Principal Components Analysis with - PowerPoint Presentation

Slide1

Principal Components Analysis with SPSS

Karl L. Wuensch

Dept of Psychology

East Carolina UniversitySlide2

When to Use PCA

You have a set of

p

continuous variables.

You want to repackage their variance into

m

components.

You will usually want

m

to be <

p

, but not always.Slide3

Components and Variables

Each component is a weighted linear combination of the variables

Each variable is a weighted linear combination of the components.Slide4

Factors and Variables

In Factor Analysis, we exclude from the solution any variance that is unique, not shared by the variables.

U

j

is the unique variance for

X

jSlide5

Goals of PCA and FA

Data reduction.

Discover and summarize pattern of intercorrelations among variables.

Test theory about the latent variables underlying a set a measurement variables.

Construct a test instrument.

There are many others uses of PCA and FA.Slide6

Data Reduction

Ossenkopp and Mazmanian (

Physiology and Behavior

,

34

: 935-941).19 behavioral and physiological variables.A single criterion variable, physiological response to four hours of cold-restraint

Extracted five factors.Used multiple regression to develop a model for predicting the criterion from the five factors.Slide7

Exploratory Factor Analysis

Want to discover the pattern of intercorrleations among variables.

Wilt et al., 2005 (thesis).

Variables are items on the SOIS at ECU.

Found two factors, one evaluative, one on difficulty of course.

Compared FTF students to DE students, on structure and means.Slide8

Confirmatory Factor Analysis

Have a theory regarding the factor structure for a set of variables.

Want to confirm that the theory describes the observed intercorrelations well.

Thurstone: Intelligence consists of seven independent factors rather than one global factor.

Often done with SEM software Slide9

Construct A Test Instrument

Write a large set of items designed to test the constructs of interest.

Administer the survey to a sample of persons from the target population.

Use FA to help select those items that will be used to measure each of the constructs of interest.

Use

Cronbach

alpha to check reliability of resulting scales. Slide10

An Unusual Use of PCA

Poulson, Braithwaite, Brondino, and Wuensch (1997,

Journal of Social Behavior and Personality

,

12

, 743-758). Simulated jury trial, seemingly insane defendant killed a man.

Criterion variable = recommended verdictGuiltyGuilty But Mentally IllNot Guilty By Reason of Insanity.Slide11

Predictor variables = jurors’ scores on 8 scales.Discriminant function analysis.

Problem with multicollinearity.

Used PCA to extract eight orthogonal components.

Predicted recommended verdict from these 8 components.

Transformed results back to the original scales.Slide12

A Simple, Contrived Example

Consumers rate importance of seven characteristics of beer.

low Cost

high Size of bottle

high Alcohol content

Reputation of brand

ColorAromaTasteSlide13

FACTBEER.SAV at http://core.ecu.edu/psyc/wuenschk/SPSS/SPSS-Data.htm

.

Analyze, Data Reduction, Factor.

Scoot beer variables into box.Slide14

Click Descriptives and then check Initial Solution, Coefficients, KMO and Bartlett’s Test of Sphericity, and Anti-image. Click Continue. Slide15

Click Extraction and then select Principal Components, Correlation Matrix, Unrotated Factor Solution, Scree Plot, and Eigenvalues Over 1. Click Continue. Slide16

Click Rotation. Select Varimax and Rotated Solution. Click Continue. Slide17

Click Options. Select Exclude Cases

Listwise

and Sorted By Size. Click Continue.

Click OK, and SPSS completes the Principal Components Analysis.Slide18

Checking for Unique Variables 1

Check the correlation matrix.

If there are any variables not well correlated with some others, might as well delete them.Slide19

Checking for Unique Variables 2

Correlation Matrix

cost size alcohol reputat color aroma taste

cost 1.00 .832 .767 -.406 .018 -.046 -.064

size .832 1.00 .904 -.392 .179 .098 .026

alcohol .767 .904 1.00 -.463 .072 .044 .012 reputat -.406 -.392 -.463 1.00 -.372 -.443 -.443 color .018 .179 .072 -.372 1.00 .909 .903

aroma -.046 .098 .044 -.443 .909 1.00 .870 taste -.064 .026 .012 -.443 .903 .870 1.00 Slide20

Checking for Unique Variables 3

Bartlett’s test of sphericity tests null that the matrix is an identity matrix, but does not help identify individual variables that are not well correlated with others.Slide21

Checking for Unique Variables 4

For each variable, check

R

2

between it and the remaining variables.

SPSS reports these as the

initial communalities whenyou do a principal axisfactor analysisDelete any variable with alow R2 .Slide22

Checking for Unique Correlations

Look at partial correlations – pairs of variables with large partial correlations share variance with one another but not with the remaining variables – this is problematic.

Kaiser’s

MSA

will tell you, for each variable, how much of this problem exists.

The smaller the

MSA, the greater the problem.Slide23

Checking for Unique Correlations 2

An

MSA

> .9 is marvelous, .8 meritorious. .7 middling, .6 mediocre, and .5 miserable.

Variables with small MSAs should be deleted

Or additional variables added that will share variance with the troublesome variables.Slide24

Checking for Unique Correlations 3

Anti-image Matrices

cost

size

alcohol

reputat

color

aroma

taste

Anti-image

Correlation

cost

.779

a

-.543

.105

.256

.100

.135

-.105

size

-.543

.550

a

-.806

-.109

-.495

.061

.435

alcohol

.105

-.806

.630

a

.226

.381

-.060

-.310

reputat

.256

-.109

.226

.763

a

-.231

.287

.257

color

.100

-.495

.381

-.231

.590

a

-.574

-.693

aroma

.135

.061

-.060

.287

-.574

.801

a

-.087

taste

-.105

.435

-.310

.257

-.693

-.087

.676

a

a. Measures of Sampling Adequacy (MSA) on main diagonal. Off diagonal are partial correlations x -1.Slide25

Extracting Principal Components 1

From

p

variables we can extract

p

components.Each of

p eigenvalues represents the amount of standardized variance that has been captured by one component.The first component accounts for the largest possible amount of variance.The second captures as much as possible of what is left over, and so on.Each is orthogonal to the others. Slide26

Extracting Principal Components 2

Each variable has standardized variance = 1.

The total standardized variance in the

p

variables =

p

.The sum of the m = p eigenvalues = p.All of the variance is extracted.For each component, the proportion of variance extracted = eigenvalue / p.Slide27

Extracting Principal Components 3

For our beer data, here are the eigenvalues and proportions of variance for the seven components:Slide28

How Many Components to Retain

From

p

variables we can extract

p

components.We probably want fewer than p

.Simple rule: Keep as many as have eigenvalues  1.A component with eigenvalue < 1 captured less than one variable’s worth of variance.Slide29

Visual Aid: Use a Scree PlotScree is rubble at base of cliff.

For our beer data,Slide30

Only the first two components have eigenvalues greater than 1.Big drop in eigenvalue between component 2 and component 3.

Components 3-7 are scree.

Try a 2 component solution.

Should also look at solution with one fewer and with one more component. Slide31

Less Subjective Methods

Parallel Analysis and

Velcier’s

MAP test.

SAS, SPSS,

Matlab scripts available at

https://people.ok.ubc.ca/brioconn/nfactors/nfactors.html Slide32

Parallel Analysis

How many components account for more variance than do components derived from random data?

Create 1,000 or more sets of random data.

Each with same number of cases

and variables

as your data set.For each set, find the eigenvalues.Slide33

For the eigenvalues from the random sets, find the 95

th

percentile for each component.

Retain as many components for which the eigenvalue from your data exceeds the 95

th

percentile from the random data sets.Slide34

Random Data Eigenvalues

Root

Prcntyle

1.000000 1.344920

2.000000 1.207526

3.000000 1.118462

4.000000 1.038794

5.000000 .973311 6.000000 .907173 7.000000 .830506

Our data yielded eigenvalues of 3.313, 2.616, and 0.575.Retain two componentsSlide35

Velicer’s MAP Test

Step by step, extract increasing numbers of components.

At each step, determine how much common variance is left in the residuals.

Retain all steps up to and including that producing the smallest residual common variance.Slide36

Velicer's Minimum Average Partial (MAP) Test:

Velicer's

Average Squared Correlations

.000000 .266624

1.000000 .440869

2.000000 .129252 3.000000 .170272 4.000000 .331686 5.000000 .486046 6.000000 1.000000 

The smallest average squared correlation is .129252 The number of components is 2Slide37

Which Test to Use?

Parallel analysis tends to

overextract

.

MAP tends to

underextract.If they disagree, increase number of random sets in the parallel analysis

And inspect carefully the two smallest values from the MAP test.May need apply the meaningfulness criterion.Slide38

Loadings, Unrotated

and Rotated

loading matrix = factor pattern matrix = component matrix.

Each loading is the Pearson

r

between one variable and one component.

Since the components are orthogonal, each loading is also a β weight from predicting X from the components.Here are the unrotated loadings for our 2 component solution: Slide39

All variables load well on first component, economy and quality vs. reputation.

Second component is more interesting, economy versus quality.Slide40

Rotate these axes so that the two dimensions pass more nearly through the two major clusters (COST, SIZE, ALCH and COLOR, AROMA, TASTE).

The number of degrees by which I rotate the axes is the angle

PSI

. For these data, rotating the axes -40.63 degrees has the desired effect. Slide41

Component 1 = Quality versus reputation.Component 2 = Economy (or cheap drunk) versus reputation.Slide42

Number of Components in the Rotated Solution

Try extracting one fewer component, try one more component.

Which produces the more sensible solution?

Error = difference in obtained structure and true structure.

Overextraction (too many components) produces less error than underextraction.

If there is only one true factor and no unique variables, can get “factor splitting.”Slide43

In this case, first unrotated factor  true factor.

But rotation splits the factor, producing an imaginary second factor and corrupting the first.

Can avoid this problem by including a garbage variable that will be removed prior to the final solution.Slide44

Explained Variance

Square the loadings and then sum them across variables.

Get, for each component, the amount of variance explained.

Prior to rotation, these are eigenvalues.

Here are the SSL for our data, after rotation:Slide45

After rotation the two components together account for (3.02 + 2.91) / 7 = 85% of the total variance. Slide46

If the last component has a small SSL, one should consider dropping it.If SSL = 1, the component has extracted one variable’s worth of variance.

If only one variable loads well on a component, the component is not well defined.

If only two load well, it may be reliable, if the two variables are highly correlated with one another but not with other variables.Slide47

Naming Components

For each component, look at how it is correlated with the variables.

Try to name the construct represented by that factor.

If you cannot, perhaps you should try a different solution.

I have named our components “aesthetic quality” and “cheap drunk.”Slide48

Communalities

For each variable, sum the squared loadings across components.

This gives you the

R

2

for predicting the variable from the components,

which is the proportion of the variable’s variance which has been extracted by the components.Slide49

Here are the communalities for our beer data. “Initial” is with all 7 components, “Extraction” is for our 2 component solution.Slide50

Orthogonal Rotations

Varimax --

minimize the complexity of the components by making the large loadings larger and the small loadings smaller within each component.

Quartimax -- makes large loadings larger and small loadings smaller within each variable.

Equamax – a compromize between these two.Slide51

Oblique Rotations

Axes drawn through the two clusters in the upper right quadrant would not be perpendicular.Slide52

May better fit the data with axes that are not perpendicular, but at the cost of having components that are correlated with one another.

More on this later.