EFA MANUAL.doc

7/28/2019 EFA MANUAL.doc

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Lab 9a -- Factor Analysis

Factor Analysis

SPSS Steps

Output -- Principal Components

Output -- Common Factor Analysis Interpretation

The area of Factor Analysis is, in many ways, diffuse and large. We will consider

only a portion of the subject. Factor Analysis is generally used to find parsimony

among several variables. The quest is usually for underlying constructs

orfactors that each explain the variation among several variables. We will consider

two mathematical models in this lab, though there are many more, and variants of

these.

SPSS Steps

Enter the variable names in an SPSS spreadsheet as usual in the "Variable View."Although it is possible, under some circumstances to use categorical variables in

factor analysis, you should use only ordinal, interval, or ratio scaled variables in this

lab. In the "Data View" enter the values for the variables for each case. Remember a

case is a row and a variable is a column. Also remember to save your data.

For the factor analysis lab, you will need a minimum of 6 variables with about 30

subjects. This is not enough for a serious factor analysis, but will be enough for your

lab. The example herein is one in which we use eight biometric

measures: height, armspan, forearem, lowerleg, weight, diameter, girth,

and width. The purpose is to attempt to represent these eight variables by a smallernumber of dimensions, or factors. This file, "PCPADEMO.sav" is included in this

folder.

To run the factor analysis lab go to the top of the spreadsheet and click on

Analyze/Data Reduction/Factor. A dialogue box will appear. Highlight the

variables in the variable list on the left that you wish to factor analyze and move them

out of the variable list on the left into the Variables: box or the right by clicking on

the right arrow between the boxes.

Now click on the Descriptives button. In the Descriptives dialog box that appears

check Univariate descriptives, Coefficients, Significance levels, and KMO

and Bartletts test of sphericity." Using the example data set, your screen should look

something like this:
http://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Factor%20Analysishttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#SPSS%20Stepshttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Output%20--%20Principal%20Componentshttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Output%20--%20Common%20Factor%20Analysishttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Interpretationhttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#SPSS%20Stepshttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Output%20--%20Principal%20Componentshttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Output%20--%20Common%20Factor%20Analysishttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Interpretationhttp://www.coe.fau.edu/faculty/morris/sta7114%20files/lab%209/9a%20instructions/factor_analysis_page.htm#Factor%20Analysis


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Now click the Extraction button in the "Factor Analysis" box. In the Extraction

dialog box (below), all you need to do is to click Scree plot. Before you click

Continue however, note that in this dialog box you can set the minimum eigenvalue

to retain (SPSS uses "Kaiser's Rule" of larger than 1.00 if you do not select another

minimum), and you can also specify the exact number of factors to retain regardless

of the eigenvalues. These are features that I leave for your experimentation. In

addition, note that there is the possibility of selecting different "Methods:" of factor

analysis here. The default method, highlighted in the image below is "Principal

components." As that is, by far, the most frequently used method, start with it.

When you click on the down arrow, you will see that there are many possible

methods. I also include the Principal Axis results (obtained by exactly the same steps

as the Principal Component results, except that the Principal Axis method was

selected at this step) which is a Common Factor method for contrast. Now click

"Continue" in the "Extraction" box and it disappears.

Now click the Rotation button in the Factor Analysis box. In this Rotation

dialog box (below) click Varimax and "Loading plot(s)." Again, before you

click the "Continue" button, note that a variety of rotations are offered. Now click

the Continue button in the Rotation box and it disappears..


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Now you are ready to click the OK button in the Factor Analysis box, and the

analysis will run.

The output file will appear, and for this example both the Principal Components

output and Principal Axis (Common Factor) output are below.

Principal Components -- Factor Analysis

Descriptive Statistics

Mean Std. Deviation Analysis N

HEIGHT 72.0000 20.03285 305

ARMSPAN 60.0000 15.02465 305

FOREARM 14.0000 5.00822 305

LOWERLEG 18.0000 10.01647 305

WEIGHT 180.0000 30.04926 305

DIAMETER 18.0000 7.01152 305

GIRTH 18.0000 7.01151 305

WIDTH 20.0000 7.01152 305

Correlation Matrix

HEIGH ARMSPA FOREAR LOWERLE WEIGH DIAMETE GIRT WIDT


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T N M G T R H H

Correlatio

n

HEIGHT 1.000 .846 .805 .859 .473 .398 .301 .382

ARMSPAN .846 1.000 .881 .826 .376 .326 .277 .415

FOREARM .805 .881 1.000 .801 .380 .319 .237 .345

LOWERLE

G.859 .826 .801 1.000 .436 .329 .327 .365

WEIGHT .473 .376 .380 .436 1.000 .762 .730 .629

DIAMETE

R.398 .326 .319 .329 .762 1.000 .583 .577

GIRTH .301 .277 .237 .327 .730 .583 1.000 .539

WIDTH .382 .415 .345 .365 .629 .577 .539 1.000

Sig. (1-

tailed)

HEIGHT .000 .000 .000 .000 .000 .000 .000

ARMSPAN .000 .000 .000 .000 .000 .000 .000

FOREARM .000 .000 .000 .000 .000 .000 .000

LOWERLE

G.000 .000 .000 .000 .000 .000 .000

WEIGHT .000 .000 .000 .000 .000 .000 .000

DIAMETE

R.000 .000 .000 .000 .000 .000 .000

GIRTH .000 .000 .000 .000 .000 .000 .000

WIDTH .000 .000 .000 .000 .000 .000 .000

KMO and Bartlett's Test

Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .845

Bartlett's Test of Sphericity

Approx. Chi-Square 2085.738

df 28

Sig. .000

Communalities

Initial Extraction

HEIGHT 1.000 .877


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ARMSPAN 1.000 .903

FOREARM 1.000 .872

LOWERLEG 1.000 .861

WEIGHT 1.000 .850

DIAMETER 1.000 .739

GIRTH 1.000 .717

WIDTH 1.000 .625

Extraction Method: Principal Component Analysis.

Total Variance Explained

Initial Eigenvalues Extraction Sums of SquaredLoadings Rotation Sums of SquaredLoadings

Component Total% of

Variance

Cumulative

%Total

% of

Variance

Cumulative

%Total

% of

Variance

Cumulative

%

1 4.673 58.411 58.411 4.673 58.411 58.411 3.497 43.717 43.717

2 1.771 22.137 80.548 1.771 22.137 80.548 2.947 36.832 80.548

3 .481 6.013 86.561

4 .421 5.268 91.829

5 .233 2.915 94.744

6 .187 2.333 97.078

7 .137 1.716 98.794

89.646E-

021.206 100.000



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Component Matrix(a)

Component

1 2

HEIGHT .859 -.372

ARMSPAN .842 -.441

FOREARM .813 -.459

LOWERLEG .840 -.395

WEIGHT .758 .525

DIAMETER .674 .533

GIRTH .617 .580

WIDTH .671 .418


a 2 components extracted.

Rotated Component Matrix(a)


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Component

1 2

HEIGHT .900 .260

ARMSPAN .930 .195

FOREARM .919 .164

LOWERLEG .899 .229

WEIGHT .251 .887

DIAMETER .181 .840

GIRTH .107 .840

WIDTH .251 .750


Rotation Method: Varimax with Kaiser Normalization.

a Rotation converged in 3 iterations.

Component Transformation Matrix

Component 1 2

1 .771 .636

2 -.636 .771



_____________________________________________________________________

_________________________

Common Factor Analysis

Communalities

Initial Extraction


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HEIGHT .816 .838

ARMSPAN .849 .889

FOREARM .801 .821

LOWERLEG .788 .808

WEIGHT .749 .888

DIAMETER .604 .640

GIRTH .562 .583

WIDTH .478 .492

Extraction Method: Principal Axis Factoring.

Total Variance Explained

Initial EigenvaluesExtraction Sums of Squared

Loadings

Rotation Sums of Squared

Loadings

Factor Total% of

Variance

Cumulative

%Total

% of

Variance

Cumulative

%Total

% of

Variance

Cumulative

%

1 4.673 58.411 58.411 4.449 55.611 55.611 3.315 41.438 41.438

2 1.771 22.137 80.548 1.510 18.875 74.486 2.644 33.049 74.486

3 .481 6.013 86.561

4 .421 5.268 91.829

5 .233 2.915 94.744

6 .187 2.333 97.078

7 .137 1.716 98.794

89.646E-

021.206 100.000


Factor Matrix(a)

Factor

1 2

HEIGHT .856 -.324

ARMSPAN .848 -.411

FOREARM .808 -.409


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LOWERLEG .831 -.342

WEIGHT .750 .571

DIAMETER .631 .492

GIRTH .569 .510

WIDTH .607 .351


a 2 factors extracted. 9 iterations required.

Rotated Factor Matrix(a)

Factor

1 2HEIGHT .872 .278

ARMSPAN .920 .204

FOREARM .887 .182

LOWERLEG .864 .248

WEIGHT .233 .913

DIAMETER .188 .778

GIRTH .129 .753

WIDTH .258 .652



a Rotation converged in 3 iterations.

Interpretation

The descriptive information shows the means and standard deviations for all of the

eight variables, as well as all possible bivariate correlations and their p values. We

note that all of the correlations are positive and significant as might be expected of

these variables.

Barlett's test of spericity is significant, thus the hypothesis that the intercorrelation

matrix involving these eight variables is an identity matrix is rejected. Thus from the

perspective of Bartlett's test, factor analysis is feasible. As Bartlett's test is almost

always significant, a more discriminating index of factor analyzability is the KMO.


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For this data set, it is .845, which is very large, so the KMO also supports factor

analysis.

Kaiser's rule of retaining factors with eigenvalues larger than 1.00 was used in this

analysis as the default. As the eigenvalues for the first two principal components (no

distinction is made in deciding dimensionality by SPSS in the principal component

and common factor analysis) with eigenvalues of 4.673 and 1.771 were retained.

The Principal Component communalities (Extraction, as the Initial are always 1.00)

range from .625 to .903, thus most of the variance of these variables was accounted

for by this two dimensional factor solution. One can see that the corresponding

Extraction communalities for the Common Factor analysis were a bit smaller (as

would be expected) but still show the majority of the variance of all variables

represented in the two factor solution. Note that the "Initial" communality estimates

for the SPSS version of a Principal Axis Common Factor Analysis are the R2 s

predicting each of the variables from all other variables -- a usual choice.

Also note the Scree Plot in the Principal Components output (the same thing is

produced in the Common Factor Analysis). The Scree Plot is a graphic aid proposed

by Cattell. It is simply a plot of the monotonically descending eigenvalues. It is

intended to help in deciding where a the "trivial" dimensions begin. One might argue

that the Kaiser Rule opting for two dimensions is fairly well supported by the Scree

Plot.

In the Principal Components Output, the Rotated Component Matrix gives thecorrelation of each variable with each factor. From the contribution of the variables

(also called a "loading") we can name these factors something like "Lankiness" and

"Heaviness." One might come up with a variety of other names that are equally

descriptive. You will note that the results of the Common Factor analysis are much

the same with loadings that are a bit smaller. One might argue that the two methods,

therefore, give the same result. However, that would be dangerous as it depends on

the number of variables, their communalities, and also we are restricting the results to

the same dimensionality in this case.

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