Factor analysis

# Factor analysis

Overview
Factor analysis is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

method used to describe variability
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

among observed, correlated variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables mainly reflect the variations in fewer such unobserved variables. Factor analysis searches for such joint variations in response to unobserved latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...

s. The observed variables are modeled as linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

s of the potential factors, plus "error
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

" terms. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. Factor analysis originated in psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...

, and is used in behavioral sciences, social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

, marketing
Marketing
Marketing is the process used to determine what products or services may be of interest to customers, and the strategy to use in sales, communications and business development. It generates the strategy that underlies sales techniques, business communication, and business developments...

, product management
Product management
Product management is an organizational lifecycle function within a company dealing with the planning, forecasting, or marketing of a product or products at all stages of the product lifecycle....

, operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, and other applied sciences that deal with large quantities of data.
Discussion

Recent Discussions
Encyclopedia
Factor analysis is a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

method used to describe variability
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

among observed, correlated variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables mainly reflect the variations in fewer such unobserved variables. Factor analysis searches for such joint variations in response to unobserved latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...

s. The observed variables are modeled as linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

s of the potential factors, plus "error
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

" terms. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. Factor analysis originated in psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...

, and is used in behavioral sciences, social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

, marketing
Marketing
Marketing is the process used to determine what products or services may be of interest to customers, and the strategy to use in sales, communications and business development. It generates the strategy that underlies sales techniques, business communication, and business developments...

, product management
Product management
Product management is an organizational lifecycle function within a company dealing with the planning, forecasting, or marketing of a product or products at all stages of the product lifecycle....

, operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, and other applied sciences that deal with large quantities of data.

Factor analysis is related to principal component analysis (PCA), but the two are not identical. Latent variable model
Latent variable model
A latent variable model is a statistical model that relates a set of variables to a set of latent variables.It is assumed that 1) the responses on the indicators or manifest variables are the result of...

s, including factor analysis, use regression modelling techniques to test hypotheses producing error terms, while PCA is a descriptive statistical technique .

### Definition

Suppose we have a set of observable random variables, with means .

Suppose for some unknown constants and unobserved random variables , where and , where , we have

Here, the are independently distributed error terms with zero mean and finite variance, which may not be the same for all . Let , so that we have

In matrix terms, we have

If we have observations, then we will have the dimensions , , and . Each column of and denote values for one particular observation, and matrix does not vary across observations.

Also we will impose the following assumptions on .
1. and are independent.

Any solution of the above set of equations following the constraints for is defined as the factors, and as the loading matrix.

Suppose . Then note that from the conditions just imposed on , we have

or

or

Note that for any orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

if we set and , the criteria for being factors and factor loadings still hold. Hence a set of factors and factor loadings is identical only up to orthogonal transformations.

### Example

The following example is a fictionalized simplification for expository purposes, and should not be taken as being realistic. Suppose a psychologist proposes a theory that there are two kinds of intelligence, "verbal intelligence" and "mathematical intelligence", neither of which is directly observed. Evidence
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...

for the theory is sought in the examination scores from each of 10 different academic fields of 1000 students. If each student is chosen randomly from a large population, then each student's 10 scores are random variables. The psychologist's theory may say that for each of the 10 academic fields, the score averaged over the group of all students who share some common pair of values for verbal and mathematical "intelligences" is some constant
Constant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...

times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.e., it is a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

of those two "factors". The numbers for a particular subject, by which the two kinds of intelligence are multiplied to obtain the expected score, are posited by the theory to be the same for all intelligence level pairs, and are called "factor loadings" for this subject. For example, the theory may hold that the average student's aptitude in the field of amphibiology is
{10 × the student's verbal intelligence} + {6 × the student's mathematical intelligence}.

Two students having identical degrees of verbal intelligence and identical degrees of mathematical intelligence may have different aptitudes in amphibiology because individual aptitudes differ from average aptitudes. That difference is called the "error" — a statistical term that means the amount by which an individual differs from what is average for his or her levels of intelligence (see errors and residuals in statistics
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

).

The observable data that go into factor analysis would be 10 scores of each of the 1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data.

### Mathematical model of the same example

In the example above, for i = 1, ..., 1,000 the ith student's scores are

where
• xk,i is the ith student's score for the kth subject
• is the mean of the students' scores for the kth subject (assumed to be zero, for simplicity, in the example as described above, which would amount to a simple shift of the scale used)
• vi is the ith student's "verbal intelligence",
• mi is the ith student's "mathematical intelligence",
• are the factor loadings for the kth subject, for j = 1, 2.
• εk,i is the difference between the ith student's score in the kth subject and the average score in the kth subject of all students whose levels of verbal and mathematical intelligence are the same as those of the ith student,

In matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

notation, we have

where
• N is 1000 students

• X is a 10 × 1,000 matrix of observable random variables,
• μ is a 10 × 1 column vector of unobservable constants (in this case "constants" are quantities not differing from one individual student to the next; and "random variables" are those assigned to individual students; the randomness arises from the random way in which the students are chosen),
• L is a 10 × 2 matrix of factor loadings (unobservable constants, ten academic topics, each with two intelligence parameters that determine success in that topic),
• F is a 2 × 1,000 matrix of unobservable random variables (two intelligence parameters for each of 1000 students),
• ε is a 10 × 1,000 matrix of unobservable random variables.

Observe that by doubling the scale on which "verbal intelligence"—the first component in each column of F—is measured, and simultaneously halving the factor loadings for verbal intelligence makes no difference to the model. Thus, no generality is lost by assuming that the standard deviation of verbal intelligence is 1. Likewise for mathematical intelligence. Moreover, for similar reasons, no generality is lost by assuming the two factors are uncorrelated
Uncorrelated
In probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero. Uncorrelatedness is by definition pairwise; i.e...

with each other. The "errors" ε are taken to be independent of each other. The variances of the "errors" associated with the 10 different subjects are not assumed to be equal.

Note that, since any rotation of a solution is also a solution, this makes interpreting the factors difficult. See disadvantages below. In this particular example, if we do not know beforehand that the two types of intelligence are uncorrelated, then we cannot interpret the two factors as the two different types of intelligence. Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence without an outside argument.

The values of the loadings L, the averages μ, and the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

s of the "errors" ε must be estimated given the observed data X and F (the assumption about the levels of the factors is fixed for a given F).

### Type of factor analysis

Exploratory factor analysis (EFA) is used to uncover the underlying structure of a relatively large set of variables. The researcher's a priori assumption is that any indicator may be associated with any factor. This is the most common form of factor analysis. There is no prior theory and one uses factor loadings to intuit the factor structure of the data.

Confirmatory factor analysis
Confirmatory factor analysis
In statistics, confirmatory factor analysis is a special form of factor analysis. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct . In contrast to exploratory factor analysis, where all loadings are free to vary,...

(CFA)
seeks to determine if the number of factors and the loadings of measured (indicator) variables on them conform to what is expected on the basis of pre-established theory. Indicator variables are selected on the basis of prior theory and factor analysis is used to see if they load as predicted on the expected number of factors. The researcher's a priori assumption is that each factor (the number and labels of which may be specified a priori) is associated with a specified subset of indicator variables. A minimum requirement of confirmatory factor analysis is that one hypothesizes beforehand the number of factors in the model, but usually also the researcher will posit expectations about which variables will load on which factors. The researcher seeks to determine, for instance, if measures created to represent a latent variable really belong together.

### Types of factoring

Principal component analysis (PCA): The most common form of factor analysis, PCA seeks a linear combination of variables such that the maximum variance is extracted from the variables. It then removes this variance and seeks a second linear combination which explains the maximum proportion of the remaining variance, and so on. This is called the principal axis method and results in orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

(uncorrelated) factors.

Canonical factor analysis, also called Rao's canonical factoring, is a different method of computing the same model as PCA, which uses the principal axis method. CFA seeks factors which have the highest canonical correlation with the observed variables. CFA is unaffected by arbitrary rescaling of the data.

Common factor analysis, also called principal factor analysis (PFA) or principal axis factoring (PAF), seeks the least number of factors which can account for the common variance (correlation) of a set of variables.

Image factoring: based on the correlation matrix of predicted variables rather than actual variables, where each variable is predicted from the others using multiple regression.

Alpha factoring: based on maximizing the reliability of factors, assuming variables are randomly sampled from a universe of variables. All other methods assume cases to be sampled and variables fixed.

Factor regression model: a combinatorial model of factor model and regression model; or alternatively, it can be viewed as the hybrid factor model, whose factors are partially known.

### Terminology

Pearson product-moment correlation coefficient
In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...

, the squared factor loading is the percent of variance in that indicator variable explained by the factor. To get the percent of variance in all the variables accounted for by each factor, add the sum of the squared factor loadings for that factor (column) and divide by the number of variables. (Note the number of variables equals the sum of their variances as the variance of a standardized variable is 1.) This is the same as dividing the factor's eigenvalue by the number of variables.

Interpreting factor loadings: By one rule of thumb in confirmatory factor analysis, loadings should be .7 or higher to confirm that independent variables identified a priori are represented by a particular factor, on the rationale that the .7 level corresponds to about half of the variance in the indicator being explained by the factor. However, the .7 standard is a high one and real-life data may well not meet this criterion, which is why some researchers, particularly for exploratory purposes, will use a lower level such as .4 for the central factor and .25 for other factors call loadings above .6 "high" and those below .4 "low". In any event, factor loadings must be interpreted in the light of theory, not by arbitrary cutoff levels.

In oblique rotation, one gets both a pattern matrix and a structure matrix. The structure matrix is simply the factor loading matrix as in orthogonal rotation, representing the variance in a measured variable explained by a factor on both a unique and common contributions basis. The pattern matrix, in contrast, contains coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s which just represent unique contributions. The more factors, the lower the pattern coefficients as a rule since there will be more common contributions to variance explained. For oblique rotation, the researcher looks at both the structure and pattern coefficients when attributing a label to a factor.

Communality: The sum of the squared factor loadings for all factors for a given variable (row) is the variance in that variable accounted for by all the factors, and this is called the communality. The communality measures the percent of variance in a given variable explained by all the factors jointly and may be interpreted as the reliability of the indicator.

Spurious solutions: If the communality exceeds 1.0, there is a spurious solution, which may reflect too small a sample or the researcher has too many or too few factors.

Uniqueness of a variable: That is, uniqueness is the variability of a variable minus its communality.

Eigenvalues:/Characteristic roots: The eigenvalue for a given factor measures the variance in all the variables which is accounted for by that factor. The ratio of eigenvalues is the ratio of explanatory importance of the factors with respect to the variables. If a factor has a low eigenvalue, then it is contributing little to the explanation of variances in the variables and may be ignored as redundant with more important factors. Eigenvalues measure the amount of variation in the total sample accounted for by each factor.

Extraction sums of squared loadings: Initial eigenvalues and eigenvalues after extraction (listed by SPSS as "Extraction Sums of Squared Loadings") are the same for PCA extraction, but for other extraction methods, eigenvalues after extraction will be lower than their initial counterparts. SPSS also prints "Rotation Sums of Squared Loadings" and even for PCA, these eigenvalues will differ from initial and extraction eigenvalues, though their total will be the same.

Factor scores (also called component scores in PCA): are the scores of each case (row) on each factor (column). To compute the factor score for a given case for a given factor, one takes the case's standardized score on each variable, multiplies by the corresponding factor loading of the variable for the given factor, and sums these products. Computing factor scores allows one to look for factor outliers. Also, factor scores may be used as variables in subsequent modeling.

### Criteria for determining the number of factors

Using one or more of the methods below, the researcher determines an appropriate range of solutions to investigate. Methods may not agree. For instance, the Kaiser criterion may suggest five factors and the scree test may suggest two, so the researcher may request 3-, 4-, and 5-factor solutions discuss each in terms of their relation to external data and theory.

Comprehensibility: A purely subjective criterion would be to retain those factors whose meaning is comprehensible to the researcher. This is not recommended .

Kaiser criterion: The Kaiser rule is to drop all components with eigenvalues under 1.0 – this being the eigenvalue equal to the information accounted for by an average single item. The Kaiser criterion is the default in SPSS
SPSS
SPSS is a computer program used for survey authoring and deployment , data mining , text analytics, statistical analysis, and collaboration and deployment ....

and most statistical software but is not recommended when used as the sole cut-off criterion for estimating the number of factors as it tends to overextract factors.

Variance explained criteria: Some researchers simply use the rule of keeping enough factors to account for 90% (sometimes 80%) of the variation. Where the researcher's goal emphasizes parsimony (explaining variance with as few factors as possible), the criterion could be as low as 50%

Scree plot: The Cattell scree test plots the components as the X axis and the corresponding eigenvalues
Eigenvalue, eigenvector and eigenspace
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

as the Y-axis. As one moves to the right, toward later components, the eigenvalues drop. When the drop ceases and the curve makes an elbow toward less steep decline, Cattell's scree test says to drop all further components after the one starting the elbow. This rule is sometimes criticised for being amenable to researcher-controlled "fudging". That is, as picking the "elbow" can be subjective because the curve has multiple elbows or is a smooth curve, the researcher may be tempted to set the cut-off at the number of factors desired by his or her research agenda.

Horn's Parallel Analysis (PA): A Monte-Carlo based simulation method that compares the observed eigenvalues with those obtained from uncorrelated normal variables. A factor or component is retained if the associated eigenvalue is bigger than the 95th of the distribution of eigenvalues derived from the random data. PA is one of the most recommendable rules for determining the number of components to retain, but only few programs include this option.

Before dropping a factor below one's cut-off, however, the researcher should check its correlation with the dependent variable. A very small factor can have a large correlation with the dependent variable, in which case it should not be dropped.

### Rotation methods

The unrotated output maximises the variance accounted for by the first and subsequent factors, and forcing the factors to be orthogonal. This data-compression comes at the cost of having most items load on the early factors, and usually, of having many items load substantially on more than one factor. Rotation serves to make the output more understandable, by seeking so-called "Simple Structure": A pattern of loadings where items load most strongly on one factor, and much more weakly on the other factors. Rotations can be orthogonal or oblique (allowing the factors to correlate).

Varimax rotation
Varimax rotation
In statistics, a varimax rotation is a change of coordinates used in principal component analysis and factor analysis that maximizes the sum of the variances of the squared loadings...

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.

Quartimax rotation is an orthogonal alternative which minimizes the number of factors needed to explain each variable. This type of rotation often generates a general factor on which most variables are loaded to a high or medium degree. Such a factor structure is usually not helpful to the research purpose.

Equimax rotation is a compromise between Varimax and Quartimax criteria.

Direct oblimin rotation is the standard method when one wishes a non-orthogonal (oblique) solution – that is, one in which the factors are allowed to be correlated. This will result in higher eigenvalues but diminished interpretability
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.-Informal definition:Assume T and S are formal theories...

of the factors. See below.

Promax rotation is an alternative non-orthogonal (oblique) rotation method which is computationally faster than the direct oblimin method and therefore is sometimes used for very large datasets.

### History

Charles Spearman
Charles Spearman
Charles Edward Spearman, FRS was an English psychologist known for work in statistics, as a pioneer of factor analysis, and for Spearman's rank correlation coefficient...

pioneered the use of factor analysis in the field of psychology and is sometimes credited with the invention of factor analysis. 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 general mental ability, or g, underlies and shapes human cognitive performance. His postulate now enjoys broad support in the field of intelligence research, where it is known as the g theory
General intelligence factor
The g factor, where g stands for general intelligence, is a statistic used in psychometrics to model the mental ability underlying results of various tests of cognitive ability...

.

Raymond Cattell
Raymond Cattell
Raymond Bernard Cattell was a British and American psychologist, known for his exploration of many areas in psychology...

expanded on Spearman's idea of a two-factor theory of intelligence after performing his own tests and factor analysis. He used a multi-factor theory to explain intelligence. Cattell's theory addressed alternate factors in intellectual development, including motivation and psychology. Cattell also developed several mathematical methods for adjusting psychometric graphs, such as his "scree" test and similarity coefficients. His research led to the development of his theory of fluid and crystallized intelligence
Fluid and crystallized intelligence
In psychology, fluid and crystallized intelligence are factors of general intelligence originally identified by Raymond Cattell...

, as well as his 16 Personality Factors
16 Personality Factors
The 16 Personality Factors, measured by the 16PF Questionnaire, were derived using factor-analysis by psychologist Raymond Cattell.Below is a table outlining this model.- Raymond Cattell's 16 Personality Factors :-Relationship to the Big Five:...

theory of personality. Cattell was a strong advocate of factor analysis and psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...

. He believed that all theory should be derived from research, which supports the continued use of empirical observation and objective testing to study human intelligence.

### Applications in psychology

Factor analysis is used to identify "factors" that explain a variety of results on different tests. For example, intelligence research found that people who get a high score on a test of verbal ability are also good on other tests that require verbal abilities. Researchers explained this by using factor analysis to isolate one factor, often called crystallized intelligence or verbal intelligence, that represents the degree to which someone is able to solve problems involving verbal skills.

Factor analysis in psychology is most often associated with intelligence research. However, it also has been used to find factors in a broad range of domains such as personality, attitudes, beliefs, etc. It is linked to psychometrics
Psychometrics
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...

, as it can assess the validity of an instrument by finding if the instrument indeed measures the postulated factors.

• Reduction of number of variables, by combining two or more variables into a single factor. For example, performance at running, ball throwing, batting, jumping and weight lifting could be combined into a single factor such as general athletic ability. Usually, in an item by people matrix, factors are selected by grouping related items. In the Q factor analysis technique, the matrix is transposed and factors are created by grouping related people: For example, liberals, libertarians, conservatives and socialists, could form separate groups.

• Identification of groups of inter-related variables, to see how they are related to each other. For example, Carroll used factor analysis to build his Three Stratum Theory
Three Stratum Theory
In 1993 John Carroll published "Human cognitive abilities: A survey of factor-analytic studies", which outlined his hierarchical, Three-Stratum Theory of cognitive abilities....

. He found that a factor called "broad visual perception" relates to how good an individual is at visual tasks. He also found a "broad auditory perception" factor, relating to auditory task capability. Furthermore, he found a global factor, called "g" or general intelligence, that relates to both "broad visual perception" and "broad auditory perception". This means someone with a high "g" is likely to have both a high "visual perception" capability and a high "auditory perception" capability, and that "g" therefore explains a good part of why someone is good or bad in both of those domains.

• "...each orientation is equally acceptable mathematically. But different factorial theories proved to differ as much in terms of the orientations of factorial axes for a given solution as in terms of anything else, so that model fitting did not prove to be useful in distinguishing among theories." (Sternberg, 1977). This means all rotations represent different underlying processes, but all rotations are equally valid outcomes of standard factor analysis optimization. Therefore, it is impossible to pick the proper rotation using factor analysis alone.
• Factor analysis can be only as good as the data allows. In psychology, where researchers often have to rely on less valid and reliable measures such as self-reports, this can be problematic.
• Interpreting factor analysis is based on using a "heuristic", which is a solution that is "convenient even if not absolutely true". More than one interpretation can be made of the same data factored the same way, and factor analysis cannot identify causality.

## Factor analysis in marketing

The basic steps are:
• Identify the salient attributes consumers use to evaluate products
In general, the product is defined as a "thing produced by labor or effort" or the "result of an act or a process", and stems from the verb produce, from the Latin prōdūce ' lead or bring forth'. Since 1575, the word "product" has referred to anything produced...

in this category.
• Use quantitative marketing research
Quantitative marketing research
Quantitative marketing research is the application of quantitative research techniques to the field of marketing. It has roots in both the positivist view of the world, and the modern marketing viewpoint that marketing is an interactive process in which both the buyer and seller reach a satisfying...

techniques (such as surveys
Statistical survey
Survey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....

) to collect data from a sample of potential customer
Customer
A customer is usually used to refer to a current or potential buyer or user of the products of an individual or organization, called the supplier, seller, or vendor. This is typically through purchasing or renting goods or services...

s concerning their ratings of all the product attributes.
• Input the data into a statistical program and run the factor analysis procedure. The computer will yield a set of underlying attributes (or factors).
• Use these factors to construct perceptual maps
Perceptual mapping
Perceptual mapping is a graphics technique used by asset marketers that attempts to visually display the perceptions of customers or potential customers. Typically the position of a product, product line, brand, or company is displayed relative to their competition.Perceptual maps can have any...

and other product positioning
Positioning (marketing)
In marketing, positioning has come to mean the process by which marketers try to create an image or identity in the minds of their target market for its product, brand, or organization....

devices.

### Information collection

The data collection stage is usually done by marketing research professionals. Survey questions ask the respondent to rate a product sample or descriptions of product concepts on a range of attributes. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colourfulness, price, or size. The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is coded and input into a statistical program such as R
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

, PSPP
PSPP
PSPP is a free software application for analysis of sampled data. It has a graphical user interface and conventional command line interface. It is written in C, uses GNU Scientific Library for its mathematical routines, and plotutils for generating graphs....

, SAS
SAS System
SAS is an integrated system of software products provided by SAS Institute Inc. that enables programmers to perform:* retrieval, management, and mining* report writing and graphics* statistical analysis...

, Stata
Stata
Stata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world...

, STATISTICA
STATISTICA
STATISTICA is a statistics and analytics software package developed by StatSoft. STATISTICA provides data analysis, data management, data mining, and data visualization procedures...

, JMP and SYSTAT.

### Analysis

The analysis will isolate the underlying factors that explain the data. Factor analysis is an interdependence technique. The complete set of interdependent relationships is examined. There is no specification of dependent variables, independent variables, or causality. Factor analysis assumes that all the rating data on different attributes can be reduced down to a few important dimensions. This reduction is possible because the attributes are related. The rating given to any one attribute is partially the result of the influence of other attributes. The statistical algorithm deconstructs the rating (called a raw score) into its various components, and reconstructs the partial scores into underlying factor scores. The degree of correlation between the initial raw score and the final factor score is called a factor loading. There are two approaches to factor analysis: "principal component analysis" (the total variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

in the data is considered); and "common factor analysis" (the common variance is considered).

Note that principal component analysis and common factor analysis differ in terms of their conceptual underpinnings. The factors produced by principal component analysis are conceptualized as being linear combinations of the variables whereas the factors produced by common factor analysis are conceptualized as being latent variables. Computationally, the only difference is that the diagonal of the relationships matrix is replaced with communalities (the variance accounted for by more than one variable) in common factor analysis. This has the result of making the factor scores indeterminate and differ depending on the method of computation. Meanwhile, factor scores produced by principal component analysis are not dependent on the method of computation. Although there have been heated debates over the merits of the two methods, a number of leading statisticians have concluded that in practice there is little difference which makes sense since the computations are quite similar despite the differing conceptual bases, especially for datasets where communalities are high and/or there are many variables, reducing the influence of the diagonal of the relationship matrix on the final result.

The use of principal components in a semantic space can vary somewhat because the components may only "predict" but not "map" to the vector space. This produces a statistical principal component use where the most salient words or themes represent the preferred basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

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• Both objective and subjective attributes can be used provided the subjective attributes can be converted into scores
• Factor Analysis can be used to identify hidden dimensions or constructs which may not be apparent from direct analysis
• It is easy and inexpensive to do

• Usefulness depends on the researchers' ability to collect a sufficient set of product attributes. If important attributes are missed the value of the procedure is reduced.
• If sets of observed variables are highly similar to each other and distinct from other items, factor analysis will assign a single factor to them. This may make it harder to identify factors that capture more interesting relationships.
• Naming the factors may require background knowledge or theory because multiple attributes can be highly correlated for no apparent reason.

## Factor analysis in physical sciences

Factor analysis has also been widely used in physical sciences such as geochemistry
Geochemistry
The field of geochemistry involves study of the chemical composition of the Earth and other planets, chemical processes and reactions that govern the composition of rocks, water, and soils, and the cycles of matter and energy that transport the Earth's chemical components in time and space, and...

, ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

, and hydrochemistry.

In groundwater quality management, it is important to relate the spatial distribution of different chemical
parameters to different possible sources, which have different chemical signatures. For example, a sulfide mine is likely to be associated with high levels of acidity, dissolved sulfates and transition metals. These signatures can be identified as factors through R-mode factor analysis, and the location of possible sources can be suggested by contouring the factor scores.

In geochemistry
Geochemistry
The field of geochemistry involves study of the chemical composition of the Earth and other planets, chemical processes and reactions that govern the composition of rocks, water, and soils, and the cycles of matter and energy that transport the Earth's chemical components in time and space, and...

, different factors can correspond to different mineral associations, and thus to mineralisation.

## Factor analysis in microarray analysis

Factor analysis can be used for summarizing high-density oligonucleotide
Oligonucleotide
An oligonucleotide is a short nucleic acid polymer, typically with fifty or fewer bases. Although they can be formed by bond cleavage of longer segments, they are now more commonly synthesized, in a sequence-specific manner, from individual nucleoside phosphoramidites...

DNA microarrays data at probe level for Affymetrix
Affymetrix
Affymetrix is a company that manufactures DNA microarrays; it is based in Santa Clara, California, United States. The company was founded by Dr. Stephen Fodor in 1992. It began as a unit in Affymax N.V...

GeneChips. In this case, the latent variable corresponds to the RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....

concentration in a sample.

## Implementation

Factor analysis has been implemented in several statistical analysis programs since the 1980s: SAS, BMDP
BMDP
BMDP is a statistical package developed in 1961 at UCLA. Based on the older BIMED program for biomedical applications, it used keyword parameters in the input instead of fixed-format cards, so the letter P was added to the letters BMD, although the name was later defined as being an abbreviation...

and SPSS
SPSS
SPSS is a computer program used for survey authoring and deployment , data mining , text analytics, statistical analysis, and collaboration and deployment ....

.
It is also implemented in the R programming language
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

(with the factanal function) and in OpenOpt
OpenOpt
OpenOpt is an open-source framework for numerical optimization, nonlinear equations and systems of them. It is licensed under the BSD license, making it available to be used in both open- and closed-code software. The package already has some essential ....

.
Rotations are implemented in the GPArotation R package.