Factor Analysis Slides

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Chapter 5

Dimension Reduction and

Extraction of Meaningful

Factors

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The FACTOR ProcedureGeneral form of the FACTOR procedure:

PROC FACTOR options;

VAR variables;RUN;


VAR variables;RUN;

Section 5.2

Exploratory Factor

Analysis

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Objectives Explain the distinctions between principal components

and common factor analyses.

Identify several factor extraction methods for factoranalysis.

Differentiate between oblique and orthogonal rotationmethods for factor analysis.

Use the FACTOR procedure to perform exploratoryfactor analysis.

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Why Perform Factor Analysis?You suspect that the variables you observe (manifestvariables) are functions of variables that you cannotobserve directly (latent variables).

Identify the latent variables to learn somethinginteresting about the behavior of your population.

Identify relationships between different latentvariables.

Show that a small number of latent variables underliesthe process or behavior you have measured to simplifyyour theory.

Explain inter-correlations among observed variables.

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Exploratory Factor Analysis

F1:Consumerconfidence

F2: Buyingpower

New Home

Buys

Durable

Goods Buys

Borrowing

Income

Import

Purchases

u1

u2

u3

u4

u5

?


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Components versus Factors, Revisited

Principal Components

the symptoms

Latent Factors

the disease

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The Common Factor Model

Y = X + Ewhere

Y manifest variables

X common factors

weights E unique factors +error variation

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Assumptions of the Common Factor Model The unique factors (residuals) are uncorrelated with

each other.

The unique factors (residuals) are uncorrelated withthe common (latent) factors.

Under these constraints, you can solve for the correlationmatrix:

or R = +U R -U =

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PCA versus Factor Analysis

The variables reflect thecommon (latent) factors andexplain shared variation inthe manifest variables.

The components are derivedfrom the variables andexplain 100% of thevariation in the data.

Not necessary that 100% ofvariance be accounted forby the extracted factors.

100% of variance accountedfor by all components.

Factor AnalysisPCA

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Limitations of Exploratory Factor AnalysisFactor scores are not linear combinations of the variables.They are estimates of latent factors. Try to avoid datafishing problems by:

1. Carefully selecting your manifest variables.

2. Using rotation to interpret the factors.

3. Performing a confirmatory factor analysis to testhypotheses about the adequacy of the factor solution.

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Factor Extraction Methods: OverviewPrincipal Factor Analysis

Computationally efficient

Most commonly used.

Maximum Likelihood Factor Analysis

Less efficient computationally; iterative procedure

Better estimates than principal factor analysis in largesamples

Hypothesis tests for number of factors.

Prior communality estimates are usually the squaredmultiple correlation of each manifest variable with all theothers.


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How Many Factors? Proportion of variance accounted for

Minimum #factors to account for 100% of thecommon variance

Scree test Find the elbow

Interpretability criteria

At least three items load on each factor

Variables within a factor share conceptual meaning

Variables between factors measure differentconstructs

Rotated factors demonstrate simple structure.

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The FACTOR Procedure, revisitedGeneral form of the FACTOR procedure:


VAR variables;RUN;


VAR variables;RUN;

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This demonstration illustrates exploratory factoranalysis using the FACTOR procedure

Using the FACTOR Procedurech5s2d1.sas

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Are Factors Correlated? Rotation Methods

Buying

Power

ConsumerConfidence

BuyingPower

ConsumerConfidence

Orthogonal

Oblique

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Rotation Methods Varimax-Orthogonal: Maximizes the variance of

columns of the factor pattern matrix.

Promax-Oblique in two steps:

1. Varimax rotation

2. Relax orthogonality constraints and rotate further.

PROC FACTOR can also perform many other rotationmethods. See the SAS/STAT Users Guide.

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Displayed Factor Analysis OutputEigenvalues

In factor analysis, the eigenvalues displayed are relatedto the reduced correlation matrix (R-U).

In PCA, eigenvalues are ofR.

Rule of eigenvalue >1 is less meaningful in

determining the number of factors to retain for factoranalysis.

Scree plot of eigenvalues is often useful in factoranalysis.


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Displayed Factor Analysis OutputFactor Pattern Matrix

The matrix of standardized regression coefficients forY =XB +E

Equal to the matrix of correlations between thevariables and the extracted (orthogonal) commonfactors.

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Displayed Factor Analysis OutputRotated Factor Pattern Matrix

The matrix of standardized regression coefficients forrotated factors

Equal to the correlations between the variables andthe rotated common factors for orthogonal rotations.

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Displayed Factor Analysis OutputStructure Matrix

Generated for oblique rotations only

The matrix of the correlations between variables androtated common factors.

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Displayed Factor Analysis OutputReference Structure Matrix


The matrix of semipartial correlations betweenvariables and common factors, removing from eachcommon factor the effects of other common factors.

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Displayed Factor Analysis OutputCorrelation between Factors


Factor plots

Final communality estimates

R2 for predicting variables from factors

Called squared canonical correlations when MLmethod is used

Variance explained by each factor

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This demonstration illustrates the FACTOR

procedure for exploratory factor analysis withrotation.

Rotated Factor Analysisch5s2d2.sas


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This exercise reinforces the concepts discussedpreviously.

Exercises

Section 5.3

Cronbachs Coefficient

Alpha for Scale Rel iab il ity

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Objectives Perform reliability analysis of scale data using

Cronbachs coefficient alpha.

Interpret output from the CORR procedure forCronbachs alpha.

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Internal Consistency of a ScaleWhen measuring a latent variable, you need a way toquantify the latent variable.

Items that load on a factor for the latent variable are oftensummed to create a scale score.

But how reliable is the scale?

The truereliability is the squared correlation betweenthe scale score (Y) and the true value of the latentvariable (T).

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Cronbachs AlphaOne way of estimating the reliability of a scale is tocompute Cronbachs alpha

where Y are the variables that make up the scale, p is thenumber of variables, and Y0 is the sum of all the variablesinY.

Be sure to reverse-code items as necessary!

0

cov( )

1 var( )

i j

i j

Y Yp

p Y

=

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Other Statistics Standardized alpha

Useful when variables have different variances

Correlation with total for each item (standardizedand raw)

Find the items that best characterize the latent

variable Alpha if item deleted for each item (standardized

and raw)

Identify well-fitting and poorly fitting items.


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The CORR ProcedureGeneral form of the CORR procedure:

PROC CORR options;

VAR variables;RUN;

PROC CORR options;

VAR variables;RUN;

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This demonstration illustrates PROC CORR forreliability analysis.

PROC CORR forReliability Analysisch5s3d1.sas

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This exercise reinforces the concepts discussedpreviously.

Exercises

Factor Analysis Slides

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