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Factor analysisFrom Wikipedia, the free encyclopedia

Factor analysis is astatisticalmethod used to describevariabilityamong observed, correlatedvariablesin

terms of a potentially lower number of unobserved 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 unobserved

variables. Factor analysis searches for such joint variations in response to unobservedlatent variables. The

observed variables are modeled aslinear combinationsof the potential factors, plus "error" terms. The

information gained about the interdependencies between observed variables can be used later to reduce the

set of variables in a dataset. Computationally this technique is equivalent tolow rank approximationof the

matrix of observed variables. Factor analysis originated inpsychometrics, and is used in behavioral

sciences,social sciences,marketing,product management,operations research, and other applied sciences

that deal with large quantities ofdata.

Factor analysis is related toprincipal component analysis(PCA), but the two are not identical.Latent variable

models, including factor analysis, use regression modelling techniques to test hypotheses producing error

terms, while PCA is a descriptive statistical technique.[1]

There has been significant controversy in the field over

the equivalence or otherwise of the two techniques (seeexploratory factor analysis versus principal

components analysis).[citation needed]

Contents

[hide]

1 Statistical model

o 1.1 Definition

o 1.2 Example

o 1.3 Mathematical model of the same example

2 Practical implementation

o 2.1 Type of factor analysis

o 2.2 Types of factoring

o 2.3 Terminology

o 2.4 Criteria for determining the number of factors

o 2.5 Rotation methods

3 Factor analysis in psychometrics

o 3.1 History

o 3.2 Applications in psychology

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o 3.3 Advantages

o 3.4 Disadvantages

4 Exploratory factor analysis versus principal components analysis

o

4.1 Arguments contrasting PCA and EFA

o 4.2 Variance versus covariance

o 4.3 Differences in procedure and results

5 Factor analysis in marketing

o 5.1 Information collection

o 5.2 Analysis

o 5.3 Advantages

o 5.4 Disadvantages

6 Factor analysis in physical sciences

7 Factor analysis in microarray analysis

8 Implementation

9 See also

10 References

11 Further reading

12 External links

[edit]Statistical model

[edit]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

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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.

2.

3. (to make sure that the factors are

uncorrelated)

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 anyorthogonal matrix 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 transformation.

[edit]Example

The following example is for expository purposes,

and should not be taken as being realistic. Suppose

a psychologist proposes a theory that there are two

kinds ofintelligence, "verbal intelligence" and

"mathematical intelligence", neither of which is

directly observed.Evidencefor the theory is sought

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in the examination scores from each of 10 different

academic fields of 1000 students. If each student is

chosen randomly from a largepopulation, 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 someconstanttimes their level of verbal

intelligence plus another constant times their level of

mathematical intelligence, i.e., it is a combination of

those two "factors". The numbers for a particular

subject, by which the two kinds of intelligence aremultiplied 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

oftaxonomyis

{10 the student's verbal intelligence} + {6 the student's

mathematical intelligence}.

The numbers 10 and 6 are the factor loadings

associated with taxonomy. 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 taxonomy 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 (seeerrors and

residuals in statistics).

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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.

[edit]Mathematical model of the

same example

In the example above, fori= 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, forj= 1, 2.

k,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

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mathematical intelligence are the

same as those of the ith student,

Inmatrixnotation, we have

Nis 1000 students

Xis a 10 1,000 matrix

ofobservable random variables,

is a 10 1 column vector

ofunobservable 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). Note

that, is anouter

productof with a 11000 row

vector of ones, yielding a 10 1000 matrix of the elements of ,

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),

Fis a 2 1,000 matrix

ofunobservable random variables

(two intelligence parameters for

each of 1000 students),

is a 10 1,000 matrix

ofunobservable random

variables.

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Observe that by doubling the scale on

which "verbal intelligence"the first

component in each column ofFis

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

areuncorrelatedwith each other. The"errors" 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 , and thevariancesof the

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"errors" must be estimated given the

observed dataXand F(the

assumption about the levels of the

factors is fixed for a given F).

[edit]Practical

implementation

This section needs additional citations

for verification. Please helpimprove

this articlebyadding citations to reliable

sources. Unsourced material may

bechallengedandremoved.(April 2012)

[edit]Type of factor analysis

Exploratory factor analysis(EFA) is

used to identify complex

interrelationships among items and

group items that are part of unified

concepts.[2]

The researcher makes no

"a priori" assumptions about

relationships among factors.[2]

Confirmatory factor

analysis(CFA) is a more complex

approach that tests the hypothesis that

the items are associated with specific

factors.[2]

CFA usesstructural equation

modelingto test a measurement

model whereby loading on the factors

allows for evaluation of relationships

between observed variables and

unobserved variables.[2]

Structural

equation modeling approaches can

accommodate measurement error,

and are less restrictive thanleast-

squares estimation.[2]

Hypothesized

models are tested against actual data,

and the analysis would demonstrate

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loadings of observed variables on the

latent variables (factors), as well as

the correlation between the latent

variables.[2]

[edit]Types of factoring

Principal component

analysis(PCA): PCA is a widely used

method for factor extraction, which is

the first phase of EFA.[2]

Factor

weights are computed in order to

extract the maximum possible

variance, with successive factoring

continuing until there is no further

meaningful variance left.[2]

The factor

model must then be rotated for

analysis.[2]

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. Canonical factor

analysis seeks factors which have the

highest canonical correlation with the

observed variables. Canonical factor

analysis 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

thecorrelation matrixof predicted

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variables rather than actual variables,

where each variable is predicted from

the others usingmultiple 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,[3]

whose factors are partially

known.

[edit]Terminology

Factor loadings: The factor loadings,

also called component loadings in

PCA, are thecorrelation

coefficientsbetween the variables

(rows) and factors (columns).

Analogous toPearson's r, 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

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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.

Inobliquerotation, 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,

containscoefficientswhich just

represent unique contributions. The

more factors, the lower the pattern

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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

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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.

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[edit]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[citation needed]

.

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 inSPSSand

moststatistical softwarebut is not

recommended when used as the sole

cut-off criterion for estimating the

number of factors as it tends to

overextract factors.[4]

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

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emphasizesparsimony(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 correspondingeigenvaluesas

theY-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

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retain,[citation needed]

but only few

programs include this option.[5]

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

thedependent variable, in which case

it should not be dropped.

[edit]Rotation methods

The unrotated output maximises the

variance accounted for by the first and

subsequent factors, and forcing the

factors to beorthogonal. 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 rotationis 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

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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. However, the

orthogonality (i.e., independence) of

factors is often an unrealistic

assumption. Oblique rotations are

inclusive of orthogonal rotation, and

for that reason, oblique rotations are a

preferred method.[6]

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 willresult in higher eigenvalues but

diminishedinterpretabilityof the

factors. See below.[clarification needed]

Promax rotation is an alternative

non-orthogonal (oblique) rotation

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method which is computationally faster

than the direct oblimin method and

therefore is sometimes used for very

largedatasets.

[edit]Factor analysis in

psychometrics

See also:g factor

[edit]History

Charles Spearmanpioneered 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, org,

underlies and shapes human cognitive

performance. His postulate now

enjoys broad support in the fieldofintelligence research, where it is

known as thegtheory.

Raymond Cattellexpanded 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

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coefficients. His research led to the

development of his theory offluid and

crystallized intelligence, as well as

his16 Personality Factorstheory of

personality. Cattell was a strong

advocate of factor analysis

andpsychometrics. 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.

[edit]Applications inpsychology

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 calledcrystallized intelligenceor

verbal intelligence, which 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

topsychometrics, as it can assess the

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validity of an instrument by finding if

the instrument indeed measures the

postulated factors.

[edit]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 hisThree

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.

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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.[edit]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[7]

). 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

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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".[8]

More than one

interpretation can be made of the

same data factored the same

way, and factor analysis cannot

identify causality.[edit]Exploratory factor

analysis versus principal

components analysis

See also:Principal component

analysisandExploratory factor

analysis

Whileexploratory factor

analysisandprincipal component

analysisare treated as synonymous

techniques in some fields of statistics,

this has been criticised (e.g. Fabrigar

et al., 1999;[9]

Suhr, 2009[10]

). In factor

analysis, the researcher makes the

assumption that an underlying causal

model exists, whereas PCA is simply a

variable reductiontechnique.

[11]Researchers have

argued that the distinctions between

the two techniques may mean that

there are objective benefits for

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preferring one over the other based on

the analytic goal.

[edit]Arguments

contrasting PCA and EFA

Fabrigar et al. (1999)[9]

address a

number of reasons used to suggest

that principal components analysis is

equivalent to factor analysis:

1. It is sometimes suggested

that principal components

analysis is computationally

quicker and requires fewer

resources than factor

analysis. Fabrigar et al.

suggest that the ready

availability of computer

resources have rendered this

practical concern irrelevant.[9]

2. PCA and factor analysis can

produce similar results. This

point is also addressed by

Fabrigar et al.; in certain

cases, whereby the

communalities are low (e.g.,

.40), the two techniques

produce divergent results. In

fact, Fabrigar et al. argue that

in cases where the data

correspond to assumptions of

the common factor model,

the results of PCA are

inaccurate results.[9]

3. There are certain cases

where factor analysis leads

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to 'Heywood cases'. These

encompass situations

whereby 100% or more of

thevariancein a measured

variable is estimated to be

accounted for by the model.

Fabrigar et al. suggest that

these cases are actually

informative to the researcher,

indicating a misspecified

model or a violation of the

common factor model. The

lack of Heywood cases in thePCA approach may mean

that such issues pass

unnoticed.[9]

4. Researchers gain extra

information from a PCA

approach, such as an

individuals score on a certain

component such

information is not yielded

from factor analysis.

However, as Fabrigar et al.

contend, the typical aim of

factor analysis i.e. to

determine the factors

accounting for the structure

of thecorrelationsbetween

measured variables doesnot require knowledge of

factor scores and thus this

advantage is negated.[9]

It is

also possible to compute

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factor scores from a factor

analysis.

[edit]Variance versus

covariance

Factor analysis takes into account

therandom errorthat is inherent in

measurement, whereas PCA fails to

do so. This point is exemplified by

Brown (2009),[12]

who indicated that, in

respect to the correlation matrices

involved in the calculations:

"In PCA, 1.00s are put in the

diagonal meaning that all of the

variance in the matrix is to be

accounted for (including

variance unique to each

variable, variance common

among variables, and error

variance). That would,

therefore, by definition, include

all of the variance in the

variables. In contrast, in EFA,

the communalities are put in

the diagonal meaning that only

the variance shared with other

variables is to be accounted for

(excluding variance unique to

each variable and error

variance). That would,

therefore, by definition, include

only variance that is common

among the variables."

Brown (2009), Principal

components analysis and

exploratory factor analysis

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Definitions, differences and

choices

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.

[edit]Differences in

procedure and results

The differences between principal

components analysis and factor

analysis are further illustrated by Suhr

(2009):

PCA results in principal

components that account for a

maximal amount of variance for

observed variables; FA account

for common variance in the

data.[10]

PCA inserts ones on the

diagonals of the correlation

matrix; FA adjusts the diagonals

of the correlation matrix with the

unique factors.[10]

PCA minimizes the sum of

squared perpendicular distance to

the component axis; FA estimates

factors which influence responses

on observed variables.[10]

The component scores in PCA

represent a linear combination of

the observed variables weighted

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byeigenvectors; the observed

variables in FA are linear

combinations of the underlying

and unique factors.[10]

In PCA, the components yielded

are uninterpretable, i.e. they do

not represent underlying

constructs; in FA, the underlying

constructs can be labeled and

readily interpreted, given an

accurate model specification.[10]

[edit]Factor analysis in

marketing

The basic steps are:

Identify the salient attributes

consumers use to

evaluateproductsin this

category.

Usequantitative marketing

researchtechniques (suchassurveys) to collect data from a

sample of

potentialcustomersconcerning

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

constructperceptual mapsand

otherproduct positioningdevices.

[edit]Information collection

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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. Thesame question is asked about all the

products in the study. The data for

multiple products is coded and input

into a statistical program such

asR,SPSS,SAS,Stata,STATISTICA,

JMP, and SYSTAT.

[edit]Analysis

The analysis will isolate the underlying

factors that explain the data using a

matrix of associations.[13]

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

some attributes may be related to

each other. The rating given to any

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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.

[edit]Advantages

Both objective and subjectiveattributes can be used provided

the subjective attributes can be

converted into scores.

Factor analysis can identify latent

dimensions or constructs that

direct analysis may not.

It is easy and inexpensive.

[edit]Disadvantages

Usefulness depends on the

researchers' ability to collect a

sufficient set of product attributes.

If important attributes are

excluded or neglected, 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 obscure factors

that represent more interesting

relationships.[clarification needed]

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Naming factors may require

knowledge of theory because

seemingly dissimilar attributes

can correlate strongly for

unknown reasons.

[edit]Factor analysis in

physical sciences

Factor analysis has also been widely

used in physical sciences such

asgeochemistry,ecology,

andhydrochemistry.[14]

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.[15]

Ingeochemistry, different factors can

correspond to different mineral

associations, and thus to

mineralisation.[16]

[edit]Factor analysis in

microarray analysis

Factor analysis can be used for

summarizing high-

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densityoligonucleotideDNA

microarraysdata at probe level

forAffymetrixGeneChips. In this case,

the latent variable corresponds to

theRNAconcentration in a sample.[17]

[edit]Implementation

Factor analysis has been implemented

in several statistical analysis programs

since the

1980s:SAS,BMDPandSPSS.[18]

It is

also implemented in

theRprogramming language(with

thefactanalfunction) and inOpenOpt.

Rotations are implemented in

the GPArotation R package.

[edit]See also

Wikimedia Commons has

media related to: Factor

analysis

Design of experiments

Formal concept analysis

Higher-order factor analysis

Independent component analysis

Multilinear PCA

Multilinear subspace learning

Non-negative matrix factorization

Perceptual mapping

Product management

Q methodology

Recommendation system

Varimax rotation

[edit]References