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Factor analysis is a statistical method used to describe variability among observed variables in terms of fewer unobserved variables called factors. The observed variables are modeled as linear combinations of the factors, plus "error" terms. The information gained about the interdependencies can be used later to reduce the set of variables in a dataset. 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.

Factor analysis is often confused with principal components analysis. The two methods are related, but distinct, though factor analysis becomes essentially equivalent to principal components analysis if the "errors" in the factor analysis model (see below) are assumed to all have the same variance.

Contents

Definition

Suppose we have a set of p observable random variables, x_1,...,x_p\, with means {\mu}_1,...,{\mu}_p\, .

Suppose for some unknown constants l_{ij}\, and k\, unobserved random variables F_j\, , where i\in {1,...,p} and j \in {1,...,k}, where k<p\, , we have

x_i-{\mu}_i=l_{i1}F_1 + ... + l_{ik}F_k + {\epsilon}_i\, .

Here {\epsilon}_i\, is independently distributed error terms with zero mean and finite variance - which may not be the same for all of them. Let V(\epsilon_i)=\psi_i\, , so that we have Cov(\epsilon_{p\times 1})=Diag({\psi}_1,...,{\psi}_p)=\Psi\, and E(\epsilon)=0\, .

In matrix terms, we have

x_{p\times 1}-{\mu}_{p\times 1}=L_{p\times k}F_{k\times 1}+{\epsilon}_{p\times 1}\, , or omitting the matrix dimensions for clarity,

x-\mu=LF+\epsilon\,


Also we will impose the following assumptions on F\, .

  1. F\, and \epsilon\, are independent.
  2. E(F) = 0\,
  3. Cov(F)=I_{k\times k}\,

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


Suppose Cov(x_{p\times 1})=\Sigma_{p\times p}\, . Then note that from the conditions just imposed on F\, , we have

Cov(x-\mu)=Cov(LF+\epsilon)\, ,

or \Sigma = L Cov(F) L^T + Cov(\epsilon)\, ,

or \Sigma = LL^T + \Psi\,

Note that for any orthogonal matrix Q\, if we set L=LQ\, and F=Q^T F\, , the criteria for being factors and factor loadings still hold. Hence a set of factors and factor loadings is identical only upto orthogonal transformations.

Example

The following example is a simplification for expository purposes, and should not be taken to be 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 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 times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.e., it is a linear combination of those two "factors". The numbers, for this 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}.

The numbers 10 and 6 are the factor loadings associated with amphibiology. Other academic subjects may have different factor loadings.

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).

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. Even the number of factors (two, in this example) 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

\begin{matrix}x_{1,i} & = & \mu_1 & + & \ell_{1,1}v_i & + & \ell_{1,2}m_i & + & \varepsilon_{1,i} \\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots \\ x_{10,i} & = & \mu_{10} & + & \ell_{10,1}v_i & + & \ell_{10,2}m_i & + & \varepsilon_{10,i} \end{matrix}

where

  • xk,i is the ith student's score for the kth subject
  • μk 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",
  • \ell_{k,j} 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 notation, we have

X = μ + LF + ε

where

  • 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),
  • F is a 2 × 1,000 matrix of unobservable random variables,
  • 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 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 variances of the "errors" ε must be estimated given the observed data X.

Factor analysis in psychometrics

History

Charles Spearman spearheaded 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.

Raymond Cattell 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, as well as his 16 Personality Factors theory of personality. Cattell was a strong advocate of factor analysis and psychometrics. 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, as it can assess the validity of an instrument by finding if the instrument indeed measures the postulated factors.

Advantages

  • 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. 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.

Disadvantages

  • "...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 have to rely on more or 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" (Richard B. Darlington). 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 this category.
  • Use quantitative marketing research techniques (such as surveys) to collect data from a sample of potential customers 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 and other product positioning 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 SPSS, SAS, Stata, 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 are examined. There is no specification of either 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 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 thus differ depending on the method used to compute them whereas those produced by principal components 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 (Velicer and Jackson, 1990) 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 (Gorsuch, 1983).

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. [ok]

Advantages

  • Both objective and subjective attributes can be usedcitation needed
  • Factor Analysis can be used to identify the hidden dimensions or constructs which may or may not be apparent from direct analysis.
  • It is not extremely difficult to do, inexpensive, and accuratecitation needed
  • There is flexibility in naming and using dimensionscitation needed

Disadvantages

  • Usefulness depends on the researchers' ability to develop a complete and accurate set of product attributes - If important attributes are missed the value of the procedure is reduced accordingly.
  • Naming of the factors can be difficult - multiple attributes can be highly correlated with no apparent reason.
  • If the observed variables are completely unrelated, factor analysis is unable to produce a meaningful pattern (though the eigenvalues will highlight this: suggesting that each variable should be given a factor in its own right).
  • If sets of observed variables are highly similar to each other but distinct from other items, Factor analysis will assign a factor to them, even though this factor will essentially capture true variance of a single item. In other words, it is not possible to know what the 'factors' actually represent; only theory can help inform the researcher on this.

Factor analysis in physical sciences

Factor analysis has also been widely used in physical sciences such as geochemistry, ecology, and hydrochemistry1 .

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.2

In geochemistry, different factors can correspond to different mineral associations, and thus to mineralisation.3

See also

Footnotes

  1. ^ Subbarao, C., Subbarao, N.V., Chandu, S.N., 1995. Characterisation of groundwater contamination using factor analysis. Environmental Geology 28, 175–180
  2. ^ Love, D., Hallbauer, D.K., Amos, A. and Hranova, R.K. 2004. Factor analysis as a tool in groundwater quality management: two southern African case studies. Physics and Chemistry of the Earth, 29, 1135-1143. [1]
  3. ^ Barton, E.S. and Hallbauer, D.K. 1996. Trace-element and U---Pb isotope compositions of pyrite types in the Proterozoic Black Reef, Transvaal Sequence, South Africa: Implications on genesis and age. Chemical Geology, 133, 173-199. [2]

References

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