Factor Analysis vs. PCA: Choosing the Right Tool

Key Takeaways

• Factor analysis and principal component analysis are two techniques for dimensionality reduction that try to find a smaller set of variables that can explain the variation and correlation among large variables.

• Factor analysis is based on a causal model that assumes that there are latent factors that influence the observed variables, while PCA is based on a mathematical transformation that does not assume any causal model or latent variables.

• Factor analysis requires more assumptions and decisions than PCA, such as choosing between EFA and CFA, deciding how many factors to extract, choosing a method for estimating the factor loadings, and choosing a method for rotating the factors.

• PCA is more straightforward and objective than factor analysis, as it only requires deciding how many components to retain and interpreting the meaning of the components based on their loadings and correlations with the original variables

• Factor analysis can be useful for identifying the latent constructs or dimensions that underlie a set of variables, while PCA can be useful for finding the best way to represent the data in a lower-dimensional space.

• The choice between factor analysis and PCA depends on the problem objective and the nature of the data. You should use factor analysis to find the latent factors or constructs that underlie your data and if you have prior knowledge or hypotheses about them. You should use PCA if you are interested in finding the best way to represent your data in a lower-dimensional space and do not have any prior knowledge or hypothesis about the latent factors or constructs.

Factor Analysis	Principal Component Analysis
Finds hidden reasons for data changes	Finds new ways to show data changes
Find reasons that are similar and different	Finds ways that are not similar
Needs more choices and steps	Needs fewer choices and steps

While working on a data analysis project, I faced a common challenge: reducing the dimensionality of my data set without losing too much information. I had many variables correlated with each other, and I wanted to find a smaller set of variables that could capture the essence of my data. I knew there were two popular techniques for dimensionality reduction: factor analysis and principal component analysis (PCA).

But I was not sure which one to use and what were the differences between them. So, I decided to research and learn more about these two methods.

In this article, I will share what I learned and how I chose the best technique for my project.

What is Factor Analysis?

Factor analysis is a statistical method that explains the variation and correlation among a large set of observed variables in terms of a smaller number of unobserved latent variables called factors. The factors are assumed to be the underlying causes that influence the observed variables; read more about Factor Analysis.

Factor analysis can be divided into exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). EFA is used when we do not have any prior knowledge or hypothesis about the factors and their relationships with the observed variables.

Exploratory factor analysis

EFA tries to discover the factors and their loadings (the coefficients that indicate how much each factor contributes to each observed variable) from the data.

Confirmatory factor analysis

CFA is used when we have some prior knowledge or hypothesis about the factors and their relationships with the observed variables. CFA tests whether the data fits the hypothesized model of factors and loadings.

Factor analysis can be useful.

• Identifying the latent constructs or dimensions that underlie a set of variables

• Reducing the number of variables for further analysis

• Testing hypotheses about the structure and meaning of the data

• Developing scales or instruments for measuring latent variables

What is Principal Component Analysis (PCA)?

Principal component analysis (PCA) is a technique for transforming a large set of correlated variables into a smaller set of uncorrelated variables called principal components. The principal components are linear combinations of the original variables that capture the maximum variation in the data.

PCA can be seen as a particular case of factor analysis, where the factors are orthogonal (uncorrelated) and account for all the variation in the data. Unlike factor analysis, PCA does not assume any underlying causal model or latent variables. PCA tries to find the best way to represent the data in a lower-dimensional space.

PCA can be useful

• Reducing the dimensionality of the data

• Simplifying the data for visualization or interpretation

• Enhancing the signal-to-noise ratio in the data

• Performing feature extraction or selection for machine learning algorithms

Difference Between Factor Analysis and Principal Component Analysis

The main difference between factor analysis and principal component analysis is that factor analysis is based on a causal model that assumes that there are latent factors that influence the observed variables, while PCA is based on a mathematical transformation that does not assume any causal model or latent variables.

As you can see, PCA tries to find orthogonal (perpendicular) components and explain as much variation as possible in the data. Factor analysis tries to find correlated factors and explain as much common variation as possible in the data while allowing for some unique variation and error terms.

Assumptions and Decisions

Another difference between factor analysis and principal component analysis is that factor analysis requires more assumptions and decisions than PCA. For example, in factor analysis, we need to:

• Choose between EFA and CFA depending on our research question and prior knowledge

• Decide how many factors to extract based on various criteria such as eigenvalues, scree plot, or parallel analysis

• Choose a method for estimating the factor loadings, such as maximum likelihood, principal axis factoring, or generalized least squares

• Choose a method for rotating the factors, such as varimax, quartimax, or oblimin

• Interpret the meaning of the factors based on their loadings and theoretical relevance

In contrast, PCA is more straightforward and objective, as it only requires us to:

• Decide how many components to retain based on the proportion of variance explained or other criteria

• Interpret the meaning of the components based on their loadings and correlations with the original variables

Use Cases and Applications of Factor Analysis

It can be applied in various fields and domains where we want to understand our data's underlying structure and meaning. Some examples are:

Psychology

It can be used to develop and validate psychological tests and scales that measure latent traits such as personality, intelligence, or attitudes. For example, the Big Five personality test is based on a factor analysis of various personality traits.

Marketing

It can be used to identify the key factors that influence consumer behaviour and preferences. For example, an analysis of customer satisfaction surveys can reveal the main dimensions of customer satisfaction and loyalty.

Education

It can evaluate the quality and validity of educational tests and assessments. For example, a student's test scores can show the extent to which the test measures different skills and abilities.

Sociology

It can be used to explore the social and cultural factors that affect human behaviour and society. For example, demographic data can reveal a population's major social groups and trends.

Use Cases and Applications of Principal Component Analysis

Principal component analysis can be applied in various fields and domains where we want to reduce the complexity and dimensionality of our data. Some examples are:

Image Processing

PCA can compress and enhance images by reducing the number of pixels or colours without losing much information. For example, PCA can be used to perform face recognition by extracting the main features of a face image.

Data Mining

PCA can be used to preprocess and transform data for machine learning algorithms by reducing the number of features or variables without losing much information. For example, PCA can detect anomalies by finding outliers in a data set.

Bioinformatics

PCA can analyze and visualize high-dimensional biological data such as gene expression or protein structure by reducing the number of dimensions without losing much information. For example, PCA can perform cluster analysis by finding groups of similar genes or proteins.

Finance

PCA can be used to analyze and model financial data such as stock prices or exchange rates by reducing the number of variables without losing much information. For example, PCA can optimise a portfolio by finding the optimal combination of assets that minimize risk and maximize return.

Which Method to Choose Based on the Problem Objective?

The choice between factor analysis and principal component analysis depends on the problem objective and the nature of the data. Here are some general guidelines that can help you decide which method to use:

• Use factor analysis if you are interested in finding the latent factors or constructs that underlie your data and you have some prior knowledge or hypothesis about them. It is more suitable for exploratory or confirmatory research questions that involve causal inference or theory testing.

• Use principal component analysis if you are interested in finding the best way to represent your data in a lower-dimensional space and have no prior knowledge or hypothesis about the latent factors or constructs. PCA is more suitable for descriptive or predictive research questions that involve data transformation or dimensionality reduction.

Of course, these are not strict rules, and there may be situations where both methods can be applied or compared. For example, you may use both factor analysis and principal component analysis to see how consistent they are in finding the underlying structure of your data, or you may want to use factor analysis as a preprocessing step before applying principal component analysis.

Conclusion

In this article, I have explained what factor and principal component analysis are, how they differ, and their use cases and applications. I have also shared with you how I chose these two methods for my data analysis project. I hope you have learned something new from this article and found it helpful for your data analysis projects.

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FAQs

What is the difference between factor analysis and principal component analysis?

Factor analysis is based on a causal model that assumes that there are latent factors that influence the observed variables, while PCA is based on a mathematical transformation that does not assume any causal model or latent variables.

How many factors or components should I extract?

This question has no definitive answer, as different criteria may yield different results. Some common criteria are eigenvalues, scree plots, parallel analysis, proportion of variance explained, or cross-validation.

How can I interpret the factors or components?

You can interpret the factors or components based on their loadings and correlations with the original variables. You can also use a rotation method to make the factors or components more interpretable and meaningful.

When should I use factor analysis or principal component analysis?

Use factor analysis if you are interested in finding the latent factors or constructs that underlie your data and have some prior knowledge or hypothesis about them. You should use PCA if you are interested in finding the best way to represent your data in a lower-dimensional space and do not have any prior knowledge or hypothesis about the latent factors or constructs.

How can I perform factor analysis or principal component analysis in RStudio?

You can use various packages and functions in RStudio to perform factor or principal component analyses. Some examples are:

• For factor analysis, you can use the function in the stats package or the fa function in the psych package.

• For principal component analysis, you can use the prcomp function in the stats package or the principal function in the psych package.

• You can also use other packages such as FactoMineR, GPArotation, or lavaan for more advanced options and features.

What are some common applications of factor analysis or principal component analysis in data analysis?

It can be applied in various fields and domains where we want to understand our data's underlying structure and meaning, such as psychology, marketing, education, or sociology. Principal component analysis can be applied in various fields and domains where we want to reduce the complexity and dimensionality of our data, such as image processing, data mining, bioinformatics, or finance.

What are some limitations or challenges of factor analysis or principal component analysis?

Factor analysis and principal component analysis are not perfect methods, and they may have some limitations or challenges, such as:

• They may not be able to capture all the nuances and complexities of the data

• They may be sensitive to outliers, missing values, multicollinearity, or non-normality

• They may be influenced by subjective choices or judgments

• They may not be generalizable or replicable across different data sets or samples

What are some alternatives or extensions of factor analysis or principal component analysis?

Besides factor or principal component analysis, many other methods can be used for dimensionality reduction or latent variable modelling. Some examples are:

• Independent component analysis (ICA): A method that tries to find components that are statistically independent rather than orthogonal

• Cluster analysis: A method that tries to find groups of similar observations rather than variables; read more.

• Structural equation modelling (SEM): A method that tries to test complex models that involve multiple factors, indicators, mediators, moderators, or outcomes

• Non-negative matrix factorization (NMF): A method that tries to find factors that are non-negative rather than arbitrary

• Multidimensional scaling (MDS): A method that tries to find a low-dimensional representation of the distances or similarities among observations rather than variables

What is the difference between PCA and factor analysis in Sklearn?

Sklearn is a Python library for machine learning and data analysis. It provides different PCA and factor analysis modules, which have different methods and parameters. For example, the PCA module has fit, transform, inverse_transform, and explained_variance_ratio_, while the factor analysis module has fit, transform, loglike, and score. The parameters for PCA include n_components, whiten, svd_solver, tol, iterated_power, random_state, and copy, while the parameters for factor analysis include n_components, tol, copy, max_iter, noise_variance_init, svd_method, iterated_power, random_state, and check_input.

What is the difference between PCA and MFA?

PCA is principal component analysis, while MFA stands for multiple factor analysis. Both are dimensionality reduction techniques that can be applied to multivariate data sets. However, PCA assumes that all variables have the same scale and weight in the analysis, while MFA allows for different groups of variables with different scales and weights. MFA is a generalization of PCA that can handle multiple tables of variables that measure different aspects of the same set of observations. Using appropriate scaling methods, MFA can also handle quantitative and qualitative variables.

What is the use of factor analysis?

It is a statistical technique that can be used for various purposes, such as:

(a) identifying the underlying structure or dimensions of a set of variables;

(b) reducing the number of variables by grouping them into factors;

(c) testing hypotheses about the relationships between variables and factors;

(d) validating the reliability and validity of measurement scales or instruments;

(e) exploring the differences or similarities among groups of observations based on their factor scores;

(f) predicting outcomes or behaviours based on factor scores.

What is PCA analysis used for?

PCA analysis is used for:

(a) simplifying complex data sets by finding a smaller set of variables that capture most of the information in the original data;

(b) visualizing high-dimensional data by projecting it onto lower-dimensional spaces;

(c) removing noise or redundancy from data by discarding components with low variance;

(d) enhancing the performance or efficiency of machine learning algorithms by reducing the dimensionality of input features;

(e) discovering patterns or trends in data by examining the principal components and their loadings.

What are the 4 types of MFA?

MFA stands for multi-factor authentication, a security method that requires users to verify their identity using two or more factors before accessing an account or service.

The four types of MFA are:

(a) something you know, such as a password, PIN, or security question;

(b) something you have, such as a token device, smart card, or mobile phone;

(c) something you are, such as a fingerprint, face scan, or voice recognition;

(d) something you do, such as a gesture, keystroke pattern, or behavioural biometric.

What are the different types of PCA?

There are different types of PCA based on different criteria, such as:

(a) the type of data being analyzed, such as continuous, binary, categorical, ordinal, functional, etc.;

(b) the type of transformation being applied to the data, such as linear, nonlinear, kernel-based, robust, sparse, etc.; (c) the objective being optimized in the analysis, such as variance maximization, entropy minimization, mutual information maximization, etc.;

(d) the type of constraint imposed on the analysis includes orthogonality preservation, sparsity induction, non-negativity enforcement, etc.

What is the difference between PCA and functional PCA?

• PCA stands for principal component analysis,

• Functional PCA stands for functional principal component analysis.

Both are dimensionality reduction techniques that can be applied to multivariate data sets. However, PCA assumes that the data are discrete observations at fixed points, while functional PCA assumes that the data are continuous functions over a domain. Functional PCA can handle irregularly spaced or sparse observations by smoothing them into smooth functions and then finding the principal components of the functional data. Functional PCA can also capture the dynamics or variability of the functions over time or space.

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Refernece
https://stats.stackexchange.com/questions/1576/what-are-the-differences-between-factor-analysis-and-principal-component-analysi
https://www.analytixlabs.co.in/blog/factor-analysis-vs-pca/
https://datascience.stackexchange.com/questions/105023/factor-analysis-vs-pca
https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html
https://www.theanalysisfactor.com/factor-analysis-1-introduction/
https://builtin.com/data-science/step-step-explanation-principal-component-analysis
https://en.wikipedia.org/wiki/Principal_component_analysis
https://en.wikipedia.org/wiki/Functional_principal_component_analysis