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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. Essentially, it models observed variables as linear combinations of potential factors plus "error" terms. The technique is used widely across various fields, including [[psychology]], [[sociology]], [[marketing]], [[business]], and [[medicine]], to identify underlying relationships between measured variables.		{{short description\|Statistical method used to describe variability among observed variables}}

	==Overview==		== Overview ==
	Factor analysis ~~aims~~ to ~~find independent latent variables~~, ~~known as factors, that can explain the patterns of correlations within a set of observed~~ variables~~. It is often used~~ in ~~data reduction to identify~~ a ~~small~~ number of factors ~~from a large set of~~ variables~~, making it easier to interpret the data. There~~ are ~~two main types~~ of ~~factor analysis: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA)~~. ~~EFA is~~ used to ~~uncover~~ the ~~underlying structure of a relatively large~~ set of variables~~, while CFA tests the hypothesis that~~ a ~~relationship between observed variables and their underlying latent constructs exists~~.		[[File:FactorPlot.svg\|thumb\|right\|A graphical representation of factor analysis results.]]
			'''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 observed variables are modeled as linear combinations of the potential factors, plus "error" terms. The information gained about the interdependencies can be used later to reduce the set of variables in a dataset.

	==~~Mathematical Formulation~~==		== History ==
	The ~~mathematical model for factor analysis can be expressed as:~~		Factor analysis originated in psychometrics and is used in behavioral sciences, social sciences, marketing, product management, operations research, and other applied sciences that deal with large quantities of data. The method was first introduced by [[Charles Spearman]] in the early 20th century.

	~~\[X~~ = ~~\mu + \Lambda F + \epsilon\]~~		== Types of Factor Analysis ==
			There are two main types of factor analysis:

	~~where \~~(X\) is the ~~vector of observed~~ variables~~, \(\mu\) is the vector of means, \(\Lambda\) is the matrix of factor loadings, \(F\)~~ is the ~~vector~~ of ~~unobserved latent factors, and \(\epsilon\) is~~ the ~~vector~~ of ~~error terms. The factor loadings represent the correlations between the~~ factors ~~and the observed variables~~.		* '''Exploratory Factor Analysis (EFA)''': Used to identify the underlying relationship between measured variables. It is often used when the researcher does not have a preconceived idea of the structure or number of factors.

	~~==Applications==~~		* '''Confirmatory Factor Analysis (CFA)''': Used to test the hypothesis that the relationships between observed variables and their underlying latent constructs exist. It is often used when the researcher has a specific idea about the structure or number of factors.
	Factor ~~analysis is applied in various domains~~ to ~~uncover~~ the underlying structure ~~in datasets:~~

	~~- In [[psychology]], it~~ is used ~~to identify factors that influence mental processes and behaviors.~~		== Applications ==
	~~- In [[marketing]], it helps in identifying underlying dimensions of consumer preferences and attitudes.~~		Factor analysis is widely used in various fields:
	~~- In [[sociology]], it is utilized to explore social attitudes and values.~~
	~~- In [[medicine]], factor analysis can reveal patterns in symptoms, aiding~~ in ~~the diagnosis and understanding of diseases.~~

	~~==Limitations==~~		* In [[psychology]], it is used to identify latent constructs such as intelligence, personality traits, and other psychological phenomena.
	~~While factor analysis is a powerful tool~~, it ~~has limitations. The interpretation of~~ factors ~~can be subjective,~~ and ~~the method relies heavily on the assumption of linearity between variables and factors~~. ~~Additionally~~, ~~the choice of the number of~~ factors ~~to retain can be somewhat arbitrary~~ and ~~influence the results~~.		* In [[marketing]], it helps in identifying underlying factors that affect consumer behavior and preferences.
			* In [[finance]], it is used to identify factors that affect stock prices and market trends.

	==~~See Also~~==		== Methodology ==
			The process of factor analysis involves several steps:

			1. '''Data Collection''': Gather data on the variables of interest.
			2. '''Correlation Matrix''': Compute the correlation matrix of the variables.
			3. '''Extraction of Factors''': Use methods such as principal component analysis or maximum likelihood to extract factors.
			4. '''Rotation''': Apply rotation methods like varimax or promax to achieve a simpler and more interpretable factor structure.
			5. '''Interpretation''': Analyze the factor loadings to interpret the factors.

			== Limitations ==
			Factor analysis has several limitations:

			* It requires a large sample size to produce reliable results.
			* The results can be sensitive to the method of extraction and rotation used.
			* It assumes linear relationships between variables and factors.

			== Related pages ==
	* [[Principal component analysis]]		* [[Principal component analysis]]
	* [[~~Cluster analysis~~]]		* [[Latent variable]]
	* [[Multivariate statistics]]		* [[Multivariate statistics]]

	~~==References==~~		[[Category:Statistical methods]]
	~~<references/>~~

	[[Category:~~Statistics~~]]
	~~[[Category:Psychometrics]]~~
	~~[[Category:Data analysis]]~~

	~~{{Template:Statistics-stub}}~~
	~~{{Template:Psychometrics-stub}}~~

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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. Essentially, it models observed variables as linear combinations of potential factors plus "error" terms. The technique is used widely across various fields, including [[psychology]], [[sociology]], [[marketing]], [[business]], and [[medicine]], to identify underlying relationships between measured variables.

==Overview==
Factor analysis aims to find independent latent variables, known as factors, that can explain the patterns of correlations within a set of observed variables. It is often used in data reduction to identify a small number of factors from a large set of variables, making it easier to interpret the data. There are two main types of factor analysis: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). EFA is used to uncover the underlying structure of a relatively large set of variables, while CFA tests the hypothesis that a relationship between observed variables and their underlying latent constructs exists.

==Mathematical Formulation==
The mathematical model for factor analysis can be expressed as:

\[X = \mu + \Lambda F + \epsilon\]

where \(X\) is the vector of observed variables, \(\mu\) is the vector of means, \(\Lambda\) is the matrix of factor loadings, \(F\) is the vector of unobserved latent factors, and \(\epsilon\) is the vector of error terms. The factor loadings represent the correlations between the factors and the observed variables.

==Applications==
Factor analysis is applied in various domains to uncover the underlying structure in datasets:

- In [[psychology]], it is used to identify factors that influence mental processes and behaviors.
- In [[marketing]], it helps in identifying underlying dimensions of consumer preferences and attitudes.
- In [[sociology]], it is utilized to explore social attitudes and values.
- In [[medicine]], factor analysis can reveal patterns in symptoms, aiding in the diagnosis and understanding of diseases.

==Limitations==
While factor analysis is a powerful tool, it has limitations. The interpretation of factors can be subjective, and the method relies heavily on the assumption of linearity between variables and factors. Additionally, the choice of the number of factors to retain can be somewhat arbitrary and influence the results.

==See Also==
* [[Principal component analysis]]
* [[Cluster analysis]]
* [[Multivariate statistics]]

==References==
<references/>

[[Category:Statistics]]
[[Category:Psychometrics]]
[[Category:Data analysis]]

{{Template:Statistics-stub}}
{{Template:Psychometrics-stub}}

Factor analysis - Revision history

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