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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Applied Marketing(Market Research Methods)
Topic 10:
Factor analysis
Dr James Abdey
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Overview
In regression, a dependent variable was clearlyidentified
In factor analysis variables are not classified as
independent nor dependent
All interdependent relationships among variables
are examined
The factor model is introduced followed by the steps
taken in factor analysis
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis
Factor analysis is a general name denoting a class
of procedures primarily used for data reduction and
summarisation
Factor analysis is an interdependence technique in
that an entire set of interdependent relationships
(correlations) is examined without making the
distinction between dependent and independent
variables
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis
Factor analysis is used in the followingcircumstances:
Identify latent variables or factors that explain thecorrelations among a set of observed variables
Reduction of dimensionality to identify a new,smaller, set of uncorrelated variables to replacethe original set of correlated variables in subsequentmultivariate analysis (regression or discriminant
analysis)
Score respondents on the reduced dimensions foruse in subsequent multivariate analysis
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: How do we achieve
this?
Many theories in behavioural and social sciences are
formulated in terms of theoretical constructs that are
not directly observed or measured, such as:
Manufacturer image Preference Buying behaviour Motivation Psychographic profile of consumers Comfort Luxury Etc.
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: How do we achieve
this?
The measurement of a construct is achieved through
one or more observable indicators (questionnaire
items)
The purpose of a factor analysis model is to describehow well the observed indicators serve as a
measurement instrument for the constructs, also
known as latent variables
In some cases, a concept may be represented by a
single latent variable, but often they are
multidimensional in nature, and so involve more
than one latent variable
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Applications
Market segmentation identify the factors forgrouping customers, for example:
Economy seekers Convenience Performance Comfort
Product research determine the brand attributes
that influence consumer choice
Advertising studies
Pricing studies
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Types of analysis
There are two types of analysis which can be
performed
Exploratory factor analysis no theory is known inadvance about the data
Confirmatory factor analysis validate a theory
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: General ideas
Factor analysis is closely related to the standard
regression model the regression relationship is
between an observed variable and the latent
variables
Distributional assumptions are made about the
residual or error terms which enable us to makeinferences
The idea is to invert the regression relationships
to learn about the latent variables when the manifestvariables are given
Since we can never observe the latent variables, we
can only ever learn about this relationship indirectly
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: General ideas
Several manifest variables will usually depend on thesame latent variable, and this dependence will
induce a correlation between them
The existence of a correlation between two indicatorsmay be taken as evidence of a common source of
influence
As long as any correlation remains, we may therefore
suspect the existence of a further common source of
influence
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Example
Managers are interested in classfying customersaccording to how they make buying decisions,
gathering data on the following variables:
X1 = Price level X2 = Store personnel X3 = Returns policy X4 = Product availability X5 = Product quality X6 = Assortment depth X7 = Assortment width X8 = In-store service X9 = Store atmosphere
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Example
We can construct a table of pairwise correlationcoefficients:
X1 X2 X3 X4 X5 X6 X7 X8 X9X1 1.00
X2 0.43 1.00
X3 0.30 0.77 1.00
X4 0.47 0.50 0.43 1.00
X5 0.77 0.41 0.31 0.43 1.00
X6 0.28 0.45 0.42 0.71 0.33 1.00
X7 0.35 0.49 0.47 0.72 0.38 0.72 1.00X8 0.24 0.72 0.73 0.43 0.24 0.31 0.44 1.00
X9 0.37 0.74 0.77 0.48 0.33 0.43 0.47 0.71 1.00
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Example
Re-ordering by magnitude of pairwise correlationcoefficients:
X3 X8 X9 X2 X6 X7 X4 X1 X5X3 1.00
X8 0.77 1.00
X9 0.77 0.71 1.00
X2 0.77 0.72 0.74 1.00
X6 0.42 0.31 0.43 0.45 1.00
X7 0.47 0.44 0.47 0.49 0.72 1.00
X4 0.43 0.43 0.48 0.50 0.71 0.72 1.00X1 0.30 0.24 0.37 0.43 0.28 0.35 0.47 1.00
X5 0.31 0.24 0.33 0.41 0.33 0.38 0.43 0.77 1.00
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis: Example
Reasons for owning a personal alarm: X1 = Feels comfortable in the hand X2 = Could be easily kept in the pocket X3 = Would fit easily into a handbag X4 = Could be easily worn on the person X5 = Could be carried easily X6 = Could be set off almost as a reflex action X7 = Would be difficult for an attacker to take it off me X8 = Could keep a very firm grip on it if attacked X9 = I would be embarrassed to carry it around X10 = Would be difficult for an attacker to switch off X11 = Solidly built
X12 = Would be difficult to break X13 = Looks as of it would give off a very loud noise X14 = Attacker might have second thoughts
Extracted factors could be size, appearance,
robustness, feel in hand
F l i
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis model
Mathematically, each variable is expressed as alinear combination of underlying factors
The covariation among the variables is described in
terms of a small number of common factors plus aunique factor for each variable
If the variables are standardised, the factor model
may be represented as:
Xi = Ai1F1 + Ai2F2 + Ai3F3 + . . .+ AimFm+ ViUi
F t l iF l i d l
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis model
Xi = i-th standardised observed variable
Aij = standardised multiple regression coefficient of
variable i on common factor j
F= common factor
Vi = standardised regression coefficient of variable i
on unique factor i
Ui = the unique factor for variable i
m= number of common factors
Factor analysisF l i d l
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis model
The unique factors are correlated with each other
and with the common factors
The common factors themselves can be
expressed as linear combinations of the
observed variables
Fi =Wi1X1 +Wi2X2 +Wi3X3 + . . .+WikXk
where
Fi = estimate of i-th factor
Wi = weight or factor score coefficient
k= number of observed variables
Factor analysisF t l i d l
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Factor analysis model
It is possible to select weights or factor scorecoefficients so that the first factor explains the
largest portion of the total variance
Then a second set of weights can be selected, sothat the second factor accounts for most of the
residual variance, subject to being uncorrelated with
the first factor
This same principle could be applied to selectingadditional weights for the additional factors
Factor analysisSt ti ti i t d ith f t
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Statistics associated with factor
analysis
Bartletts test of sphericity A test statistic used to examine the hypothesis that
the variables are uncorrelated in the population In other words, the population correlation matrix is an
identity matrix; each variable correlates perfectly with
itself ( = 1) but has no correlation with the othervariables ( = 0)
Correlation matrix A correlation matrix is a lower triangle matrix
showing the simple correlations, r, between allpossible pairs of variables included in the analysis
The diagonal elements, which are all 1, are usuallyomitted
Factor analysisStatistics associated ith factor
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Statistics associated with factor
analysis
Communality Communality is the amount of variance a variable
shares with all the other variables being considered This is also the proportion of variance explained by
the common factors
Eigenvalue The eigenvalue represents the total variance
explained by each factor
Factor analysisStatistics associated with factor
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Factor analysis
Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Statistics associated with factor
analysis
Factor loadings Factor loadings are simple
correlations between the variables and the factors
Factor loading plot A factor loading plot is a plot
of the original variables using the factor loadings as
coordinates Factor matrix A factor matrix contains the factor
loadings of all the variables on all the factors
extracted
Factor scores Factor scores are composite scoresestimated for each respondent on the derived factors
Percentage of variance The percentage of the
total variance attributed to each factor
Factor analysisStatistics associated with factor
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y
Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Statistics associated with factor
analysis
Kaiser-Meyer-Olkin (KMO) measure of samplingadequacy
An index used to examine the appropriateness offactor analysis
High values (between 0.5 and 1.0) indicate factor
analysis is appropriate Values below 0.5 imply that factor analysis may not
be appropriate
Residuals The differences between the observed
correlations, as given in the input correlation matrix,
and the reproduced correlations, as estimated fromthe factor matrix
Scree plot A scree plot is a plot of the eigenvalues
against the number of factors in order of extraction
Factor analysisFormulate the problem
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y
Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Formulate the problem
The objectives of factor analysis should be identified
The variables to be included in the factor analysis
should be specified based on past research,
theory and judgement of the researcher
It is important that the variables be appropriately
measured on an interval or ratio scale
An appropriate sample size should be used
As a rough guideline, there should be at least four or
five times as many observations (sample size) as
there are observed variables
Factor analysisConstruct the correlation matrix
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Construct the correlation matrix
The analytical process is based on a matrix of
correlations between the variables Bartletts test of sphericity can be used to test the
null hypothesis that the variables are uncorrelated in
the population; in other words, the population
correlation matrix is an identity matrix
If this hypothesis cannot be rejected, then the
appropriateness of factor analysis should be
questioned, since the variables seem to be
uncorrelated
Small values of the KMO statistic indicate that thecorrelations between pairs of variables cannot be
explained by other variables and that factor analysis
may not be appropriate
Factor analysisDetermine the method of factor
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variables
Determine the model fit
Determine the method of factor
analysis
In principal components analysis, the total
variance in the data is considered
The diagonal of the correlation matrix consists of
unities, and full variance is brought into the factor
matrix
Principal components analysis is recommended
when the primary concern is to determine the
minimum number of factors that will account formaximum variance in the data for use in
subsequent multivariate analysis
The factors are called principal components
Factor analysisDetermine the method of factor
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7/28/2019 10 Factor Analysis
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Determine the method of factor
analysis
In common factor analysis, the factors are
estimated based only on the common variance
Communalities are inserted in the diagonal of the
correlation matrix
This method is appropriate when the primary
concern is to identify the underlying dimensions
and the common variance is of interest
This method is also known as principal axis
factoring
Factor analysisDetermine the number of factors
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7/28/2019 10 Factor Analysis
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Determine the number of factors
A priori determination Sometimes, because of prior knowledge, the
researcher knows how many factors to expect andthus can specify the number of factors to beextracted beforehand
Determination based on eigenvalues In this approach, only factors with eigenvalues
greater than 1.0 are retained An eigenvalue represents the amount of variance
associated with the factor Hence, only factors with a variance greater than 1.0
are included
Factors with variance less than 1.0 are no better thana single variable, since, due to standardisation, eachvariable has a variance of 1.0
If the number of variables is less than 20, thisapproach will result in a conservative number offactors
Factor analysisDetermine the number of factors
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7/28/2019 10 Factor Analysis
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Determine the number of factors
Determination based on scree plot A scree plot is a plot of the eigenvalues against the
number of factors in order of extraction The point before the scree begins denotes the true
number of factors
Determination based on percentage of variance In this approach the number of factors extracted is
determined so that the cumulative percentage ofvariance extracted by the factors reaches a
satisfactory level It is recommended that the factors extracted shouldaccount for at least 60% of the variance
Factor analysisRotating factors
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Rotating factors
Although the initial or unrotated factor matrix
indicates the relationship between the factors andindividual variables, it seldom results in factors
that can be interpreted, because the factors are
correlated with many variables
Therefore, through rotation, the factor matrix is
transformed into a simpler one that is easier tointerpret
In rotating the factors, we would like each factor to
have non-zero, or significant, loadings or coefficients
for only some of the variables Likewise, we would like each variable to have
non-zero or significant loadings with only a few
factors, if possible with only one
Factor analysisRotating factors
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7/28/2019 10 Factor Analysis
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Dr James Abdey
Overview
Factor analysis
Factor analysis modelStatistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Rotating factors
The rotation is called orthogonal rotation if the axes
are maintained at right angles
The most commonly used method for rotation is the
varimax procedure
This is an orthogonal method of rotation that
minimises the number of variables with high loadings
on a factor, thereby enhancing the interpretability of
the factors
Orthogonal rotation results in factors that are
uncorrelated
Factor analysisRotating factors
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Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
Rotating factors
The rotation is called oblique rotation when the
axes are not maintained at right angles, and the
factors are correlated
Sometimes, allowing for correlations among factorscan simplify the factor pattern matrix
Oblique rotation should be used when factors in
the population are likely to be strongly correlated
Factor analysisInterpret factors
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Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
p
A factor can then be interpreted in terms of the
variables that have high loadings on it
Another useful aid in interpretation is to plot the
variables, using the factor loadings ascoordinates
Variables at the end of an axis are those that have
high loadings on only that factor and hence describe
the factor
Factor analysisCalculate factor scores
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Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
The factor scores for the i-th factor may be
estimated as follows:
Fi =Wi1X1 +Wi2X2 +Wi3X3 + . . .+WikXk
Factor analysisSelect surrogate variables
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Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
g
By examining the factor matrix, one could select for
each factor the variable with the highest loading onthat factor
That variable could then be used as a surrogate
variable for the associated factor
However, the choice is not as easy if two or more
variables have similarly high loadings
In such a case, the choice between these variables
should be based on theoretical and measurement
considerations
Factor analysisDetermine the model fit
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Dr James Abdey
Overview
Factor analysis
Factor analysis model
Statistics associated with
factor analysis
Formulate the problem
Correlation matrix
Determine the method of
factor analysis
Rotating factors
Interpret factors
Calculate factor scores
Select surrogate variablesDetermine the model fit
The correlations between the variables can be
reproduced from the estimated correlations between
the variables and the factors
The differences between the observed correlations(as given in the input correlation matrix) and the
reproduced correlations (as estimated from the factor
matrix) can be examined to determine model fit
These differences are called residuals
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