of the statistical packages that I use wrap these two concepts together.

However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use one over the other. A real example would be incredibly useful. Principal component analysis involves extracting linear composites of observed variables. Factor analysis is based on a formal model predicting observed variables from theoretical latent factors. In psychology these two techniques are often applied in the construction of multi-scale tests to determine which items load on which scales.

They typically yield similar substantive conclusions for a discussion see Comrey Factor-Analytic Methods of Scale Development in Personality and Clinical Psychology. This helps to explain why some statistics packages seem to bundle them together. I have also seen situations where "principal component analysis" is incorrectly labelled "factor analysis".

Run factor analysis if you assume or wish to test a theoretical model of latent factors causing observed variables. Run principal component analysis If you want to simply reduce your correlated observed variables to a smaller set of important independent composite variables.

This undoubtedly results in a lot of confusion about the distinction between the two. The bottom line is that these are two different models, conceptually. In PCA, the components are actual orthogonal linear combinations that maximize the total variance. In FA, the factors are linear combinations that maximize the shared portion of the variance--underlying "latent constructs". That's why FA is often called "common factor analysis". FA uses a variety of optimization routines and the result, unlike PCA, depends on the optimization routine used and starting points for those routines.

Simply there is not a single unique solution. In R, the factanal function provides CFA with a maximum likelihood extraction. It's simply not the same model or logic. I'm not sure if you would get the same result if you used SPSS's Maximum Likelihood extraction either as they may not use the same algorithm. For better or for worse in R, you can, however, reproduce the mixed up "factor analysis" that SPSS provides as its default. Here's the process in R. With the exception of the sign, which is indeterminate.

That result could also then be rotated using any of R's available rotation methods. There are numerous suggested definitions on the web. Here is one from a on-line glossary on statistical learning :.

Constructing new features which are the principal components of a data set. The principal components are random variables of maximal variance constructed from linear combinations of the input features. Equivalently, they are the projections onto the principal component axes, which are lines that minimize the average squared distance to each point in the data set.

To ensure uniqueness, all of the principal component axes must be orthogonal. PCA is a maximum-likelihood technique for linear regression in the presence of Gaussian noise on both inputs and outputs. A generalization of PCA which is based explicitly on maximum-likelihood. Like PCA, each data point is assumed to arise from sampling a point in a subspace and then perturbing it with full-dimensional Gaussian noise. The difference is that factor analysis allows the noise to have an arbitrary diagonal covariance matrix, while PCA assumes the noise is spherical.

In addition to estimating the subspace, factor analysis estimates the noise covariance matrix. You are right about your first point, although in FA you generally work with both uniqueness and communality. I don't quite follow your points, though. Rotation of principal axes can be applied whatever the method is used to construct latent factors. In fact, most of the times this is the VARIMAX rotation orthogonal rotation, considering uncorrelated factors that is used, for practical reasons easiest interpretation, easiest scoring rules or interpretation of factor scores, etc.

PROMAX might probably better reflect the reality latent constructs are often correlated with each other , at least in the tradition of FA where you assume that a latent construct is really at the heart of the observed inter-correlations between your variables. From a psychometrical perspectice, FA models are to be preferred since they explicitly account for measurement errors, while PCA doesn't care about that. Briefly stated, using PCA you are expressing each component factor as a linear combination of the variables, whereas in FA these are the variables that are expressed as linear combinations of the factors including communalities and uniqueness components, as you said.

The top answer in this thread suggests that PCA is more of a dimensionality reduction technique, whereas FA is more of a latent variable technique. This is sensu stricto correct. But many answers here and many treatments elsewhere present PCA and FA as two completely different methods, with dissimilar if not opposite goals, methods and outcomes.

I disagree; I believe that when PCA is taken to be a latent variable technique, it is quite close to FA, and they should better be seen as very similar methods. Also, can PCA be a substitute for factor analysis? There I argue that for simple mathematical reasons the outcome of PCA and FA can be expected to be quite similar, given only that the number of variables is not very small perhaps over a dozen. See my [long! Here I would like to show it on an example.

Here is how the correlation matrix looks like:. There are small deviations here and there, but the general picture is almost identical, and all the loadings are very similar and point in the same directions. This is exactly what was expected from the theory and is no surprise; still, it is instructive to observe. For a much prettier PCA biplot of the same dataset, see this answer by vqv.

Factor loadings were computed by an "iterated principal factors" algorithm until convergence 9 iterations , with communalities initialized with partial correlations. Once the loadings converged, the scores were calculated using Bartlett's method. This yields standardized scores; I scaled them up by the respective factor variances given by loadings lengths. A basic, yet a kind of painstaking, explanation of PCA vs Factor analysis with the help of scatterplots, in logical steps.

I thank amoeba who, in his comment to the question, has encouraged me to post an answer in place of making links to elsewhere. So here is a leisure, late response. We center them subtract the mean and do a scatterplot. Then we perform PCA on these centered data. The key property of PCA is that P1 - called 1st principal component - gets oriented so that the variance of data points along it is maximized. In PCA, we typically discard weak last components: we thus summarize data by few first extracted components, with little information loss.

P1's variance is 1. So, we discarded P2 and expect that P1 alone can reasonably represent the data. This is actually a "regressional model" where observed variables are predicted back by the latent variable if to allow calling a component a "latent" one P1 extracted from those same variables. Look at the plot Fig. P1 axis is shown tiled with its values P1 scores in green these values are the projections of data points onto P1.

Some arbitrary data points were labeled A, B, The connector "error" length squared is sum of the two errors squared, according to Pythagorean. Now, what is characteristic of PCA is that if we compute E1 and E2 for every point in the data and plot these coordinates - i. And it does: the cloud is plotted on the same picture as the beige cloud - and you see it actually forms axis P2 of Fig. No wonder, you may say. It is so obvious: in PCA , the discarded junior component s is what precisely decompose s in the prediction errors E, in the model which explains restores original variables V by the latent feature s P1.

Errors E together just constitute the left out component s. Here is where factor analysis starts to differ from PCA. Formally, the model predicting manifest variables by the extracted latent feature s is the same in FA as in PCA; [ Eq.

While latent P1 had its native variance. In both models - Eq. OK, back to the thread. E1 and E2 are uncorrelated in factor analysis; thus, they should form a cloud of errors either round or elliptic but not diagonally oriented.

While in PCA their cloud formed straight line coinciding with diagonally going P2. Both ideas are demonstrated on the pic:. Note that errors are round not diagonally elongated cloud in FA. Factor latent in FA is oriented somewhat different, i. On the pic, factor line is strangely conical a bit - it will become clear why in the end. Variables correlated, which is seen in the diagonally elliptical shape of the data cloud. P1 skimmed the maximal variance, so the ellipse is co-directed to P1.

Consequently P1 explained by itself the correlation; but it did not explain the existing amount of correlation adequately; it looked to explain variation in data points, not correlatedness. Actually, it over-accounted for the correlation, the result of which was the appearance of the diagonal, correlated cloud of errors which compensate for the over-account. Factor F can do it alone; and the condition when it becomes able to do it is exactly where errors can be forced to be uncorrelated.

Since the error cloud is round no correlatedness - positive or negative - has remained after the factor was extracted, hence it is the factor which skimmed it all. As a dimensionality reduction, PCA explains variance but explains correlations imprecisely. FA explains correlations but cannot account by the common factors as much data variation as PCA can. Because they explain correlation well mathematically.

## The Fundamental Difference Between Principal Component Analysis and Factor Analysis

James H. Watson, Seung C. Horenstein, Julian, Bernanke, Ben S.

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.

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

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.

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.

#### A worked example

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3 comments

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

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

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