Factor Analysis with - PowerPoint Presentation

Slide1

Factor Analysis with SPSS

Karl L. Wuensch

Dept.

of Psychology

East Carolina UniversitySlide2

What is a Common Factor?It is an abstraction, a hypothetical construct

that

relates

to at least two of our measurement variables.

We want to estimate the common factors that contribute to the variance in our variables.

Is this an act of discovery or an act of invention?Slide3

What is a Unique Factor?

It is a factor that contributes to the variance in only one variable.

There is one unique factor for each variable.

The unique factors are unrelated to one another and unrelated to the common factors.

We want to exclude these unique factors from our solution.Slide4

Iterated Principal Factors AnalysisThe 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, 1’s with estimates of communalities.

Initial estimate of communality =

R

2

between one variable and all others.Slide5

Lets Do ItUsing the beer data, change the extraction method to principal axis.Slide6

Look at the Initial CommunalitiesThey were all 1’s for our PCA.

They sum to 5.675.

We have eliminated 7 – 5.675 = 1.325 units of unique variance.Slide7

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.Slide8

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 5.6.After excluding 1.4 units of unique variance, we have extracted 5.6 units of common variance.

That is 5.6 / 7 = 80% of the total variance in our seven variables.Slide9

We have packaged those 5.6 units of common variance into two factors:Slide10

Our Rotated Factor LoadingsNot much different from those for the PCA.Slide11

Reproduced and Residual Correlation Matrices

Correlations between variables result from their sharing common underlying factors.

Try to reproduce the original correlation matrix from the correlations between factors and variables (the loadings).

The difference between the reproduced correlation matrix and the original correlation matrix is the residual matrix.Slide12

We want these residuals to be small.Check “Reproduced” under “Descriptive” in the Factor Analysis dialogue box, to get both of these matrices: Slide13

Nonorthogonal (Oblique) Rotation

The axes will not be perpendicular, the factors will be correlated with one another.

the factor

loadings

(in the

pattern matrix

) will no longer be equal to the correlation between each factor and each variable.

They will still equal the beta weights, the A’s in Slide14

Promax rotation is available in SAS.First a Varimax

rotation is performed.

Then the axes are rotated obliquely.

Here are the beta weights, in the “Pattern Matrix,” the correlations in the “Structure Matrix,” and the

correlations

between factors:Slide15

Beta Weights CorrelationsSlide16

Exact Factor ScoresYou can compute, for each subject, estimated factor scores.

Multiply each standardized variable score by the corresponding standardized scoring coefficient.

For our first subject,

Factor 1 = (-.294)(.41) + (.955)(.40) + (-.036)(.22) + (1.057)(-.07) + (.712)(.04) + (1.219)(.03)

+ (-1.14)(.01) = 0.23

.Slide17

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. Slide18

Here are the scoring coefficients:Look back at the data sheet and you will see the estimated factor scores.Slide19

R2 of the Variables With Each Factor

These are treated as indicators of the internal consistency of the solution.

.70 and above is good.

They are in the main diagonal of this matrix

Factor Score Covariance Matrix

Factor 1 2

1 .

966

.003

2 .003

.953

Slide20

R2 of the Variables With Each Factor 2

These squared multiple correlation coefficients are equal to the variance of the factor scores. Slide21

Use the Factor ScoresLet us see how the factor scores are related to the SES and Group variables.

Use multiple regression to predict SES from the factor scores.Slide22
Slide23

Also, use independent t to compare groups on mean factor scores.Slide24

Unit-Weighted Factor ScoresDefine subscale 1 as simple sum or mean of scores on all items loading well (> .4) on Factor 1.

Likewise for Factor 2, etc.

Suzie Cue’s answers are

Color, Taste, Aroma, Size, Alcohol, Cost, Reputation

80, 100, 40, 30, 75, 60, 10

Aesthetic Quality = 80+100+40-10 = 210

Cheap Drunk = 30+75+60-10 = 155 Slide25

It may be better to use factor scoring coefficients (rather than loadings) to determine unit weights.Grice (2001) evaluated several techniques and found the best to be assigning a unit weight of

1 to each variable that has a scoring coefficient at least 1/3 as large as the largest for that factor.

Using this rule, we would not include Reputation on either subscale and would drop Cost from the second subscale.Slide26

Item Analysisand Cronbach’s

Alpha

Are our subscales reliable?

Test-Retest reliability

Cronbach’s Alpha – internal consistency

Mean split-half reliability

With correction for attenuation

Is a conservative estimate of reliabilitySlide27

AQ = Color + Taste + Aroma – ReputationMust negatively weight Reputation prior to item analysis.Transform, Compute,NegRep = -1

Reputat.Slide28

Analyze, Scale, Reliability AnalysisSlide29

StatisticsScale if item deleted.

Continue, OKSlide30

Shoot for an alpha of at least .70 for research instruments.Slide31

Note that deletion of the Reputation item would increase alpha to .96.Slide32

Comparing Two Groups’ Factor StructureEyeball Test

Same number of well defined factors in both groups?

Same variables load well on same factors in both groups?Slide33

Pearson rJust correlate the loadings for one factor in one group with those for the corresponding factor in the other group.If there are many small loadings,

r

may be large due to the factors being similar on small loadings despite lack of similarity on the larger loadings.Slide34

CC, Tucker’s coefficient of congruenceFollow the instructions in the document Comparing Two Groups’ Factor Structures: Pearson

r

and the Coefficient of

Congruence

CC of .85 to .94

corresponds

to similar factors, and .95 to 1 as essentially identical factors.Slide35

Cross-ScoringObtain scoring coefficients for each group.For each group, compute factor scores using coefficients obtained from the analysis for that same group (SG) and using coefficients obtained from the analysis for the other group (OG).

Correlate SG factor scores with OG factor scores. Slide36

Catell’s Salient Similarity IndexFactors (one from one group, one from the other group) are compared in terms of similarity of loadings.

Catell’s

Salient Similarity Index,

s

, can be transformed to a

p

value testing the null that the factors are not related to one another.

See my document

Cattell’s

s

for details.Slide37

Required Number of Subjects and VariablesRules 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.

It depends – see the references in the handout. Slide38

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.Slide39

If communalities are low, need more subjects.If communalities are high (> .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. Slide40

What I Have Not Covered TodayLOTS.

For a

brief introduction to reliability, validity, and scaling,

see

Document

or

Slideshow

.

For

an SAS version of this workshop, see

Document

or

Slideshow

.Slide41

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