Factor analysis is a technique used to reduce a large number of variables into smaller quantities called factors. This technique extracts the maximum common variance from all variables and places them in a common score. As an index of all the variables, we can use this score for a more detailed analysis. Factor analysis is part of the general linear model (GLM) and this method also assumes several assumptions: there is a linear relationship, there is no multicollinearity. It also includes relevant variables in the analysis and there is a true correlation between variables and factors. There are several methods available, but principal component analysis is used more frequently.

**Why use Factor Analysis?**

**Why use Factor Analysis?**

Factor analysis is a useful tool to investigate variable relationships for complex concepts such as socioeconomic status, dietary patterns, or psychological scales. It enables researchers to investigate concepts that cannot be easily measured directly by collapsing a large number of variables into a few interpretable underlying factors.

**What is a Factor?**

**What is a Factor?**

The key concept of factor analysis is that multiple observed variables have similar response patterns because they are all associated with one latent (i.e. not directly measured) variable. For example, people may similarly answer questions about income, education, and occupation, which are associated with latent variable socioeconomic status.

In each factor analysis, there are the same number of factors as variables. Each factor captures a certain amount of the overall variance in the observed variables, and the factors are always listed in order of how much variation they account for. The eigenvalue is a measure of how much of the variance of the observed variables a factor explains. Any factor with an eigenvalue ≥1 explains more variance than a single observed variable.

The factors that explain the least amount of variation are generally ruled out. The relationship of each variable to the underlying factor is expressed by the factor load.

**Types of ****Factor Análisis**

**Types of**

**Factor Análisis**

There are different types of methods used to extract the factor from the data set:

- Principal Component Analysis - This is the most common method used by researchers. It begins to extract the maximum variance and places them in the first factor. After that, that variance explained by the first factors is removed, and then it begins to extract the maximum variance for the second factor. This process goes to the last factor.
- Common factor analysis: The second method most preferred by researchers, extracts the common variance and factors them in. This method does not include the single variance of all variables.
- Image factorization: This method is based on the correlation matrix. The regression method is used to predict the factor in image factorization.
- Maximum Likelihood Method: This method also works on the correlation metric, but uses the maximum likelihood method for factoring.
- Other factor analysis methods: Alpha factorization exceeds least squares. The weight square is another regression-based method used for factoring.

**Charge Factor**

**Charge Factor**

The factor load is basically the correlation coefficient for the variable and the factor. The factor load shows the variance explained by the variable in that particular factor.

Eigenvalues: Eigenvalues are also called characteristic roots. The eigenvalues show the variance explained by that particular factor of the total variance.

Factor Score: The factor score is also called the component score. This score is for all rows and columns, which can be used as an index of all variables and can be used for further analysis. We can standardize this score by multiplying a common term. With this factor score, whatever analysis we do, we will assume that all variables will behave like factor scores and will move.

Criteria for determining the number of factors: according to the Kaiser criterion, eigenvalues are a good criterion for determining a factor. If the eigenvalues are greater than one, we should consider that a factor and if the eigenvalues are less than one, then we should not consider it as a factor. According to the variance extraction rule, it must be greater than 0.7. If the variance is less than 0.7, then we shouldn't consider that a factor.

Rotation method: The rotation method makes it more reliable to understand the output. The eigenvalues do not affect the rotation method, but the rotation method affects the eigenvalues or the percentage of variance extracted.

**Execution of ****Factor Analysis**

**Execution of**

**Factor Analysis**

As a data analyst, the goal of a factor analysis is to reduce the number of variables to explain and interpret the results. Factor extraction involves choosing the type of model and the number of factors to extract. Factor rotation occurs after factors are extracted, with the goal of achieving a simple structure to improve interpretability.

**Confirmatory** **Factor Analysis**

**Confirmatory**

**Factor Analysis**

Confirmatory factor analysis allows the researcher to determine if there is a relationship between a set of observed variables (also known as overt variables) and their underlying constructs. It is similar to exploratory factor analysis. With confirmatory factor analysis we can specify the number of factors necessary. Although it is technically applicable to any discipline, it is generally used in the social sciences.

**I****mplementation of Confirmatory Factor Analysis**

**I**

**mplementation of Confirmatory Factor Analysis**

Conduct a literature review to help you choose an appropriate model. For example, we can choose a diagram or equations.

Determine if unique values are possible for estimating the population parameter.

Data collection.

Perform an initial data analysis to verify problems such as missing data, collinearity, or outliers.

Estimate population parameters.

Determine if the model you chose works. If the model is unacceptable, we should consider conducting an Exploratory Factor Analysis.

**Exploratory Factor Analysis**

Exploratory Factor Analysis is used to find the underlying structure of a large set of variables. Reduce the data to a much smaller set of summary variables. It is almost identical to confirmatory factor analysis. Both techniques can be used to confirm or explore. The similarities are:

Evaluate the internal reliability of a measurement.

Examine factors or theoretical constructions represented by sets of items. Assume that the factors are not correlated.

Investigate the quality of individual articles.

**Procrustes Generalized Analysis**

**Procrustes Generalized Analysis**

Procrustes analysis is a way to compare two sets of configurations or shapes. Originally developed to combine two factor analysis solutions, the technique was extended to Procrustes generalized analysis so that more than two ways could be compared. Shapes are aligned with an objective shape or with each other. This analysis uses geometric transformations (that is, isotropic rescaling, reflection, rotation, or translation) of matrices to compare data sets.

The consensus matrix is the result of the averages of all the input matrices. Matrices formed during the Generalized Procrusts Analysis process can be entered into Principal Component Analysis and projected in a two-dimensional space for easy-to-understand results.

**Use in Sensory Profiles**

Procrustes generalized analysis is a way of finding an underlying structure in the sensory profile, which is divided into two categories: conventional profile and free choice profile. With the conventional profile, a fixed set of descriptive terms is provided for evaluators. Evaluators are usually highly trained people. For example, you could ask three experts for their opinions on the body, aroma and taste of four brands of wine. Fixed descriptions can include crisp, angular, and buttery.

Procrustes generalized analysis is a way of finding an underlying structure in the sensory profile, which is divided into two categories: conventional profile and free choice profile. With the conventional profile, a fixed set of descriptive terms is provided for evaluators. Evaluators are usually highly trained people. For example, you could ask three experts for their opinions on the body, aroma and taste of four brands of wine. Fixed descriptions can include crisp, angular, and buttery.

**Latent Variables**

A latent or variable variable is generally considered to be a variable that is not directly measurable or observable. For example, a person's level of neurosis, awareness, or openness are latent variables. Although you cannot see these underlying variables (they are not part of the data set of an experiment), they can cause effects on your experimental results. Latent variables are also known as hypothetical constructions.

Latent variables are sometimes used in statistical modeling techniques such as factor analysis, where they can be inferred through modeling techniques. They are always present in almost all regression analyzes, because all additive error terms are not measurable (and therefore are latent). Statistical modeling methods that are often used to identify latent variables include: EM algorithms, factor analysis, hidden Markov models, latent semantic analysis, principal component analysis, and structural equation models. A latent variable can also be present (and included in a model) when there is no objective to actually measure that.

**Manifest Variables**

Manifest variables (also called observable variables) can be directly measured or observed. They are the opposite of latent variables. For example, age and gender are observable variables. However, it is rare that you can be 100% sure of a variable. Even gender, if you look at it, is not 100% certain, because people can lie about their shape, disguise their real gender, or be a transgender person. Therefore, latent variables should be used whenever possible.

**Conclusions**

Factor analysis is a method of modeling observed variables and their covariance structure, in terms of a smaller number of unobservable (latent) underlying factors. Factors are generally considered general concepts or ideas that can describe an observed phenomenon. For example, a basic desire to attain a certain social level could explain most consumer behavior. These unobserved factors are more interesting to the social scientist than the observed quantitative measurements.

Factor analysis is generally an exploratory-descriptive method that requires many subjective judgments. It is a widely used and often controversial tool because models, methods, and subjectivity are so flexible that debates about interpretations can ensue. The method is similar to the main components, although the factor analysis is more elaborate. In a sense, factor analysis is an investment of principal components. In factor analysis we model the observed variables as linear functions of the factors. With respect to the principal components, we create new variables that are linear combinations of the observed variables.

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**Bibliographic References**

Bryant, F. B., & Yarnold, P. R. (1995). Principal components analysis and exploratory and confirmatory factor analysis. In L. G. Grimm & P. R. Yarnold (Eds.), * Reading and understanding multivariate analysis*. Washington, DC: American Psychological Association.

Dunteman, G. H. (1989). * Principal components analysis*. Newbury Park, CA: Sage Publications.

Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. * Psychological Methods, 4*(3), 272-299.