Factor Analysis (FA) (1)
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Transcript of Factor Analysis (FA) (1)
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Factor Analysis (FA)
Factor analysis is an interdependence technique whose primary
purpose is to define the underlying structure among the
variables in the analysis.
The purpose of FA is to condense the information contained in
a number of original variables into a smaller set of new
composite dimensions or variates (factors) with a minimum loss
of information.
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Factor analysis decision processStage 1: Objectives of factor analysis
Key issues:
Specifying the unit of analysis
R factor analysis- Correlation matrix of the variables to summarize the
characteristics.
Q factor analysis- Correlation matrix of the individual respondents
based on their characteristics. Condenses large number of people intodistinctly different group.
Achieving data summarization vs. data reduction
Data summarization- It is the definition of structure. Viewing the set of
variables at various levels of generalization, ranging from the most
detailed level to the more generalized level. The linear composite of
variables is called variate or factor.
Data reduction- Creating entirely a new set of variables and completely
replace the original values with empirical value (factor score).
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Variable selection
The researcher should always consider the conceptual underpinnings of
the variables and use judgment as to the appropriateness of the variables
for factor analysis.
Using factor analysis with other multivariate techniques
Factor scores as representatives of variables will be used for further
analysis.
Stage 2: Designing a factor analysis
It involves three basic decisions:
Correlations among variables or respondents (Q type vs. R type)
Variable selection and measurement issues- Mostly performed on metricvariables. For nonmetric variables, define dummy variables (0-1) and
include in the set of metric variables.
Sample size- The sample must have more observations than variables.
The minimum sample size should be fifty observations. Minimum 5 and
hopefully at least 10 observations per variable is desirable.
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Stage 3: Assumptions in factor analysis
The assumptions are more conceptual than statistical.
Conceptual issues- 1) Appropriate selection of variables 2)
Homogeneous sample. Statistical issues- Ensuring the variables are sufficiently intercorrelated
to produce representative factors.
Measure of intercorrelation:
Visual inspection of Correlations greater than .30 in substantial
cases in correlation matrix , the factor analysis is appropriate.
If partial correlation are high, indicating no underlying factors,
then factor analysis is inappropriate.
Bartlett test of sphericity- A test for the presence of correlation
among the variables. A statistically significant Bartletts test of
sphericity (sig. >.05) indicates that sufficient correlation existamong the variables to proceed.
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Measure of sampling adequacy (MSA)- This index ranges from
0 to 1, reaching 1 when each variable is perfectly predicted
without error by the other variables. The measures can be
integrated with following guidelines: Kaiser-Meyer Measure of Sampling Adequacy
in the .90s marvelous
in the .80s meritorious
in the .70s middling
in the .60s mediocre
in the .50s miserable
below .50 unacceptable
MSA values must exceed .50 for both the overall test and each
individual variable
Variables with value less than .50 should be omitted from the
factor analysis.
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Stage 4: Deriving factors and assessing overall fit
Apply factor analysis to identify the underlying structure of
relationships.
Two decisions are important: Selecting the factor extraction method
Common factor analysis
Principal component analysis
Concept of Partitioning the variance of a variable
Common variance- Variance in the variable shared with all other
variables in the analysis. The variance is based on variablescorrelations
with other variables. Communality of variable estimates common
variance.
Specific variance- AKA unique variance. This variance of variable cannot
be explained by the correlations to the other variables but is associated
uniquely with a single variable.
Error variance- It is due to unreliability in the data-gathering process,
measurement error, or a random component in the measured
phenomenon.
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Component factor analysis- AKA principal components analysis.
Considers the total variance and derives factors that contain
small proportions of unique variance and in some instances
error variance.
Common factor analysis- Considers only the common or shared
variance, assuming that both the unique and error variance are
not of interest in defining the structure of the variables.
Diagonal value
Unity
Variance
Communality
Variance extracted
Variance excluded
Total variance
common
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Suitability of factor extraction method Component factor analysis is appropriate when data reduction is primary
concern.
Common factor analysis is appropriate when primary objective is toidentify the latent dimensions or constructs represented in the originalvalue.
Criteria for the number of factors to extract
Latent root criterion
It applies to both extraction method. This criteria assumes that any individual factor should account for the
variance of at least a single variable if it is to be retained for interpretation.
In component analysis each variable contribute a value of 1 to the latentroots or eigen values.
So, factors having eigen values greater than 1 are considered significant and
selected.
Eigen value- It represents the amount of variance accountedfor by the factor. It is column sum of squared loading for afactor.
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Scree test criterion
This is plotting the latent roots against the number of
factors in their order of extraction. The shape of the resulting curve is used to evaluate the
cutoff point.
The point at which the curve begins to straighten out is
considered to indicate the maximum numbers of factorsto extract.
As a general rule, the scree test results in at least one
and sometimes two or three more factors being
considered for inclusion than does the latent root
criterion.
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0
1
2
3
4
5
0 5 10Number
Scree plot of eigenvalues after factor
Factor
Eigen values
Scree criterion
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Stage 5: Interpreting the factors
Three processes of factor interpretation
Estimate the factor matrix
Initial unrotated factor matrix is computed.
It contains factor loadings for each variable on each factor.
Factor loadings are the correlation of each variable on each factor.
Higher loadings making the variable representative of the factor.
Factor rotation Rotational method is employed to achieve simpler and theoretically
more meaningful factor solutions.
The reference axes of the factors are turned about the origin until
some position has been reached.
There are two types of rotation:
Orthogonal factor rotation
Oblique factor rotation.
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Rotating Factors
F1
F1
F2
F2
Factor 1 Factor 2
x1 0.5 0.5
x2 0.8 0.8
x3 -0.7 0.7
x4 -0.5 -0.5
Factor 1 Factor 2
x1 0 0.6
x2 0 0.9
x3 -0.9 0
x4 0 -0.9
2
1
3
4
2
1
3
4
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Orthogonal Rotation Oblique Rotation
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When to use Factor Analysis?
Data Reduction
Identification of underlying latent structures- Clusters of correlated variables are termed factors
Example: Factor analysis could potentially be used to identify
the characteristics (out of a large number ofcharacteristics) that make a person popular.
Candidate characteristics: Level of social skills, selfishness, howinteresting a person is to others, the amount of time they spendtalking about themselves (Talk 2) versus the other person (Talk1), their propensity to lie about themselves.
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The R-Matrix
Meaningful clusters of large correlation
coefficients between subsets of variables
suggests these variables are measuring
aspects of the same underlying
dimension.
Factor 1:
The better your social skills,
the more interesting and
talkative you tend to be.
Factor 2:
Selfish people are likely to lie
and talk about themselves.
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What is a Factor?
Factors can be viewed as classification axes alongwhich the individual variables can be plotted.
The greater the loadingof variables on a factor,the more the factor explains relationships amongthose variables.
Ideally, variables should be strongly related to (orload on)only one factor.
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Graphical Representation of a
factor plot
Note that each variable
loads primarily on only
one factor.
Factor loadings tell use about
the relative contribution that a
variable makes to a factor
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Mathematical Representation
of a factor plot
Yi= b1X1i+b2X2i+ bnXn+ i
Factori= b1Variable1i+b2Variable2i+ bnVariablen+ i
The equation describing a linear model can be
applied to the description of a factor.
The bs in the equation represent the factorloadings observed in the factor plot.
Note: there is no intercept in the equation since the lines intersection at zero and hence
the intercept is also zero.
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Mathematical Representation
of a factor plot
Sociabilityi= b1Talk 1i+b2Social Skillsi+ b3interesti
+ b4Talk 2 + b5Selfishi+ b6Liari + i
There are two factors underlying thepopularity construct: general
sociability and consideration.
We can construct equations that describe each factor in terms of the
variables that have been measured.
Considerationi= b1Talk 1i+b2Social Skillsi+
b3interesti+ b4Talk 2 + b5Selfishi+ b6Liari + i
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Mathematical Representation
of a factor plot
Sociabilityi= 0.87Talk 1i+0.96Social Skillsi+ 0.92Interesti+ 0.00Talk 2 -
0.10Selfishi+ 0.09Liari + i
The values of the bsin the two equations differ, depending on
the relative importance of each variable to a particular factor.
Considerationi= 0.01Talk 1i- 0.03Social Skillsi+ 0.04interesti+ 0.82Talk 2 +
0.75Selfishi+ 0.70Liari + i
Ideally, variables should have very high b-values for one factor and very low
b-values for all other factors.
Replace values of b with the co-ordinate of each variable on the graph.
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Factor Loadings
The bvalues represent the weights of a variable on a factor and aretermed Factor Loadings.
These values are stored in a Factor pattern matrix(A). Columns display the factors (underlying constructs) and rows
display how each variable loads onto each factor.
VariablesFactors
Sociability Consideration
Talk 1 0.87 0.01
Social Skills 0.96 -0.03
Interest 0.92 0.04
Talk 2 0.00 0.82
Selfish -0.10 0.75
Liar 0.09 0.70
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Factor Scores Once factors are derived, we can estimate each
persons Factor Scores(based on their scores for eachfactors constituent variables).
Potential uses for Factor Scores.
- Estimate a persons score on one or more factors.- Answer questions of scientific or practical interest (e.g.,Are females are
more sociable than males? using the factors scores for sociability).
Methods of Determining Factor Scores- Weighted Average (simplest, but scale dependent)
- Regression Method (easiest to understand; most typically used)
- Bartlett Method (produces scores that are unbiased and correlate only with theirown factor).
- Anderson-Rubin Method (produces scores that are uncorrelated andstandardized)
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Approaches to Factor Analysis
Exploratory Reduce a number of measurements to a smaller number of indices or
factors (e.g., Principal Components Analysis or PCA).
Goal: Identify factors based on the data and to maximize the amountof variance explained.
Confirmatory Test hypothetical relationships between measures and more abstract
constructs.
Goal: The researcher must hypothesize, in advance, the number of
factors, whether or not these factors are correlated, and which itemsload onto and reflect particular factors. In contrast to EFA, where allloadings are free to vary, CFA allows for the explicit constraint ofcertain loadings to be zero.
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Communality
Understanding variance in an R-matrix Total variance for a particular variable has two
components:
Common Variance variance shared with other variables.
Unique Variance
variance specific to that variable (includingerror or random variance).
Communality The proportion of common (or shared) variance present in a
variable is known as the communality. A variable that has no unique variance has a communality of 1;
one that shares none of its variance with any other variable hasa communality of 0.
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Factor Extraction: PCA vs. Factor Analysis
Principal Component Analysis. A data reduction technique that representsa set of variables by a smaller number of variables called principal components.
They are uncorrelated, and therefore, measure different, unrelated aspects or
dimensions of the data.
Principal Componentsare chosen such that the first one accounts for as much of
the variation in the data as possible, the second one for as much of the
remaining variance as possible, and so on.
Useful for combining many variables into a smaller number of subsets.
Factor Analysis. Derives a mathematical model from which factors areestimated.
Factors are linear combinations that maximize the shared portion of the
variance underlying latent constructs.
May be used to identify the structure underlying such variables and to estimate
scores to measure latent factors themselves.
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Factor Extraction: Eigenvalues & Scree Plot
Eigenvalues Measure the amount of variation accounted for by each factor.
Number of principal components is less than or equal to the number of
original variables. The first principal component accounts for as much of
the variability in the data as possible. Each succeeding component has the
highest variance possible under the constraint that it be orthogonal to
(i.e., uncorrelated with) the preceding components.
Scree Plots
Plots a graph of each eigenvalue (Y-axis) against the factor with
which it is associated (X-axis).
By graphing the eigenvalues, the relative importance of each factor
becomes apparent.
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Factor Retention Based on Scree Plots
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Kaiser (1960) recommends retaining all factors with
eigenvalues greater than 1.
- Based on the idea that eigenvalues represent the amount
of variance explained by a factor and that an eigenvalueof 1 represents a substantial amount of variation.
- Kaisers criterion tends to overestimate the number of
factors to be retained.
Factor Retention: Kaisers Criterion
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Students often become stressed about statistics
(SAQ) and the use of computers and/or SPSS to
analyze data.
Suppose we develop a questionnaire to measurethis propensity (see sample items on the following
slides; the data can be found in SAQ.sav).
Does the questionnaire measure a single construct?
Or is it possible that there are multiple aspectscomprising students anxiety toward SPSS?
Doing Factor Analysis: An Example
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Doing Factor Analysis: Some
Considerations
Sample size is important! A sample of 300 or more
will likely provide a stable factor solution, but
depends on the number of variables and factors
identified.
Factors that have four or more loadings greater than
0.6 are likely to be reliable regardless of sample
size.
Correlations among the items should not be too low
(less than .3) or too high (greater than .8), but the
pattern is what is important.
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c
E
Factor Extraction
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Scree Plot for theSAQ Data
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Table of Communalities Before
and After Extraction
Component Matrix Before Rotation(loadings of each variable onto each factor)
Note: Loadings less than
0.4 have been omitted.
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Factor Rotation
To aid interpretation it is possible to maximize theloading of a variable on one factor while
minimizing its loading on all other factors.
This is known as Factor Rotation.
Two types: Orthogonal (factors are uncorrelated)
Oblique (factors intercorrelate)
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Orthogonal Rotation Oblique Rotation
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Rotated Com ponent Matrixa
.800
.684
.647
.638
.579
.550
.459
.677
.661
-.567
.473 .523
.516
.514
.496
.429
.833
.747
.747
.648 .645
.586
.543
.427
I have little experience of computers
SPSS alw ays cra shes w hen I try to use it
I worr y that I w ill cause irreparable damage because
of my incompetenece w ith computersA ll computers hate me
Computers have minds of their ow n and deliberately
go w rong w henever I use them
Computers are useful only for playing games
Computers are out to get me
I can't sleep for thoughts of eigen vec tors
I wake up under my duvet thinking that I am trapped
under a normal distribtion
Standard deviations excite me
People try to tell you that SPSS makes statisticseasier to understand but it doesn't
I dream that Pearson is attacking me w ith cor relation
coefficients
I w eep openly at the mention of central tendency
Statiscs makes me cry
I don't understand s tatistics
I have never been good at mathematics
I slip into a coma w henever I see an equation
I did badly at mathematics at s chool
My friends are better at statistics than meMy friends are better at SPSS than I am
If I'm good at stat istics my f riends w ill think I'm a nerd
My friends w ill think I'm s tupid for not being able to
cope w ith SPSS
Everybody looks at me when I use SPSS
1 2 3 4
Component
Extraction Method: Principal Component Analys is.
Rotation Method: Varimax w ith Kaiser Normalization.
Rotation converged in 9 iterations.a.
Orthogonal
Rotation (varimax)Fear of Computers
Fear of Statistics
Fear of Math
Peer Evaluation
Note: Varimax rotation is the
most commonly used
rotation. Its goal is to
minimize the complexity of
the components by making
the large loadings larger and
the small loadings smallerwithin each component.
Quartimax rotation makes
large loadings larger and
small loadings smaller within
each variable. Equamax
rotation is a compromise that
attempts to simplify both
components and variables.
These are all orthogonal
rotations, that is, the axes
remain perpendicular, so the
components are not
correlated.
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Oblique
Rotation: PatternMatrix
Pattern Matrixa
.706
.591
-.511
.405
.400
.643
.621
.615
.507
.885
.713
.653
.650
.588
.585
.412 .462
.411
-.902
-.774
-.774
I can't sleep for thoughts of eigen vectors
I wake up under my duvet thinking that I am trapped
under a normal distribtion
Standard deviations exc ite me
I dream that Pearson is attacking me w ith correlation
coefficients
I w eep openly at the mention of central tendency
Statiscs makes me cry
I don't understand statistics
My friends are better a t SPSS than I am
My friends are better at statistics than me
If I'm good at statistics my f riends w ill think I'm a nerd
My friends w ill think I'm stupid f or not being able to
cope w ith SPSSEverybody looks at me w hen I use SPSS
I have little exper ience of computers
SPSS alw ays cras hes w hen I try to use it
All computers hate me
I w orry that I w ill cause irreparable damage because
of my incompetenece w ith computers
Computers have minds of their ow n and deliberately
go w rong whenever I use them
Computers are useful only for playing games
People try to tell you that SPSS makes s tatisticseasier to understand but it doesn't
Computers are out to get me
I have never been good at mathematics
I slip into a coma w henever I see an equat ion
I did badly at mathematics at school
1 2 3 4
Component
Extraction Method: Principal Component A nalysis.
Rotation Method: Oblimin w ith Kaiser Normalization.
Rotation converged in 29 iterations.a.
Fear of Statistics
Fear of Computers
Fear of Math
Peer Evaluation
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Reliability:A measure should consistently reflect the construct it is measuring
Test-Retest Method
What about practice effects/mood states?
Alternate Form Method
Expensive and Impractical
Split-Half Method Splits the questionnaire into two random halves,
calculates scores and correlates them.
Cronbachs Alpha
Splits the questionnaire (or sub-scales of a questionnaire)into all possible halves, calculates the scores, correlatesthem and averages the correlation for all splits.
Ranges from 0 (no reliability) to 1 (complete reliability)
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Reliability: Fear of Computers Subscale
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Reliability: Fear of Statistics Subscale
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Reliability: Fear of Math Subscale
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Reliability: Peer Evaluation Subscale
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Reporting the ResultsA principal component analysis (PCA) was conducted on the 23 items with
orthogonal rotation (varimax). Bartletts test of sphericity, 2(253) = 19334.49,
p< .001, indicated that correlations between items were sufficiently large for
PCA. An initial analysis was run to obtain eigenvalues for each component in
the data. Four components had eigenvalues over Kaisers criterion of 1 and
in combination explained 50.32% of the variance. The scree plot was slightly
ambiguous and showed inflexions that would justify retaining either 2 or 4factors.
Given the large sample size, and the convergence of the scree plot and
Kaisers criterion on four components, four components were retained in the
final analysis. Component 1 represents a fear of computers, component 2 a
fear of statistics, component 3 a fear of math, and component 4 peer
evaluation concerns.The fear of computers, fear of statistics, and fear of math subscales of the
SAQ all had high reliabilities, all Chronbachs = .82. However, the fear of
negative peer evaluation subscale had a relatively low reliability, Chronbachs
= .57.
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Step 1: Select Factor Analysis
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Step 2: Add all variables to be included
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Step 3: Get descriptive statistics & correlations
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Step 4: Ask for Scree Plot and set extraction options
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Step 5: Handle missing values and sort coefficients bysize
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Step 6: Select rotation type and set rotationiterations
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Step 7: Save Factor Scores
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Communalities
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Variance Explained
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Scree Plot
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Rotated Component Matrix: Component 1
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Rotated Component Matrix: Component 2
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Component 1: Factor Score
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Component (Factor): Score Values
Rename Components According to
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Rename Components According to
Interpretation
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