Factor analysis

Factor analysis is commonly used in psychometrics, personality psychology, biology, marketing, product management, operations research, finance, and machine learning.

[note 1] Evidence for the hypothesis is sought in the examination scores from each of 10 different academic fields of 1000 students.

—is measured, and simultaneously halving the factor loadings for verbal intelligence makes no difference to the model.

These diagonal elements of the reduced correlation matrix are called "communalities" (which represent the fraction of the variance in the observed variable that is accounted for by the factors): The sample data

With the advent of high-speed computers, the minimization problem can be solved iteratively with adequate speed, and the communalities are calculated in the process, rather than being needed beforehand.

In the model, the error covariance is stated to be a diagonal matrix and so the above minimization problem will in fact yield a "best fit" to the model: It will yield a sample estimate of the error covariance which has its off-diagonal components minimized in the mean square sense.

[4] Structural equation modeling approaches can accommodate measurement error and are less restrictive than least-squares estimation.

[4] Principal component analysis (PCA) is a widely used method for factor extraction, which is the first phase of EFA.

A number of objective methods have been developed to solve this problem, allowing users to determine an appropriate range of solutions to investigate.

Horn's parallel analysis (PA):[8] A Monte-Carlo based simulation method that compares the observed eigenvalues with those obtained from uncorrelated normal variables.

PA is among the more commonly recommended rules for determining the number of components to retain,[7][9] but many programs fail to include this option (a notable exception being R).

[10] However, Formann provided both theoretical and empirical evidence that its application might not be appropriate in many cases since its performance is considerably influenced by sample size, item discrimination, and type of correlation coefficient.

Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies.

[citation needed] Variance explained criteria: Some researchers simply use the rule of keeping enough factors to account for 90% (sometimes 80%) of the variation.

PCA can be considered as a more basic version of exploratory factor analysis (EFA) that was developed in the early days prior to the advent of high-speed computers.

Both PCA and factor analysis aim to reduce the dimensionality of a set of data, but the approaches taken to do so are different for the two techniques.

[27] From the point of view of exploratory analysis, the eigenvalues of PCA are inflated component loadings, i.e., contaminated with error variance.

Researchers have argued that the distinctions between the two techniques may mean that there are objective benefits for preferring one over the other based on the analytic goal.

Factor analysis has been used successfully where adequate understanding of the system permits good initial model formulations.

PCA employs a mathematical transformation to the original data with no assumptions about the form of the covariance matrix.

The objective of PCA is to determine linear combinations of the original variables and select a few that can be used to summarize the data set without losing much information.

"For this reason, Brown (2009) recommends using factor analysis when theoretical ideas about relationships between variables exist, whereas PCA should be used if the goal of the researcher is to explore patterns in their data.

[41] He discovered that school children's scores on a wide variety of seemingly unrelated subjects were positively correlated, which led him to postulate that a single general mental ability, or g, underlies and shapes human cognitive performance.

[44] Thurstone introduced several important factor analysis concepts, including communality, uniqueness, and rotation.

[46][47] Raymond Cattell was a strong advocate of factor analysis and psychometrics and used Thurstone's multi-factor theory to explain intelligence.

The best known cultural dimensions models are those elaborated by Geert Hofstede, Ronald Inglehart, Christian Welzel, Shalom Schwartz and Michael Minkov.

The data for multiple products is coded and input into a statistical program such as R, SPSS, SAS, Stata, STATISTICA, JMP, and SYSTAT.

Factor analysis assumes that all the rating data on different attributes can be reduced down to a few important dimensions.

For example, a sulfide mine is likely to be associated with high levels of acidity, dissolved sulfates and transition metals.

[55] Factor analysis can be used for summarizing high-density oligonucleotide DNA microarrays data at probe level for Affymetrix GeneChips.