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Data Analysis with SPSSIntroducing Exploratory

Factor Analysis

- Bidin Yatim
- Phd (2005,Exeter)
- MSc (1984, Aston)
- BSc (1982, Nottingham)
- Department of Statistics
- Faculty of Quantitative Sciences

Topic 9

Exploratory Factor Analysis Introduction

- Factor analysis attempts to identify underlying

variables, or factors, that explain the pattern

of correlations within a set of observed

variables. Factor analysis is often used in data

reduction to identify a small number of factors

that explain most of the variance observed in a

much larger number of manifest variables.

Exploratory Factor Analysis Introduction

- Statistical technique for dealing with

interdependencies between multiple variables ie

if variables are interrelated without designating

some dependent and others independent - Many variables reduced (grouped) into a smaller

number of factors (Dimension reduction method) - To accomplish the same objective as PCA/ MDS.

The factor analysis procedure offers a high

degree of flexibility

- Seven methods of factor extraction.
- Five methods of rotation
- Three methods of computing factor scores scores

can be saved as variables for further analysis.

Alternative Methods of Factor Extraction

- Principal Component Analysis
- Maximum likelihood method
- Principal axis
- Image
- Alpha
- Generalized least squares
- Unweighted least squares

Factor Analysis Extraction

- Method to specify the method of factor

extraction - Principal Components Analysis to form

uncorrelated linear combinations of the observed

variables. The first component has maximum

variance. Successive components explain

progressively smaller portions of the variance

and are all uncorrelated with each other. It is

used to obtain the initial factor solution. It

can be used when a correlation matrix is

singular. - Unweighted Least-Squares Method minimizes the

sum of the squared differences between the

observed and reproduced correlation matrices

ignoring the diagonals. - Generalized Least-Squares Method minimizes the

sum of the squared differences between the

observed and reproduced correlation matrices.

Correlations are weighted by the inverse of their

uniqueness, so that variables with high

uniqueness are given less weight than those with

low uniqueness. - Maximum-Likelihood Method produces parameter

estimates that are most likely to have produced

the observed correlation matrix if the sample is

from a multivariate normal distribution. The

correlations are weighted by the inverse of the

uniqueness of the variables, and an iterative

algorithm is employed. - Principal Axis Factoring extracts factors from

the original correlation matrix with squared

multiple correlation coefficients placed in the

diagonal as initial estimates of the

communalities. These factor loadings are used to

estimate new communalities that replace the old

communality estimates in the diagonal. Iterations

continue until the changes in the communalities

from one iteration to the next satisfy the

convergence criterion for extraction. - Alpha considers the variables in the analysis to

be a sample from the universe of potential

variables. It maximizes the alpha reliability of

the factors. - Image Factoring developed by Guttman and based

on image theory. The common part of the variable,

called the partial image, is defined as its

linear regression on remaining variables, rather

than a function of hypothetical factors. - Analyze to specify either a correlation matrix

or a covariance matrix. - Extract can either retain all factors whose

eigenvalues exceed a specified value or retain a

specific number of factors. - Display. to request the unrotated factor

solution and a scree plot of the eigenvalues. - Scree plot A plot of the variance associated

with each factor. Used to determine how many

factors should be kept. Typically the plot shows

a distinct break between the steep slope of the

large factors and the gradual trailing of the

rest (the scree).

Factor Analysis Rotation

- Method Allows you to select the method of factor

rotation. - Orthogonal
- Varimax minimizes number of variables with high

loadings on a factor - Quartimax Method minimizes the number of

factors needed to explain each variable. It

simplifies the interpretation of the observed

variables. - Equamax Method combination of the varimax

method, which simplifies the factors, and the

quartimax method, which simplifies the variables.

The number of variables that load highly on a

factor and the number of factors needed to

explain a variable are minimized. - Oblique (nonorthogonal)
- Direct Oblimin Method When delta equals 0 (the

default), solutions are most oblique. As delta

becomes more negative, the factors become less

oblique. To override the default delta of 0,

enter a number less than or equal to 0.8. - Promax Rotation Allows factors to be

correlated. It can be calculated more quickly

than a direct oblimin rotation, so it is useful

for large datasets. - Display Allows to include output on the rotated

solution, as well as loading plots for the first

two or three factors. - Factor Loading Plot Three-dimensional factor

loading plot of the first three factors. For a

two-factor solution, a two-dimensional plot is

shown. The plot is not displayed if only one

factor is extracted. Plots display rotated

solutions if rotation is requested.

Factor Analysis Scores

- Save as variables Creates one new variable for

each factor in the final solution using- - Bartlett Scores The scores produced have a mean

of 0. The sum of squares of the unique factors

over the range of variables is minimized. - Anderson-Rubin Method modification of the

Bartlett method which ensures orthogonality of

the estimated factors. The scores produced have a

mean of 0, a standard deviation of 1, and are

uncorrelated. - Display factor score coefficient matrix Shows

the coefficients by which variables are

multiplied to obtain factor scores. Also shows

the correlations between factor scores.

Types of Factor Analysis

- Exploratory
- Confirmatory

Uses of Factor Analysis

- Instrument Development
- Theory Development
- Data Reduction
- Model Testing
- Comparing Models

Example

- What underlying attitudes lead people to

respond to the questions on a political survey as

they do? - Examining the correlations among the survey

items reveals that there is significant overlap

among various subgroups of items--questions about

taxes tend to correlate with each other,

questions about military issues correlate with

each other, and so on. - With factor analysis, you can investigate the

number of underlying factors and, in many cases,

you can identify what the factors represent

conceptually. Additionally, you can compute

factor scores for each respondent, which can then

be used in subsequent analyses. For example, you

might build a logistic regression model to

predict voting behavior based on factor scores.

Assumptions

- Interval/ ratio level data
- Bivariate normal distribution for each pair of

variables and observations should be independent.

- Linear relationships
- Substantial correlations among variables (can be

tested using Bartletts sphericity test) - Categorical data (such as religion or country of

origin) are not suitable for factor analysis.

Data for which Pearson correlation coefficients

can sensibly be calculated should be suitable for

factor analysis.

Assumptions

- The factor analysis model specifies that

variables are determined by common factors (the

factors estimated by the model) and unique

factors (which do not overlap between observed

variables) the computed estimates are based on

the assumption that all unique factors are

uncorrelated with each other and with the common

factors.

Sample Size

- 10 subjects per variable. To some, subject to

variable ratio (STV) should at least be 51. - Every analysis should have 100 to 200 subjects

Steps

- Obtain correlation matrix for the data.
- Apply EFA
- Decide on the number of factors/ components to be

retained. - Interpreting the factors/components. Use rotation

if necessary - Obtain factor score for further analysis

Two concepts about variables crucial in

understanding EFA

- Common factor
- a hypothetical construct that affects at least

two of our measurement variables - We want to estimate the common factors that

contribute to the variance in our variables. - Unique variance
- factor that contributes to the variance in only

one variable. - Only one unique factor for each variable.
- Unique factors are unrelated to one another and

unrelated to the common factors. - Want to exclude these unique factors from our

solution.

Two concepts about variables crucial in

understanding EFA

- Communalities (h2)1-uniqueness - the sum over

all factors of the squared factor loading for a

variable, it indicates the portion of variance of

the variable that is accounted for by the set of

factors (or in which a variable has in common

with the other variables in the analysis) Small

numbers indicate lack of shared variance. - Uniquenessspecific variance error variance, is

that portion of the total variance that is

unrelated to other variables

Total Variance

- Total Variance Error variance Common Variance

Specific Variance - NOTE If we have 10 original variables, and the

variables are standardized, total variance 10.

Eigenvalue

- Indicates the portion of the total variance of a

correlation matrix that is explained by a factor

Iterated Principal Factors Analysis

- The most common type of FA.
- Also known as principal axis FA.
- We eliminate the unique variance by replacing, on

the main diagonal of the correlation matrix, 1s

with estimates of communalities. - Initial estimate of communality R2 between one

variable and all others.

Lets Do It AnalyzegtData ReductiongtFactorgtExtract

ion

- Using the CerealFA data, change the extraction

method to principal axis.

SPSS - Factor AnalysisOptions

- Missing values
- Exclude cases listwise
- Exclude cases pairwise
- Replace with mean
- Coefficient display format
- Sorted by size
- suppress absolute values less than .10

Correlation Matrix

- Examine matrix
- Correlations should be .30 or higher
- Kaiser-Meyer-Olkin (KMO) Measure of Sampling

Adequacy - Bartlett's Test of Sphericity

Correlation Matrix

- Bartlett's Test of Sphericity
- Tests hypothesis that correlation matrix is an

identity matrix. - Diagonals are ones
- Off-diagonals are zeros
- Significant result indicates matrix is not an

identity matrix, therefore EFA can be used.

Correlation Matrix

- Kaiser-Meyer Olkin (KMO)
- measure of sampling adequacy.
- index for comparing magnitudes of observed

correlation coefficients to magnitudes of partial

correlation coefficients - small values indicate correlations between pairs

of variables cannot be explained by other

variables

Kaiser-Meyer-Olkin (KMO)

- Marvelous - - - - - - .90s
- Meritorious - - - - - .80s
- Middling - - - - - - - .70s
- Mediocre - - - - - - - .60s
- Miserable - - - - - - .50s
- Unacceptable - - - below .50

Look at the KMO and Bartletts test

- Bartletts test of sphericity is significant i.e.

null hypothesis that correlation matrix is an

identity is rejected. - Kaiser-Meyer-Olkin measure of sampling adequacy

is gt0.8, meritorious. - Factor analysis is

appropriate

Look at the Initial Communalities

- They sum to
- We have eliminated 25 units of unique

variance.

Iterate!

- Using the estimated communalities, obtain a

solution. - Take the communalities from the first solution

and insert them into the main diagonal of the

correlation matrix. - Solve again.
- Take communalities from this second solution and

insert into correlation matrix.

Solve again

- Repeat this, over and over, until the changes in

communalities from one iteration to the next are

trivial. - Our final communalities sum to .
- After excluding units of unique variance, we

have extracted units of common variance. - That is / 25 of the total variance in our

25 variables.

How many factor to retain?

Criteria For Retention Of Factors

- Eigenvalue greater than 1
- Single variable has variance equal to 1
- Plot of total variance - Scree plot
- Gradual trailing off of variance accounted for is

called the scree. - Note cumulative of variance of rotated factors

We have packaged those 58.05 into 4 factors

Before rotation

Rotated factor loading

Rotation produces

- Factor Pattern Matrix
- High low factor loadings are more apparent
- generally used for interpretation
- Factor Structure Matrix
- correlations between factors and variables

Interpretation of Rotated Matrix

- Loadings of .40 or higher
- Name each factor based on 3 or 4 variables with

highest loadings. - Do not expect perfect conceptual fit of all

variables.

- SPSS will not only give you the scoring

coefficients, but also compute the estimated

factor scores for you. - In the Factor Analysis window, click Scores and

select Save As Variables, Regression, Display

Factor Score Coefficient Matrix.

Here are the scoring coefficients.Look back at

the data sheet and you will see the estimated

factor scores.

Use the Factor Scores

- In multiple regression
- independent t to compare groups on mean factor

scores. - Or even in ANOVA

Required Number of Subjects and Variables

- Rules of Thumb (not very useful)
- 100 or more subjects.
- at least 10 times as many subjects as you have

variables. - as many subjects as you can, the more the better.

- Start out with at least 6 variables per expected

factor. - Each factor should have at least 3 variables that

load well. - If loadings are low, need at least 10 variables

per factor. - Need at least as many subjects as variables. The

more of each, the better. - When there are overlapping factors (variables

loading well on more than one factor), need more

subjects than when structure is simple.

- If communalities are low, need more subjects.
- If communalities are high (gt .6), you can get by

with fewer than 100 subjects. - With moderate communalities (.5), need 100-200

subjects. - With low communalities and only 3-4 high loadings

per factor, need over 300 subjects. - With low communalities and poorly defined

factors, need over 500 subjects.

What I Have Not Covered Today

- LOTS.
- For a general introduction to measurement

(reliability and validity), see

http//core.ecu.edu/psyc/wuenschk/docs2210/Researc

h-3-Measurement.doc

Multivariate Analysis Summary

- Multivariate analysis is hard, but useful if it

is important to extract as much information from

the data as possible. - For classification problems, the common methods

provide different approximations to the Bayes

discriminant. - There is considerably empirical evidence that, as

yet, no uniformly most powerful method exists.

Therefore, be wary of claims to the contrary!

Further reading

- Hair, Anderson, Tatham Black, (HATB)

Multivariate Data Analysis, 5th edn

Thats All Friends

- See U Again
- Some Other Days
- Have a Nice Time With SPSS

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