# Statistical Factor Analysis And Related Methods Pdf

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For example, a basic desire of obtaining a certain social level might explain most consumption behavior. These unobserved factors are more interesting to the social scientist than the observed quantitative measurements.

## Exploratory Factor Analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. A2 is stating that these latent variables do not influence one another, which might be too strong a condition. The remaining variance is called specific variance.

The standard way is to use principal components that uses eigenvalues and eigenvectors to bring the estimate of the total communality as close as possible to the total of the observed variances check example 2. Just one extra note. The values of the loadings are not unique in fact they are infinite. Rotation methods can be described as orthogonal , which do not allow the resulting factors to be correlated, and oblique , which do allow the resulting factors to be correlated.

Both methods have the aim of reducing the dimensionality of a vector of random variables. Also both methods assume that the modelling subspace is linear Kernel PCA is a more recent techniques that try dimensionality reduction in non-linear spaces. But while Factor Analysis assumes a model that may fit the data or not , PCA is just a data transformation and for this reason it always exists. Furthermore while Factor Analysis aims at explaining covariances or correlations, PCA concentrates on variances.

The items range in value from 1 to 5, which represent a scale from Strongly Dislike to Strongly Like. Package stats has a function factanal can be used to perform factor analysis:. The output maximizes variance for the 1st and subsequent factors, while all are orthogonal to each other. Eg, varimax rotation is an orthogonal rotation of the factor axes to maximize the variance of the squared loadings of a factor column on all the variables rows in a factor matrix, which has the effect of differentiating the original variables by extracted factor.

Each factor will tend to have either large or small loadings of any particular variable. A varimax solution yields results which make it as easy as possible to identify each variable with a single factor. This is the most common rotation option. Looking at both plots we see that the courses og Geology, Biology and Chemistry all have high factor loadings around 0. This suggests that statistics is less related to the concept of Math than Algebra and Calculus.

We could also try an oblique rotation Stats might share some with the Science factor. Notice that our factors are correlated at 0. The choice of an oblique rotation allowed for the recognition of this relationship. This next plot is the Cattell scree plot. As one moves to the right, toward later components, the eigenvalues drop. Another package, nFactors , also offers a suite of functions to aid in this decision. More methods can be found here.

Each subject score is positively correlated with each of the scores in the other subjects, indicating that there is a general tendency for those who do well in one subject to do well in others. The highest correlations are between the three mathematical subjects and to a slightly lesser extent, between the three humanities subjects, suggesting that there is more in common within each of these two groups than between them.

In order to reduce the dimension of the problem and to explain the observed correlations through some related latent factors we fit a factor model using the principal factor method. First of all we need to compute an initial estimate of the communalities by calculating the multiple correlation coefficient of each variable with the remaining ones. We obtain it as a function of the diagonal elements of the inverse correlation matrix.

Its decomposition through the spectral theorem shows that only two eigenvalues are positive. This means that two factors might be enough in order to explain the observed correlations. The first factor seems to measure overall ability in the six subjects, while the second contrasts humanities and mathematics subjects. Communalities are, for each variable, the part of its variance that is explained by the common factors. To estimate the communalities we need to sum the square of the factor loadings for each subject:.

Of course, the larger the communality the better does the variable serve as an indicator of the associated factors. Since the elements out of the diagonal are fairly small and close to zero we can conclude that the model fits adequately the data.

The following correlation matrix are from 10 different intelligence tests between scores of 75 students. By looking at the correlation matrix one can see a strong correlation between the 10 tests: all the correlation values are positive and mostly varies between 0.

Record the percentage of variability in each variable that is explained by the model communalities :. Such a rotation works on the factor loadings increasing the differences between lower weights, letting them converge to zero, and the higher weights, letting them converge to one. Introduction Factor Analysis vs. Introduction Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors.

Some egs: Varimax: a rotation that seeks to maximize the variance of the squared loading for each factor ie, make them as large as possible to capture as most signal as possible Quartimax : seeks to maximize the variance of the squared loadings for each variable, and tends to produce factors with high loadings for all variables. Factor Analysis vs.

PCA Both methods have the aim of reducing the dimensionality of a vector of random variables. The chi square statistic is 2. The p-value is 0. The degrees of freedom for the null model are 15 and the objective function was 2. Determining the Number of Factors to Extract A crucial decision in exploratory factor analysis is how many factors to extract. Determine Number of Factors to Extract install. Example 3 The following correlation matrix are from 10 different intelligence tests between scores of 75 students.

The chi square statistic is

## Factor Analysis: A Short Introduction, Part 1

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 is a useful tool for investigating variable relationships for complex concepts such as socioeconomic status, dietary patterns, or psychological scales. It allows researchers to investigate concepts that are not easily measured directly by collapsing a large number of variables into a few interpretable underlying factors. The key concept of factor analysis is that multiple observed variables have similar patterns of responses because they are all associated with a latent i. For example, people may respond similarly to questions about income, education, and occupation, which are all associated with the latent variable socioeconomic status. In every factor analysis, there are the same number of factors as there are variables.

This seminar is the first part of a two-part seminar that introduces central concepts in factor analysis. Part 1 focuses on exploratory factor analysis EFA. Although the implementation is in SPSS, the ideas carry over to any software program. Part 2 introduces confirmatory factor analysis CFA. Click on the preceding hyperlinks to download the SPSS version of both files. The SAQ-8 consists of the following questions:.

## Factor analysis

In statistical terms, factor analysis is a method to model the population covariance matrix of a set of variables using sample data. Factor analysis is used for theory development, psychometric instrument development, and data reduction. Figure 1.

In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model. This is a preview of subscription content, access via your institution.

### What is factor analysis and how does it simplify research findings?

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed variables mainly reflect the variations in two unobserved underlying variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modelled as linear combinations of the potential factors, plus " error " terms. Simply put, the factor loading of a variable quantifies the extent to which the variable is related with a given factor. A common rationale behind factor analytic methods is that the information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset.

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Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. A2 is stating that these latent variables do not influence one another, which might be too strong a condition. The remaining variance is called specific variance. The standard way is to use principal components that uses eigenvalues and eigenvectors to bring the estimate of the total communality as close as possible to the total of the observed variances check example 2. Just one extra note.

Preliminaries Matrixes, Vector Spaces The Ordinary Principal Components Model Statistical Testing of the Ordinary Principal Components Model Extensions of.

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