Post on 19-Nov-2014
Factor Analysis
SPSS for Windows® Intermediate & Advanced Applied Statistics
Zayed University Office of Research SPSS for Windows® Workshop Series
Presented by
Dr. Maher KhelifaAssociate Professor
Department of Humanities and Social SciencesCollege of Arts and Sciences
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Understanding Factor Analysis
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This workshop discusses factor analysis as an exploratory and confirmatory multivariate technique.
Understanding Factor Analysis
Factor analysis is commonly used in: Data reduction Scale development The evaluation of the psychometric quality of a
measure, and The assessment of the dimensionality of a set of
variables.
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Understanding Factor Analysis
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Regardless of purpose, factor analysis is used in: the determination of a small number of factors based on a
particular number of inter-related quantitative variables.
Unlike variables directly measured such as speed, height, weight, etc., some variables such as egoism, creativity, happiness, religiosity, comfort are not a single measurable entity.
They are constructs that are derived from the measurement of other, directly observable variables .
Understanding Factor Analysis
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Constructs are usually defined as unobservable latent variables. E.g.: motivation/love/hate/care/altruism/anxiety/worry/stress/product
quality/physical aptitude/democracy /reliability/power.
Example: the construct of teaching effectiveness. Several variables are used to allow the measurement of such construct (usually several scale items are used) because the construct may include several dimensions.
Factor analysis measures not directly observable constructs by measuring several of its underlying dimensions.
The identification of such underlying dimensions (factors) simplifies the understanding and description of complex constructs.
Understanding Factor Analysis
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Generally, the number of factors is much smaller than the number of measures.
Therefore, the expectation is that a factor represents a set of measures.
From this angle, factor analysis is viewed as a data-reduction technique as it reduces a large number of overlapping variables to a smaller set of factors that reflect construct(s) or different dimensions of contruct(s).
Understanding Factor Analysis
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The assumption of factor analysis is that underlying dimensions (factors) can be used to explain complex phenomena.
Observed correlations between variables result from their sharing of factors.
Example: Correlations between a person’s test scores might be linked to shared factors such as general intelligence, critical thinking and reasoning skills, reading comprehension etc.
Ingredients of a Good Factor Analysis Solution
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A major goal of factor analysis is to represent relationships among sets of variables parsimoniously yet keeping factors meaningful.
A good factor solution is both simple and interpretable.
When factors can be interpreted, new insights are possible.
Application of Factor Analysis
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This workshop will examine three common applications of factor analysis: Defining indicators of constructs Defining dimensions for an existing measure Selecting items or scales to be included in a measure.
Application of Factor Analysis
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Defining indicators of constructs:
Ideally 4 or more measures should be chosen to represent each construct of interest.
The choice of measures should, as much as possible, be guided by theory, previous research, and logic.
Application of Factor Analysis
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Defining dimensions for an existing measure: In this case the variables to be analyzed are chosen by
the initial researcher and not the person conducting the analysis.
Factor analysis is performed on a predetermined set of items/scales.
Results of factor analysis may not always be satisfactory: The items or scales may be poor indicators of the
construct or constructs. There may be too few items or scales to represent each
underlying dimension.
Application of Factor Analysis
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Selecting items or scales to be included in a measure. Factor analysis may be conducted to determine what
items or scales should be included and excluded from a measure.
Results of the analysis should not be used alone in making decisions of inclusions or exclusions. Decisions should be taken in conjunction with the theory and what is known about the construct(s) that the items or scales assess.
Steps in Factor Analysis
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Factor analysis usually proceeds in four steps: 1st Step: the correlation matrix for all variables is
computed 2nd Step: Factor extraction 3rd Step: Factor rotation 4th Step: Make final decisions about the number of
underlying factors
Steps in Factor Analysis: The Correlation Matrix
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1st Step: the correlation matrix Generate a correlation matrix for all variables Identify variables not related to other variables If the correlation between variables are small, it is
unlikely that they share common factors (variables must be related to each other for the factor model to be appropriate).
Think of correlations in absolute value. Correlation coefficients greater than 0.3 in absolute
value are indicative of acceptable correlations. Examine visually the appropriateness of the factor
model.
Steps in Factor Analysis: The Correlation Matrix
Bartlett Test of Sphericity: used to test the hypothesis the correlation matrix is an
identity matrix (all diagonal terms are 1 and all off-diagonal terms are 0).
If the value of the test statistic for sphericity is large and the associated significance level is small, it is unlikely that the population correlation matrix is an identity.
If the hypothesis that the population correlation matrix is an identity cannot be rejected because the observed significance level is large, the use of the factor model should be reconsidered.
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Steps in Factor Analysis: The Correlation Matrix
The Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy: is an index for comparing the magnitude of the observed
correlation coefficients to the magnitude of the partial correlation coefficients.
The closer the KMO measure to 1 indicate a sizeable sampling adequacy (.8 and higher are great, .7 is acceptable, .6 is mediocre, less than .5 is unaccaptable ).
Reasonably large values are needed for a good factor analysis. Small KMO values indicate that a factor analysis of the variables may not be a good idea.
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Steps in Factor Analysis:Factor Extraction
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2nd Step: Factor extractionThe primary objective of this stage is to determine the factors.Initial decisions can be made here about the number of factors
underlying a set of measured variables.Estimates of initial factors are obtained using Principal
components analysis.The principal components analysis is the most commonly used
extraction method . Other factor extraction methods include:Maximum likelihood methodPrincipal axis factoringAlpha methodUnweighted lease squares methodGeneralized least square methodImage factoring.
Steps in Factor Analysis:Factor Extraction
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In principal components analysis, linear combinations of the observed variables are formed.
The 1st principal component is the combination that accounts for the largest amount of variance in the sample (1st extracted factor).
The 2nd principle component accounts for the next largest amount of variance and is uncorrelated with the first (2nd extracted factor).
Successive components explain progressively smaller portions of the total sample variance, and all are uncorrelated with each other.
Steps in Factor Analysis:Factor Extraction
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we need to represent the data, we use 2 statistical criteria: Eigen Values, and The Scree Plot.
The determination of the number of factors is usually done by considering only factors with Eigen values greater than 1.
Factors with a variance less than 1 are no better than a single variable, since each variable is expected to have a variance of 1.
Total Variance Explained
Comp
onent
Initial Eigenvalues
Extraction Sums of Squared
Loadings
Total
% of
Variance
Cumulativ
e % Total
% of
Variance
Cumulativ
e %
1 3.046 30.465 30.465 3.046 30.465 30.465
2 1.801 18.011 48.476 1.801 18.011 48.476
3 1.009 10.091 58.566 1.009 10.091 58.566
4 .934 9.336 67.902
5 .840 8.404 76.307
6 .711 7.107 83.414
7 .574 5.737 89.151
8 .440 4.396 93.547
9 .337 3.368 96.915
10 .308 3.085 100.000
Extraction Method: Principal Component Analysis.
Steps in Factor Analysis:Factor Extraction
The examination of the Scree plot provides a visual of the total variance associated with each factor.
The steep slope shows the large factors.
The gradual trailing off (scree) shows the rest of the factors usually lower than an Eigen value of 1.
In choosing the number of factors, in addition to the statistical criteria, one should make initial decisions based on conceptual and theoretical grounds.
At this stage, the decision about the number of factors is not final.
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Steps in Factor Analysis:Factor Extraction
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Component Matrixa
Component
1 2 3
I discussed my frustrations and feelings with person(s) in school .771 -.271 .121
I tried to develop a step-by-step plan of action to remedy the problems .545 .530 .264
I expressed my emotions to my family and close friends .580 -.311 .265
I read, attended workshops, or sought someother educational approach to correct the
problem
.398 .356 -.374
I tried to be emotionally honest with my self about the problems .436 .441 -.368
I sought advice from others on how I should solve the problems .705 -.362 .117
I explored the emotions caused by the problems .594 .184 -.537
I took direct action to try to correct the problems .074 .640 .443
I told someone I could trust about how I felt about the problems .752 -.351 .081
I put aside other activities so that I could work to solve the problems .225 .576 .272
Extraction Method: Principal Component Analysis.
a. 3 components extracted.
Component Matrix using Principle Component Analysis
Steps in Factor Analysis:Factor Rotation
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3rd Step: Factor rotation.In this step, factors are rotated.
Un-rotated factors are typically not very interpretable (most factors are correlated with may variables).
Factors are rotated to make them more meaningful and easier to interpret (each variable is associated with a minimal number of factors).
Different rotation methods may result in the identification of somewhat different factors.
Steps in Factor Analysis:Factor Rotation
The most popular rotational method is Varimax rotations.
Varimax use orthogonal rotations yielding uncorrelated factors/components.
Varimax attempts to minimize the number of variables that have high loadings on a factor. This enhances the interpretability of the factors.
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Steps in Factor Analysis:Factor Rotation
Other common rotational method used include Oblique rotations which yield correlated factors.
Oblique rotations are less frequently used because their results are more difficult to summarize.
Other rotational methods include: Quartimax (Orthogonal) Equamax (Orthogonal) Promax (oblique)
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Steps in Factor Analysis:Factor Rotation
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linking the factor to the measured variables in the rotated factor matrix. Rotated Component Matrixa
Component
1 2 3
I discussed my frustrations and feelings with person(s) in school .803 .186 .050
I tried to develop a step-by-step plan of action to remedy the problems .270 .304 .694
I expressed my emotions to my family and close friends .706 -.036 .059
I read, attended workshops, or sought someother educational approach to
correct the problem
.050 .633 .145
I tried to be emotionally honest with my self about the problems .042 .685 .222
I sought advice from others on how I should solve the problems .792 .117 -.038
I explored the emotions caused by the problems .248 .782 -.037
I took direct action to try to correct the problems -.120 -.023 .772
I told someone I could trust about how I felt about the problems .815 .172 -.040
I put aside other activities so that I could work to solve the problems -.014 .155 .657
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization.
a. Rotation converged in 5 iterations.
Steps in Factor Analysis:Making Final Decisions
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4th Step: Making final decisions The final decision about the number of factors to choose is the
number of factors for the rotated solution that is most interpretable.
To identify factors, group variables that have large loadings for the same factor.
Plots of loadings provide a visual for variable clusters. Interpret factors according to the meaning of the variables
This decision should be guided by: A priori conceptual beliefs about the number of factors from past
research or theory Eigen values computed in step 2. The relative interpretability of rotated solutions computed in step 3.
Assumptions Underlying Factor Analysis
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Assumption underlying factor analysis include. The measured variables are linearly related to the factors +
errors. This assumption is likely to be violated if items limited response
scales (two-point response scale like True/False, Right/Wrong items).
The data should have a bi-variate normal distribution for each pair of variables.
Observations are independent. The factor analysis model assumes that variables are
determined by common factors and unique factors. All unique factors are assumed to be uncorrelated with each other and with the common factors.
Obtaining a Factor Analysis
Click: Analyze and
select Dimension
Reduction Factor A factor
Analysis Box will appear
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Obtaining a Factor Analysis
Move variables/scale items to Variable box
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Obtaining a Factor Analysis
Factor extraction
When variables are in variable box, select: Extractio
n
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Obtaining a Factor Analysis
When the factor extraction Box appears, select:
Scree Plot
keep all default selections including: Principle component
Analysis Based on Eigen
Value of 1, and Un-rotated factor
solution
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Obtaining a Factor Analysis
During factor extraction keep factor rotation default of: None Press
continue
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Obtaining a Factor Analysis
During Factor Rotation:
Decide on the number of factors based on actor extraction phase and enter the desired number of factors by choosing:
Fixed number of factors and entering the desired number of factors to extract.
Under Rotation Choose Varimax
Press continue Then OK
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Bibliographical References
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Bibliographical References
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Bibliographical References
SPSS Base 7.0 Application Guide (1996). Chicago, IL: SPSS Inc. SPSS Base 7.5 For Windows User’s Guide (1996). Chicago, IL: SPSS Inc. SPSS Base 8.0 Application Guide (1998). Chicago, IL: SPSS Inc. SPSS Base 8.0 Syntax Reference Guide (1998). Chicago, IL: SPSS Inc. SPSS Base 9.0 User’s Guide (1999). Chicago, IL: SPSS Inc. SPSS Base 10.0 Application Guide (1999). Chicago, IL: SPSS Inc. SPSS Base 10.0 Application Guide (1999). Chicago, IL: SPSS Inc. SPSS Interactive graphics (1999). Chicago, IL: SPSS Inc. SPSS Regression Models 11.0 (2001). Chicago, IL: SPSS Inc. SPSS Advanced Models 11.5 (2002) Chicago, IL: SPSS Inc. SPSS Base 11.5 User’s Guide (2002). Chicago, IL: SPSS Inc. SPSS Base 12.0 User’s Guide (2003). Chicago, IL: SPSS Inc. SPSS 13.0 Base User’s Guide (2004). Chicago, IL: SPSS Inc. SPSS Base 14.0 User’s Guide (2005). Chicago, IL: SPSS Inc.. SPSS Base 15.0 User’s Guide (2007). Chicago, IL: SPSS Inc. SPSS Base 16.0 User’s Guide (2007). Chicago, IL: SPSS Inc. SPSS Statistics Base 17.0 User’s Guide (2007). Chicago, IL: SPSS Inc. Tabachnik, B.G., & Fidell, L.S. (2001). Using multivariate statistics (4th Ed). Boston, MA: Allyn
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