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Exploratory Factor Analysis (EFA)

Mary Ann Coughlin Springfield College

William KnightBowling Green State University

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Welcome & Agenda

� First of a three part series� Today Exploratory Factor Analysis� Next Confirmatory Factor Analysis� Final Structural Equation Modeling

� Agenda:� Overview of EFA� Case Study Application with Senior Survey� Statistical Basis for EFA� Interpreting EFA output� Next Steps

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Factor Analysis

� Purpose:� Factor analysis explores the inter-

relationships among variables to discover if those variables can be grouped into a smaller set of underlying factors.

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Applications of Factor Analysis

� Three primary applications of Factor Analysis include:

� Explore data for patterns. Often a researcher is unclear if items or variables have a discernible patterns. Factor Analysis can be done in an Exploratory fashion to reveal patterns among the inter-relationships of the items.

� Data Reduction. Factor analysis can be used to reduce a large number of variables into a smaller and more manageable number of factors. Factor analysis can create factor scores for each subject that represents these higher order variables.

� Confirm hypothesis of factor structure. In measurement research when a researcher wishes to validate a scale with a given or hypothesized factor structure, Confirmatory Factor Analysis is used.

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5

Case Study

Exploratory Factor AnalysisExploratory Factor AnalysisExploratory Factor AnalysisExploratory Factor Analysis

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Case StudyExploratory Factor Analysis

� Data� Annually, the Office of Institutional Research conducts a

survey of graduating seniors. This instrument contains a bank of 27-items that attempt to measure various aspects of outcomes of an undergraduate education.

� Purpose:� The researcher wishes to determine if these 27-items can

be organized or grouped into a smaller set of underlying factors.

� These 27-items are just too detailed for presentation purposes.

� Analysis:� Exploratory Factor Analysis

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SPSS Analysis

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Assumptions of Factor Analysis

� Assumes that the observed variables are a linear combination of some underlying of hypothetical or unobservable factors.

� Some of the factors are assumed to be common to two or more variables and some are assumed to be unique to each variable.

� The factors or unobserved variables are assumed to independent of one another.

� All variables in a factor analysis must consist of at least an ordinal scale. Nominal data are not appropriate for factor analysis.

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Basic Assumptions

� Two statistics on the SPSS output allow you to look at some of the basic assumptions.� Kaiser-Meyer-Olkin Measure of Sampling Adequacy, and� Bartlett's Test of Sphericity

� Kaiser-Meyer-Olkin Measure of Sampling Adequacy generally indicates whether or not the variables are able to be grouped into a smaller set of underlying factors.� High values (close to 1.0) generally indicate that a factor

analysis may be useful with your data. � If the value is less than .50, the results of the factor analysis

probably won't be very useful.� Let us look at ours for our Factor analysis.

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SPSS Output

� So clearly our data support the use of factor analysis and suggest that the data may be grouped into a smaller set of underlying factors.

� What does Bartlett’s Test of Sphericityexplore?

KMO and Bartlett's Test

.920

4992.010

351

.000

Kaiser-Meyer-Olkin Measure of SamplingAdequacy.

Approx. Chi-Square

df

Sig.

Bartlett's Test ofSphericity

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SPSS Output

� Bartlett’s Test of Sphericity compares your correlation matrix to an identity matrix� An identity matrix is a correlation matrix with 1.0 on the

principal diagonal and zeros in all other correlations.� So clearly you want your Bartlett value to be significant as you

are expecting relationships between your variables, if a factor analysis is going to be appropriate!

� Problem with Bartlett’s test occurs with large n’s as small correlations tend to be statistically significant – so test may not mean much!

KMO and Bartlett's Test

.920

4992.010

351

.000

Kaiser-Meyer-Olkin Measure of SamplingAdequacy.

Approx. Chi-Square

df

Sig.

Bartlett's Test ofSphericity

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Procedures for Factor Analysis

� The first step in completing a factor analysis is to measure the inter-relationships among the items. � This step leads to the determination of the

appropriate number of factors.

� When initially determining the appropriate number of factors, one factor is identified for each variable. � Obviously the researcher expects that the

number of useful factors will be substantially less.� However, if no relationship exists between the

variables then each variable would make it's own unique factor.

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Procedures for Factor Analysis

� Multiple different statistical procedures exist by which the number of appropriate number of factors can be identified.� These procedures are called "Extraction Methods."� By default SPSS does what is called a Principal

Components Extraction method.� This Principal Components Method is simpler and until

more recently was considered the appropriate method for Exploratory Factor Analysis.

� Statisticians now advocate for a different extraction method due to a flaw in the approach that Principal Components utilizes for extraction.

� Let us briefly explore this issue.

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Principal Components Extraction

� In the Principal Components Analysis, inter-item correlation coefficient Matrix is what is analyzed to explore the inter-relationships between the items to determine if the items can be grouped together to represent a smaller set of underlying factors.

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Principal Components Extraction

� The correlation (R) matrix represents the relationships between all items and is a complete matrix with 1.0 on the diagonal.� The 1.0 indicates the perfect

relationship the variable has with itself.

� The upper and lower elements of the matrix are mirror images.

� On the right is a sample correlation matrix representing the relationship between six items on an attitude scale.

� This matrix is much easier to interpret than the one for our analysis, which would be a 27 X 27 matrix!!

Item 1 Item 2 Item 3 Item 4 Item 5 Item 6

Item 1 1.000 .678 .780 .240 .189 .065

Item 2 .678 1.000 .858 .380 .345 .188

Item 3 .780 .858 1.000 .189 .243 .058

Item 4 .240 .380 .189 1.000 .789 .834

Item 5 .189 .345 .243 .789 1.000 .657

Item 6 .065 .188 .058 .834 .657 1.000

Correlation "R" Matrix

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R Matrix

� This R matrix reports:� strong positive relationships among items I, 2, and 3,

as well as strong positive relationships among items 4, 5, and 6;

� in addition, items 1, 2, and 3 have weak relationships with items 4, 5, and 6;

� this pattern would be the first indication that a two-factor model might be appropriate for this data.

Item 1 Item 2 Item 3 Item 4 Item 5 Item 6

Item 1 1.000 .678 .780 .240 .189 .065

Item 2 .678 1.000 .858 .380 .345 .188

Item 3 .780 .858 1.000 .189 .243 .058

Item 4 .240 .380 .189 1.000 .789 .834

Item 5 .189 .345 .243 .789 1.000 .657

Item 6 .065 .188 .058 .834 .657 1.000

C orrelation "R" Matrix

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Principal Components Extraction

� The problem with the Principal Components Analysis is that correlation matrix has 1.0 on the diagonal.

� As a result the initial communalities among the items are set or assumed to be 1.0.

� So what are communalities and why is the initial communalities in Principal Component Analysis set to 1.0.

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Principal Components Extraction

� A communality (C) is the extent to which an item correlates with all other items.� Thus in Principal Components Extraction method

when the initial communalities are set to 1.0, then all of the variability of each item is accounted for in the analysis.

� Of course some of the variability is explained and some is unexplained.

� So in the Principal Components analysis with these initial communalities set to 1.0, you are trying to find both the common factor variance and the unique or error variance.

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Principal Components Extraction

� In a two factor model, the initial communality of Item 1 can be described by the following formula:

� Item 1 = aFactor 1 + aFactor 2 + U item 1

� Where

� aFactor 1 + aFactor 2 represent the explained common factor variance for the two factors, and

� U item 1 represents the uniqueness of the item or error variance.

� The presence of the uniqueness in the model is what makes the initial communalities add up to 1.0.

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Principal Components Extraction

� Statisticians have indicated that assuming that all of the variability of the items whether explained or unique can be accounted for in the analysis is flawed and definitely should not be used in an exploratory factor model.

� Some researchers suggest Principal Axis Factoring as the appropriate method for factor extraction using Exploratory Factor Analysis.

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Principal Axis Factoring� In Principal Axis factor extraction, the amount of variability each

item shares with all other items is determined and this value isinserted into the R matrix replacing the 1.0 on the diagonals.

� As a result, Principal Axis factoring is only analyzing common factor variability; removing the uniqueness or unexplained variability from the model.

� In a two factor model, the initial communality of Item 1 can be described by the following formula:

� Item 1 = aFactor 1 + aFactor 2

� Where:� aFactor 1 + aFactor 2 represent the explained common

factor variance for the two factors, and

� the uniqueness of the item or error variance or U item 1 is no longer in the formula.

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Principal Axis Factoring

� Thus, when reviewing communalities,� As the values of the communalities decrease, then

the more unexplained variability or uniqueness exists within that item.

� As a result, lower communalities indicate that the item does not add to the proposed factor structure.

� In layperson's terms, an item with a lower communality is a problem item, because the item is not common to the proposed factor structure; rather it is more unique or outside the factor structure.

� Now let us look at our initial communalities from our case study.

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Note:

1) These communalities are from a Principal Components Analysis.

2) They are shown for illustrative purpose to compare with the Principal Axis method which is the recommended procedure.

3) You should notice that:a) The initial communalities are set to

1.0, due to the use of the correlation matrix and the 1.0 on the diagonal.

b) Also the extracted communalities will be higher using the Principal Components analysis, due to the inclusion of the uniqueness of each item.

SPSS Analysis

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SPSS Analysis

PAF Extraction

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Note:1) These communalities are from a

Principal Axis Analysis, which is the recommended procedure.

SPSS Analysis

This item has a low extracted

communality.Thus, the item does not add to the proposed factor structure. By itself a low communality

does not mean an item should be dropped, but it does mean that the item is problematic.

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Eigenvalues

� After the initial factor structure has been extracted, the next step is to calculate eigenvalues.� Remember, the initial factor structure starts with as

many factors as there are items in the analysis.

� Eigenvalues represent the amount of variance in the data that is explained by the factor with which it is associated.

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Eigenvalues

� Common Characteristics of Eigenvalues� The factors are extracted in order by the amount

of variance that they explain. Therefore, the first factor will have the highest eigenvalue, the second the next highest, etc.

� The first few factors generally explain the majority of the variance with the last few explaining only a very small proportion of variance.

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Eigenvalues

� Common Characteristics of Eigenvalues� At this stage in the analysis, the researcher must

determine the number of meaningful factors.� Determining the number of factors is not a

straightforward task, because the decision is ultimately subjective.

� Serveral criteria do exist for determining the number of factors, but most of these are just empirical guidelines.

� Let us look at one such guideline –� Eigen One Rule or Kaiser-Guttman rule!

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Eigenvalues

� Eigen One Rule or Kaiser-Guttman rule!� This rule instructs you to keep only those factors

whose eigenvalues are greater than 1.0 and discard the rest.

� The logic of this rule is that by selecting 1.0 as the criteria for retaining the factor, the variance accounted for by the factor must be at least as large as the variance of a single standardized variable.� Remember, standardized variables (z values) have a

mean of 0 and a standard deviation of 1.0. � And if standard deviation is 1.0 – then variance must

also be 1.0.

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SPSS Analysis

6 Factors exist

that explain a

variance

equivalent to at

least one

standardized

variable

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Eigenvalues

� Criticism of Eigen One Rule-

� Many statistician criticize the eigen one rule as still being somewhat arbitrary and based solely on the data.

� These statistician feel that theory and logic should also be considered in this process.

� So as a result they suggest that the eigen one rule be used solely as a guideline.

� Before we proceed to discuss the factor matrix and how eigen values are calculated, let us look at one more way of determining the appropriate number of factors to extract.

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Scree Plot

� A scree plot can be used to visually determine the number of useful factors to be extracted.

� A scree plot is a line graph with� Eigen values plotted on the Y or vertical axis, and� The factors are plotted on the horizontal or X-axis.

� A scree plot should form the intersection of two lines� One line should be an initial steep line of useful factors and � The second line should be a gradual trailing line of factors

that should be eliminated.

� The plot is called a 'Scree' Plot because it often looks like a 'scree' slope, where rocks have fallen down and accumulated on the side of a mountain, where the scree refers to the debris that has fallen!

� Let us look at our scree plot!

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SPSS Scree Plot!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Factor Number

0

2

4

6

8

10

Eigenvalue

Initial Steep line!

Gradual Trailing

line!

? Where do the two lines intersect ?

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Interpretation of Scree Plot

� To interpret our Scree plot we want to find the point where the two lines intersect.� Keep all factors that fall on the initial steep line and� Discard the factors that are found in the gradual

trailing line or scree!

� The problem with our data is that the intersection between the two lines is not clear.� That intersection could occur anywhere between

factors 6 through 10.� So now what?

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Interpretation of Extracted Factors

� Well given that:� the application of the eigen one rule had us with 6

extracted factors, and � we do not have any theory that is driving the

creation of this instrument, and � that the scree plot could be interpreted as

supporting 6 meaningful factors, � it would make the most amount of sense to proceed with

the interpretation based upon the eigen one rule.

� Let us go back and review our SPSS output with the extracted factors.

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SPSS Analysis

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Extraction

� Notice that all statistics produced up through the initial eigenvalues were calculated for all items.

� Remember, exploratory factor analysis starts with the assumption that if the items can not be logically grouped into a smaller set of underlying factors, then each item would make a unique factor.� The eigenvalues represent the percent of explained

variance that is due to each factor.� Eigenvalues are calculated by squaring and then summing

the factor loadings for each item within each factor.

� Thus to understand eigenvalues we must review the factor matrix.

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Factor Matrix

� After the appropriate number of factors has been determined, a factor matrix is calculated. This matrix identifies the relationship between the variables and the factors.

� In general, this matrix reads like a correlation matrix and the factor loadings are similar in structure and value to correlation coefficients.

� Before, interpreting this initial factor matrix, we must be aware of some general problems with this initial matrix.

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Three Problems with Initial Factor Matrix

� Due to the fact that the first factor explains the most amount of variance, most of the variables will have at least some relationship with this first factor. � Thus, this factor becomes very generalized and difficult to interpret.

� Due to first factor being a general factor, many variables may load on more than one factor (double loading). � While this complexity, is not a problem statistically, it does add

needlessly to the complexity of the factor structure.

� Many factors may be bi-polar. A bi-polar factor is one in which both significant positive and negative loadings exist.� A negative loading is like a negative correlation coefficient. � Often negative loadings may be found which are due to the actual

values in the data. � Bipolar factors can create negative loadings which make no apparent

sense from the data.

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Initial Factor Matrix

� Let us take a look at our initial factor matrix.

� It presents some of these problems.

� As we interpret the factor matrix, remember factor loadings are like correlation coefficients.

� As a general guideline, factor loadings greater than .40 indicate that an item is related or associated with a given factor.

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Factor Matrix

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Factor Matrix

� Before the rotation, process we have many items that are loading on our first generalized factor.

� In addition, we have a few items with double factor loadings (i.e., items that load on more than one factor).

� Factor two is bipolar, with both positive and negative factor loadings.

� These problems are easier to identify if we sort the matrix and suppress values that are low (< .30).

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Factor Matrix

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Rotation of the Factor Matrix

� The factor matrix is rotated to address the three common problems with the initial matrix.

� Three common procedures exist for rotation:� Orthogonal –

� Varimax� Quartimax,

� Oblique

� Each method varies in how the rotation is accomplished.

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Orthogonal Rotation -- Varimax

� Orthogonal rotation is easiest of the three methods to describe conceptually. However, it is also the most limited in terms of it's application.

� Orthogonal rotation has a restriction which states that the factors may only be rotated in such a manner that the factors are kept at right angles to each other.

� This restriction follows the assumption that no association exists between the factors. For many applications, the factors are logically correlated, thus the limited application of this procedure.

� Graphically, the rotation procedures are illustrated as follows:

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Orthogonal Rotation

� Note:� The number lines

represent the factor loadings on two factors.

� The right angle restrictions represents the fact no relationship is assumed to exist between the two factors.

Factor 1

Factor 2

.40 .80-.40-.80

+1.0

.60

- .60

-1.0

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Orthogonal Rotation

Factor 1

Factor 2

.40 .80-.40-.80

+1.0

.60

- .60

-1.0

XX

X

X

XX

X XX X

X X

X

Note:1) X represents the variables

and plots the factor loadings for the two factors.

2) Some negative factor loadings appear for some variables on factor 1.

3) The factors are now rotated or shifted to improve the relationships between the variables and each factor, keeping the right angle restriction.

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Orthogonal Rotation

Factor 1

Factor 2

.40

.80

-.40

-.80

+1.0

.60

- .60

-1.0

XX

X

X

XX

X XX X

X X

X

Note:1) The rotation increased the

strength of the relationship between many of the variables and the factors.

2) Also, the rotation eliminated the negative factor loadings.

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Factor Transformation Matrix

� The factor transformation matrix describes the specific rotation applied to your factor solution.� This matrix is used to compute the rotated factor

matrix from the initial (unrotated) factor matrix.� For example:

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SPSS Analysis

Varimax Rotation

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Factor Transformation Matrix

� If the off-diagonal elements are close to zero, the rotation was relatively small.

� If the off-diagonal elements are large (greater than ± .5), a larger rotation was applied.

Factor Transformation Matrix

.652 .398 .422 .331 .244 .263

.377 -.675 -.221 .093 .556 -.190

-.568 .035 .305 .625 .358 -.255

-.298 .217 -.267 -.257 .607 .598

.144 .574 -.556 .040 .183 -.553

.033 -.089 -.547 .651 -.319 .408

Factor1

2

3

4

5

6

1 2 3 4 5 6

Extraction Method: Principal Axis Factoring. Rotation Method: Varimax with Kaiser Normalization.

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Next Step

� Our next step would be to interpret the factor matrix, which we ran with the values sorted by size and suppressed values less than.30.

� But before we do let us finish our discussion of other rotation procedures.

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Orthogonal -- Uncorrelated Model

� Remember, EFA with varimax rotation is an uncorrelated model.

� This concept is maintained with orthogonal rotation, as the factors remain at 90o angles, which assume the factors are uncorrelated.

� In addition, varimax rotation adds one additional feature to orthogonal rotation as it maximizes the amount of variance explained by each of the factors.

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Other Rotation Procedures

� Oblique Rotation:� Factors are rotated without maintaining the right

angle restrictions.� This allows the researcher to account for the

relationships that may logically exist between the factors.� The closer the rotation angle is to 0, the higher the

correlation between the factors.� As the rotation angle approaches 90 degrees, the

relationship between the factors nears 0.� SPSS calls this Direct OBLIM!

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Oblique Rotation

Factor 1

Factor 2.40.80

-.40-.80

+1.0

.60

- .60

-1.0

X

X

XX

X X

X XX X

XX

X

Note:1) The right angle restriction is no

longer held. 2) In the current example, only a

slight relationship is found between the factor as the factors are still at about 80 degree angles.

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Other Rotation Procedures

� Quartimax is another form of orthogonal rotation.� It is used when you are expecting one

factor and� You are testing to see if there is one factor!

� For our case study – we have run Varimax let us look at the output and interpret our factor analysis

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Factor Matrix

� We start by interpreting the factor matrix!

� So you know, a factor matrix contains the relationship between each item and each factor.� And the relationships are referred to as factor loadings.

� We use the rotated factor matrix, not the initial factor matrix, to interpret these relationships.� Because the rotated factor matrix address those three

common problems.

� One set of procedures for interpreting the rotated factor matrix is to use Thurstone's Simple Structure:� The term simple structure implies that items should be

related to only one factor.� These guidelines are helpful, if you are trying to create or

revise a scale.

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Factor Matrix� Thurstone's Rules state:

� Select items that relate strongly to the proposed factor (i.e., factor loadings of .40 or above).

� Delete or drop items that are double loaded (i.e., .40 or above on more than one factor).

� Delete items that are unique or do not load on any factor (i.e.,all factor loadings are below .40).

� Delete items that load high on a factor that was not the proposed factor.

� Some statisticians acknowledge that factor complexity is inevitable.� As a result, some researchers will maintain items with double

factor loadings as long as the items would logically belong to both factors.

� If greater than .05 percent difference exists between the two factor loadings, the item can be considered to primarily belonging to the factor with the higher factor loading.

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Factor Matrix

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Factor Matrix

� To better understand this rotated matrix, we can use the computer to organize our factor matrix, by telling SPSS to sort the matrix and suppress values that are less than .40.� You may want to consider suppressing values less

than .30, so that you can see what items are close to loading on some factors.

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Factor Matrix

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Factor Matrix

� So what have we found?� What do our factors represent?

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Interpretation of the Factors

� Analysis of our Rotated Factor matrix from the exploratory factor analysis revealed a total of six factors.

� After discussion and consultation, these factors were named:� Intellectual� Moral� Self-Development ~ Self-Awareness� Leadership� Technology� Humanities

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Factor Matrix

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Interpretation of the Factors

� The naming of the factors is the responsibility of the researcher.� When naming the factors you may consider items that

have higher factor loadings as being more representative of the factor than items with lower factor loadings.

� Naming of the factors is an important and difficult stage in the analysis.

� In some cases, the researcher has some predetermined structure that is used in this phase.� However, if a proposed factor structure does exist, the researcher

would eventually want to complete a confirmatory factor analysis.

� When naming factors, do not do so in isolation.� Be sure that you do so in the context of your research and

organization

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Making Connections

� Now that we have interpreted and named our factors, let us go back and make a few connections between the factor matrix, factor loadings, communalities, and eigenvalues.

� Let us start with the communalities� Do you remember what a communality

is?

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SPSS Analysis A communality is

the proportion of a

variable's variance

explained by the

factor structure

So what is the

difference between

initial and extraction

communality and why

are they different?

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Initial Communality

� Initial Communality is equal to:� R2

1.23456789 – k

� You can easily calculate this with the SPSS REGRESSION program � entering as the dependent variable the variable that you are

trying to calculate the communality for (e.g., variable 1) and � Then enter all the remaining variables in the factor analysis

as the predictor variables

� Doesn’t all this make sense given our definition of a communality?

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SPSS Regression AnalysisVariables Entered/Removed b

Understand the role of science and technology,Develop feminist awarenenss, Read or speacka foreign language, Form close friendships,Gain in-depth knowlegde of a field, Usecomputers for basic tasks (word processing),Place current problems in historicalprospective, Lead and supervise tasks andgroups of people, Think analytically andlogically, Use computers for complex tasks(graphing), Acquire broad knowledge in the Artsand Sciences, Appreciate art, literature, music,drama, Relate well to people of different races,nations, Communicate well orally, Acquire newskills and knowledge on my own, Plan andexecute complex projects, Identify moral andethical issues, Function independently withoutsupervision, Function effectively as a memberof a team, Evaluate and choose betweenalternative courses, Understand myself, myabilities, interests, Formulate creative / originalideas and solutions, Synthesize and integrateideas and information, Develop awareness ofsocial problems, Establish a course of action toaccomplish goals, Develop self-esteem/self-confidence

a

. Enter

Model1

Variables EnteredVariablesRemoved Method

All requested variables entered.a.

Dependent Variable: Write effectivelyb.

Here I have entered all

my 26 other variables as

the predictor variables!

Here I have entered my first

variable write effectively as

my dependent variable,

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SPSS Regression AnalysisModel Summary

.605a .366 .334 .530Model1

R R SquareAdjusted R

SquareStd. Error of the

Estimate

Predictors: (Constant), Understand the role of science andtechnology, Develop feminist awarenenss, Read or speack a foreignlanguage, Form close friendships, Gain in-depth knowlegde of afield, Use computers for basic tasks (word processing), Placecurrent problems in historical prospective, Lead and supervise tasksand groups of people, Think analytically and logically, Usecomputers for complex tasks (graphing), Acquire broad knowledge inthe Arts and Sciences, Appreciate art, literature, music, drama,Relate well to people of different races, nations, Communicate wellorally, Acquire new skills and knowledge on my own, Plan andexecute complex projects, Identify moral and ethical issues, Functionindependently without supervision, Function effectively as a memberof a team, Evaluate and choose between alternative courses,Understand myself, my abilities, interests, Formulate creative /original ideas and solutions, Synthesize and integrate ideas andinformation, Develop awareness of social problems, Establish acourse of action to accomplish goals, Develop self-esteem/self-confidence

a. Notice that the R2 for

this analysis is equal to

the communality for the

first item in the factor

analysis – write

effectively!

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SPSS AnalysisSee they match!

So, now how do we

calculate extracted

communalities, and

and why are they

different?

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Communalities

� We can manipulate the factor matrices, both initial and rotated to further illustrate the calculations of the communalities.� If we take either the initial factor or the rotated

factor matrix and square the factor loadings; � the extracted communalities for each item are the sum of

the squared factor loadings across the factors or rows.� Why doesn’t it matter whether I use the initial or

rotated factor matrix?� Because in either matrix all the common variance that the

item shares with the factor structure is the same, it is just “rotated” or distributed differently across the factors in the rotated factor matrix!

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Initial Factor Matrix

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75

1 SQ -1 2 SQ -2 3 SQ -3 4 SQ -4 5 SQ - 5 6 SQ -6 Communalities

Write effectively .5228 .2733 -.0335 .0011 -.2744 .0753 -.0476 .0023 .0938 .0088 .0200 .0004 .3612Communicate well orally .5814 .3380 .0421 .0018 -.0763 .0058 -.1123 .0126 -.0910 .0083 .0512 .0026 .3691

Acquire new skills and .5579 .3113 .1744 .0304 -.2315 .0536 -.0736 .0054 -.0147 .0002 .0235 .0006 .4015

Think analytically and logically .5415 .2932 .1927 .0371 -.2793 .0780 -.1290 .0166 .1639 .0269 -.0864 .0075 .4593

Formulate creative / original ideas .6325 .4000 .1175 .0138 -.2018 .0407 -.1519 .0231 .0382 .0015 .0987 .0098 .4889

Evaluate and choose .6307 .3978 .0982 .0096 .0465 .0022 -.1634 .0267 .0671 .0045 .1621 .0263 .4670

Lead and supervise .5196 .2699 .1063 .0113 .4529 .2051 -.1507 .0227 .0012 .0000 .3380 .1142 .6233

Relate well to people of different .5171 .2674 -.1464 .0214 .2766 .0765 .0084 .0001 .0271 .0007 -.0343 .0012 .3673Function effectively as a member of a team .5244 .2750 .0646 .0042 .4101 .1682 -.0392 .0015 .0620 .0038 .1826 .0334 .4861

Use computers for basic tasks .3788 .1435 .1350 .0182 .2078 .0432 .1711 .0293 -.0037 .0000 -.1772 .0314 .2656

Use computers for complex tasks .2057 .0423 .4090 .1672 .2383 .0568 .2928 .0857 .1362 .0186 -.1119 .0125 .3832

Place current problems in .4787 .2291 -.2436 .0594 -.1143 .0131 .0308 .0010 .1562 .0244 .1582 .0250 .3519

Identify moral and ethical issues .5927 .3513 -.2461 .0605 .0254 .0006 .0741 .0055 .2150 .0462 .0100 .0001 .4643

Understand myself, my abilities, interests .6342 .4023 -.2287 .0523 .0305 .0009 -.1656 .0274 -.1891 .0358 -.1279 .0164 .5351Function independently without supervision .6201 .3846 .0436 .0019 .0482 .0023 -.0414 .0017 -.2092 .0438 -.1101 .0121 .4464

Gain in-depth knowlegde of a field .4336 .1880 .1694 .0287 -.1718 .0295 .0200 .0004 .0277 .0008 -.0131 .0002 .2476

Plan and execute complex projects .5695 .3244 .2557 .0654 -.1332 .0177 .0168 .0003 -.0145 .0002 .0359 .0013 .4093

Read or speak a foreign language .2299 .0529 -.0971 .0094 -.1040 .0108 .2365 .0559 -.2020 .0408 .1069 .0114 .1813

Appreciate art, literature, music, drama .4621 .2135 -.2769 .0767 -.1792 .0321 .2963 .0878 -.2758 .0761 .2462 .0606 .5468

Acquire broad knowledge .4443 .1974 .0836 .0070 -.1043 .0109 .3322 .1104 -.1012 .0102 .0758 .0057 .3416

Develop feminist awarenenss .3796 .1441 -.4281 .1833 -.0193 .0004 .1229 .0151 .1584 .0251 -.0355 .0013 .3692Develop awareness of social problems .5736 .3290 -.4395 .1931 .0748 .0056 .1468 .0216 .2813 .0792 -.1206 .0145 .6430

Develop self-esteem /self-confidence .7160 .5127 -.1546 .0239 .0628 .0039 -.0748 .0056 -.1341 .0180 -.1700 .0289 .5930

Form close friendships .4726 .2234 -.1496 .0224 .2055 .0422 -.0921 .0085 -.2318 .0537 -.2191 .0480 .3982

Establish a course of action .7169 .5140 .0826 .0068 .0267 .0007 -.0755 .0057 -.0470 .0022 -.0866 .0075 .5369

Synthesize and integrate ideas .6611 .4371 .1493 .0223 -.1332 .0178 -.0485 .0024 .0827 .0068 -.0807 .0065 .4928Understand the role of science .5087 .2588 .4309 .1857 .0601 .0036 .2694 .0726 .0171 .0003 -.0599 .0036 .5246

Eigenvalues -- Extracted 7.7742 1.3151 .9976 .6478 .5369 .4829

Factor

Communalities

76

SPSS AnalysisSee they match!

So, why are they

different?

39

77

The Difference!

� The initial communalities represent the amount of variance of the variable that is explained by all the variables in the model � Hence it can be replicated by the R2 value, and � Since these are initial values prior to extraction, it represents

the amount of variance of the variable that is shared by all other variables.

� Remember, the initial assumption in EFA is that we will have as many factors as we have variables.

� The extracted communalities represent the amount of variance of the variable that is explained by the factor structure.� Thus it is calculated by squaring and summing the factor

loadings.

78

Eigenvalues

� We can also manipulate the initial and rotated factor matrix and further illustrate the calculations of the eigenvalues.� If we take the initial factor matrix and square the

factor loadings:� The extracted eigenvalues are the sum of the squared

factor loadings across the items or columns.

40

79

1 SQ -1 2 SQ -2 3 SQ -3 4 SQ -4 5 SQ - 5 6 SQ -6 Communalities

Write effectively .5228 .2733 -.0335 .0011 -.2744 .0753 -.0476 .0023 .0938 .0088 .0200 .0004 .3612Communicate well orally .5814 .3380 .0421 .0018 -.0763 .0058 -.1123 .0126 -.0910 .0083 .0512 .0026 .3691

Acquire new skills and .5579 .3113 .1744 .0304 -.2315 .0536 -.0736 .0054 -.0147 .0002 .0235 .0006 .4015

Think analytically and logically .5415 .2932 .1927 .0371 -.2793 .0780 -.1290 .0166 .1639 .0269 -.0864 .0075 .4593

Formulate creative / original ideas .6325 .4000 .1175 .0138 -.2018 .0407 -.1519 .0231 .0382 .0015 .0987 .0098 .4889

Evaluate and choose .6307 .3978 .0982 .0096 .0465 .0022 -.1634 .0267 .0671 .0045 .1621 .0263 .4670

Lead and supervise .5196 .2699 .1063 .0113 .4529 .2051 -.1507 .0227 .0012 .0000 .3380 .1142 .6233

Relate well to people of different .5171 .2674 -.1464 .0214 .2766 .0765 .0084 .0001 .0271 .0007 -.0343 .0012 .3673Function effectively as a member of a team .5244 .2750 .0646 .0042 .4101 .1682 -.0392 .0015 .0620 .0038 .1826 .0334 .4861

Use computers for basic tasks .3788 .1435 .1350 .0182 .2078 .0432 .1711 .0293 -.0037 .0000 -.1772 .0314 .2656

Use computers for complex tasks .2057 .0423 .4090 .1672 .2383 .0568 .2928 .0857 .1362 .0186 -.1119 .0125 .3832

Place current problems in .4787 .2291 -.2436 .0594 -.1143 .0131 .0308 .0010 .1562 .0244 .1582 .0250 .3519

Identify moral and ethical issues .5927 .3513 -.2461 .0605 .0254 .0006 .0741 .0055 .2150 .0462 .0100 .0001 .4643

Understand myself, my abilities, interests .6342 .4023 -.2287 .0523 .0305 .0009 -.1656 .0274 -.1891 .0358 -.1279 .0164 .5351Function independently without supervision .6201 .3846 .0436 .0019 .0482 .0023 -.0414 .0017 -.2092 .0438 -.1101 .0121 .4464

Gain in-depth knowlegde of a field .4336 .1880 .1694 .0287 -.1718 .0295 .0200 .0004 .0277 .0008 -.0131 .0002 .2476

Plan and execute complex projects .5695 .3244 .2557 .0654 -.1332 .0177 .0168 .0003 -.0145 .0002 .0359 .0013 .4093

Read or speak a foreign language .2299 .0529 -.0971 .0094 -.1040 .0108 .2365 .0559 -.2020 .0408 .1069 .0114 .1813

Appreciate art, literature, music, drama .4621 .2135 -.2769 .0767 -.1792 .0321 .2963 .0878 -.2758 .0761 .2462 .0606 .5468

Acquire broad knowledge .4443 .1974 .0836 .0070 -.1043 .0109 .3322 .1104 -.1012 .0102 .0758 .0057 .3416

Develop feminist awarenenss .3796 .1441 -.4281 .1833 -.0193 .0004 .1229 .0151 .1584 .0251 -.0355 .0013 .3692Develop awareness of social problems .5736 .3290 -.4395 .1931 .0748 .0056 .1468 .0216 .2813 .0792 -.1206 .0145 .6430

Develop self-esteem /self-confidence .7160 .5127 -.1546 .0239 .0628 .0039 -.0748 .0056 -.1341 .0180 -.1700 .0289 .5930

Form close friendships .4726 .2234 -.1496 .0224 .2055 .0422 -.0921 .0085 -.2318 .0537 -.2191 .0480 .3982

Establish a course of action .7169 .5140 .0826 .0068 .0267 .0007 -.0755 .0057 -.0470 .0022 -.0866 .0075 .5369

Synthesize and integrate ideas .6611 .4371 .1493 .0223 -.1332 .0178 -.0485 .0024 .0827 .0068 -.0807 .0065 .4928Understand the role of science .5087 .2588 .4309 .1857 .0601 .0036 .2694 .0726 .0171 .0003 -.0599 .0036 .5246

Eigenvalues -- Extracted 7.7742 1.3151 .9976 .6478 .5369 .4829

Factor

Eigenvalues

80

Eigenvalues

� We can also manipulate the initial and rotated factor matrix and further illustrate the calculations of the eigenvalues.� This time we will take the final (rotated) factor

matrix and square the factor loadings:� The rotated eigenvalues are the sum of the squared factor

loadings across the items or columns.

41

81

Rotated Factor Matrix

Rotated Factor Matrix a

.512 .263 .094 .027 -.007 .142

.460 .119 .265 .207 .037 .168

.581 .070 .141 .090 .095 .146

.643 .150 .091 .006 .118 -.026

.625 .146 .145 .199 .031 .124

.485 .175 .176 .397 .087 .067

.178 .089 .189 .724 .149 .048

.125 .335 .337 .307 .165 .070

.160 .190 .208 .567 .240 .038

.122 .118 .247 .108 .402 .053

.082 -.034 -.024 .112 .601 -.030

.303 .433 .040 .166 -.063 .198

.288 .542 .168 .185 .098 .126

.328 .275 .547 .159 -.056 .156

.372 .100 .455 .170 .165 .189

.442 .074 .080 .039 .160 .117

.537 .042 .127 .148 .230 .168

.076 .079 .078 .003 .038 .402

.178 .248 .141 .088 -.054 .650

.269 .124 .064 .050 .277 .413

.082 .560 .148 .025 -.037 .158

.158 .732 .233 .091 .102 .099

.370 .313 .543 .165 .096 .162

.122 .163 .569 .143 .077 .084

.496 .195 .386 .223 .204 .112

.587 .202 .217 .113 .208 .072

.380 -.013 .089 .138 .571 .164

Write effectively

Communicate well orally

Acquire new skills and knowledge on my own

Think analytically and logicallyFormulate creative / original ideas and solutions

Evaluate and choose between alternative courses

Lead and supervise tasks and groups of people

Relate well to people of different races, nations

Function effectively as a member of a team

Use computers for basic tasks (word processing)

Use computers for complex tasks (graphing)

Place current problems in historical prospective

Identify moral and ethical issuesUnderstand myself, my abilities, interests

Function independently without supervision

Gain in-depth knowlegde of a field

Plan and execute complex projects

Read or speack a foreign language

Appreciate art, literature, music, drama

Acquire broad knowledge in the Arts and Sciences

Develop feminist awarenenss

Develop awareness of social problemsDevelop self-esteem /self-confidence

Form close friendships

Establish a course of action to accomplish goals

Synthesize and integrate ideas and information

Understand the role of science and technology

1 2 3 4 5 6

Factor

Extraction Method: Principal Axis Factoring. Rotation Method: Varimax with Kaiser Normalization.

Rotation converged in 7 iterations.a.

82

Eigenvalues

1 SQ -1 2 SQ -2 3 SQ -3 4 SQ -4 5 SQ - 5 6 SQ -6 C ommunalities

Write effectively .5122 .2623 .2628 .0691 .0941 .0089 .0275 .0008 -.0074 .0001 .1417 .0201 .3612

C ommunicate well o rally .4602 .2118 .1192 .0142 .2655 .0705 .2072 .0429 .0369 .0014 .1685 .0284 .3691

A cquire new skills and .5814 .3380 .0698 .0049 .1413 .0200 .0899 .0081 .0953 .0091 .1464 .0214 .4015

Think analytically and lo gically .6433 .4138 .1496 .0224 .0913 .0083 .0060 .0000 .1184 .0140 -.0259 .0007 .4593

Formulate creative / o riginal ideas .6251 .3908 .1455 .0212 .1448 .0210 .1990 .0396 .0307 .0009 .1239 .0154 .4889

Evaluate and choo se .4854 .2356 .1750 .0306 .1764 .0311 .3971 .1577 .0865 .0075 .0668 .0045 .4670

Lead and supervise .1779 .0316 .0888 .0079 .1887 .0356 .7236 .5236 .1490 .0222 .0482 .0023 .6233

Relate well to people of different .1250 .0156 .3348 .1121 .3367 .1134 .3069 .0942 .1648 .0272 .0696 .0048 .3673

Function effectively as a member of a team .1600 .0256 .1903 .0362 .2084 .0434 .5672 .3218 .2400 .0576 .0381 .0015 .4861

Use computers for basic tasks .1224 .0150 .1178 .0139 .2468 .0609 .1083 .0117 .4015 .1612 .0533 .0028 .2656

Use computers for complex tasks .0816 .0067 -.0339 .0012 -.0236 .0006 .1124 .0126 .6011 .3613 -.0299 .0009 .3832

Place current problems in .3035 .0921 .4332 .1877 .0397 .0016 .1657 .0274 -.0626 .0039 .1981 .0392 .3519

Identify mo ral and ethical issues .2882 .0831 .5415 .2933 .1678 .0282 .1853 .0343 .0981 .0096 .1259 .0159 .4643

Understand myself, my abilities, interests .3277 .1074 .2747 .0755 .5471 .2994 .1595 .0254 -.0557 .0031 .1559 .0243 .5351

Function independently without supervision .3718 .1382 .0998 .0100 .4545 .2066 .1700 .0289 .1645 .0271 .1887 .0356 .4464

Gain in-depth knowlegde of a field .4416 .1950 .0736 .0054 .0796 .0063 .0394 .0016 .1598 .0255 .1171 .0137 .2476

Plan and execute complex pro jects .5373 .2887 .0416 .0017 .1272 .0162 .1476 .0218 .2295 .0527 .1680 .0282 .4093

Read o r speak a foreign language .0762 .0058 .0791 .0063 .0776 .0060 .0029 .0000 .0375 .0014 .4022 .1618 .1813

Appreciate art , literature, music, drama .1785 .0319 .2483 .0616 .1414 .0200 .0884 .0078 -.0542 .0029 .6500 .4225 .5468

A cquire broad knowledge .2691 .0724 .1239 .0154 .0635 .0040 .0497 .0025 .2766 .0765 .4133 .1708 .3416

Develop feminist awarenenss .0818 .0067 .5601 .3137 .1479 .0219 .0255 .0006 -.0372 .0014 .1577 .0249 .3692

Develop awareness o f so cial problems .1583 .0251 .7317 .5354 .2328 .0542 .0907 .0082 .1016 .0103 .0987 .0097 .6430

Develop self-esteem /self-confidence .3701 .1369 .3134 .0982 .5434 .2952 .1650 .0272 .0956 .0091 .1619 .0262 .5930

Form clo se fr iendships .1218 .0148 .1628 .0265 .5687 .3235 .1426 .0203 .0773 .0060 .0841 .0071 .3982

Establish a course of action .4961 .2461 .1949 .0380 .3863 .1493 .2228 .0496 .2036 .0414 .1118 .0125 .5369

Synthesize and integrate ideas .5865 .3440 .2019 .0408 .2166 .0469 .1127 .0127 .2079 .0432 .0720 .0052 .4928

Understand the ro le of science .3800 .1444 -.0126 .0002 .0891 .0079 .1385 .0192 .5709 .3259 .1642 .0270 .5246

Eigenvalues - - Rotated 3.8796 2.0431 1.9009 1.5008 1.3027 1.1274

Factor

42

83

SPSS Analysis

Total Variance Explained

8.311 30.783 30.783 7.774 28.793 28.793 3.880 14.369 14.369

1.859 6.885 37.667 1.315 4.871 33.664 2.043 7.568 21.936

1.556 5.762 43.430 .998 3.695 37.359 1.901 7.040 28.976

1.252 4.638 48.067 .648 2.399 39.758 1.501 5.558 34.535

1.095 4.054 52.121 .537 1.989 41.746 1.303 4.825 39.359

1.007 3.731 55.852 .483 1.788 43.535 1.127 4.176 43.535

.977 3.619 59.471

.905 3.352 62.823

.795 2.946 65.769

.769 2.850 68.618

.718 2.661 71.279

.686 2.540 73.819

.668 2.475 76.293

.632 2.342 78.636

.580 2.147 80.782

.567 2.098 82.881

.537 1.987 84.868

.518 1.920 86.788

.468 1.734 88.521

.453 1.679 90.201

.437 1.618 91.819

.425 1.575 93.393

.403 1.491 94.884

.376 1.392 96.276

.372 1.376 97.652

.333 1.233 98.885

.301 1.115 100.000

Factor1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %

Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings

Extraction Method: Principal Axis Factoring.

84

Next Steps

� How do we use the factors to analyze for differences across individuals or groups?

� We need scores that can be used to represent the factors.

43

85

Factor Scores

� Factor scores are like predicted scores for each individual score for each factor.

� Regression factor scores are calculated as:� the case's standardized score (Z score) on each variable,

multiplies by the corresponding factor loading of the variables for the given factor, and sums these products.

� These regression factor scores will have a mean of 0, but a variance equal to the squared multiple correlations between the estimated factor scores and the true factor values.

� These scores can be correlated with each other even when the factors are orthogonal

� As a result these regression scores can be difficult to interpret, as they will go beyond the normal range of standardized scores (-3.0 -- +3.0).

86

Other Methods for Calculating Factor Scores

� Other methods exist for calculating factor scores:

� The Andersen-Rubin method creates a factor score that is based upon a standardized scale.� Meaning that they will have a mean of zero and as standard

deviation of 1.0.� Making these scores easier to interpret.

� Some researcher still prefer to use factor scores that are based upon the mean or sum of the items in the factor.� These scores are generally easier to interpret as they are based

upon the scale of the variables.� However, these scores do not take into consideration the fact that

some of variables contribute more to explaining the variance of each of the factors (i.e., they have higher factor loadings).

� Of course the advantage of factor scores is that they allow the researcher to compare differences between various subgroups (e.g., gender) on the factor scores.

44

87

SPSS Analysis

Factor Scores

88

Review and Next Steps:

� So far, we have --� Provided an Overview of EFA� Completed a Case Study Application with Senior

Survey� Discussed the Statistical Basis for EFA� Interpreted EFA output

� Next steps –� Confirmatory Factor Analysis� Model testing

� Remember , today was the first of a three part series� Next Confirmatory Factor Analysis� Final Structural Equation Modeling

45

89

For More Information

� In case you missed it: recorded version and slides available at www.spss.com/airseries

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� Please fill out the post event survey

Questions!