Structural Equation Modeling

And Related Techniques

Handout 5

Factor Analysis Hands on

Michael Biderman

Department of Psychology

University of Tennessee at Chattanooga

Factor analysis of Faked Goldberg scale testlets

The file testletdata040510 contains 15 Faking conditions testlets from the dataset described in the previous handouts.

Exercise 1. Perform an EFA of the Faking testlets, FETL1, FETL2, . . . FOTL3. Don’t include the FJTL testlets, those are not personality testlets.

Suggestions, requests Analyze -> Reduction -> Factor ...

fetl1

fetl2

fetl3

fatl1

fatl2

etc.

etc.

Check the Eigenvalues over 1 box to let SPSS identify the number of factors.

Recall that we did that in the EFA of the Honest testlets, and we obtained 5 factors with eigenvalues >= 1.
Exercise 1 Suggestions continued

Request an oblique (correlated factors) solution, since we already know the factors are correlated.

Output you should see . . .

Some of the Agreeableness (FATL1, FATL2, FATL3) and Openness (FOTL1, FOTL2, FOTL3) testlets had pretty low communalities.

Exercise 1 likely ouput continued

This output definitely does NOT suggest that there are 5 factors.

Inspection of eigenvalues > 1 suggest only three.

The scree plot suggests only 1 factor, perhaps 3, definitely not 5.

Exercise 1 likely output continued

Since the program automatically retained only 3 factors, the pattern matrix above doesn’t really make any sense.

Exercise 2. Redo this EFA, requesting 5 factors.

Some of the output you should see . . .

Note that the communality estimates were generally larger in this 5 factor solution.

Five factors should account for more of the variance than three.

Exercise 2 Output: Faking testlets, 5 factor oblique EFA

The pattern of loadings onto 5 factors was generally what we would expect.

Exercise 2 Factor Correlations

The factors based on the Faked testlets were MUCH more highly correlated than those based on the Honest testlets. Can you guess why?

Exercise 3. Perform a CFA on the Faked testlets.

Download the partially complete AMOS model. Connect it with the testletdata040510 file and complete it.

The model you download should look like the following . . .

Add the appropriate curved arrows.

Fix values of appropriate single-headed arrows.

Exercise 3 Output: Faked Condition CFA: Standardized view

Your model should look like the following . . .

The loadings are OK, as are the reliabilities of most of the testlets.

The goodness of fit isn’t terrible. The RMSEA is nearly “good”.

But look at those factor correlations!! Yikes!

What’s going on?

The Faking Factor

Summary of the situation:

The EFA doesn’t suggest 5 factors, even though we know that there should be some indication of 5 factors, since we’re dealing with the Big 5 personality dimensions.

The CFA gives only a fairly good 5 factor solution, although the correlations between the supposedly orthogonal Big 5 dimensions are HUGE.

One hypothesis that answers the above question is that there is a 6^th factor affecting all the testlets – a faking factor.

Participants were instructed to fake every item. Some were good fakers. Others were not. The good fakers had higher testlet scores than the poor fakers. These differences in ability to fake affected all items and caused the scores on all the testlets to be (much) more highly correlated than they should have been due simply to the personality dimensions.

The individual differences in ability to fake represent a 6^th influence on the testlet scores, over and above the influences of the 5 personality dimensions.

Note that this 6^th factor influenced ALL the testlets, not just 3 as was the case with each of the personality dimensions.

The interesting aspect of this situation is that the suggestions of a 6^th factor are very oblique.

The only indication from the EFA was the predominance of the 1^st factor in the scree plot, suggesting not 6 factors, but 1.

The only indications in the CFA are the generally high correlations among the Big 5 factors.

Basically the CFA results suggest that there are correlations among the testlets from different dimensions that can’t be accounted for unless we assume that there are very high correlations among the dimension factors.

Inspection of the modification indices gives no hint either. There is not yet a “add-a-factor-affecting-all-observed-variables” modification index available in AMOS or any other SEM program.

Exercise 4: Add a faking factor to the model.

Your input model should look like this

All that remains is to fix one of the Faking Ability regression weights. Fix the one to FETL1 and run the model.

The Dark Side of CFA (and SEM)

Setting the regression of FETL1 onto Faking Ability to 1 results in a model that “doesn’t converge”. Argh!!

The reason is that the estimates of parameters are obtained using iterative “hill climbing” methods.

For some models, there is no “top” of the hill – all the ground is level, and the program probably would iterate forever.

Exercise 5. Experiment by setting different regression arrows from Faking Ability to the various testlets to 1. Set only one at a time. Record the result from each. If the model converges, write the chi-square value.

Testlet Result. Include Chi-square if it’s printed.

Failure to converge.

FETL1 _____________________________________

FETL2 _____________________________________

FETL3 _____________________________________

FATL1 _____________________________________

FATL2 _____________________________________

FATL3 _____________________________________

FCTL1 _____________________________________

FCTL2 _____________________________________

FCTL3 _____________________________________

FSTL1 _____________________________________

FSTL2 _____________________________________

FSTL3 _____________________________________

FOTL1 _____________________________________

FOTL2 _____________________________________

FOTL3 _____________________________________

Exercise 4: One of the successful solutions (FA -> FETL3 = 1.)

Note 1)the chi-square is nearly “not significant”

2)the RMSEA value is in the “good fit” range.

3)the Faking ability factor loads positively on all testlets.

4)the correlations between the factors are much smaller than they were without the faking factor.

Life is good!

The moral of this story: Don’t give up. Try everything you can think of to get a usable solution.

Exercise 6. Factor Analysis of Caldwell Mor Barak scale

Perform an EFA on the Caldwell diversity items.

The file is caldwellnm040516. It’ll have to be downloaded.

The questionnaire was responded to by 196 female African American managers who accessed it through a web site.

The items in each scale are

Perception of Organizational Fairness

OFF1R 13 Reverse of - I feel I have been treated differently here

because of my race.

OFF2 14 Managers here have a track record of hiring and promoting em

OFF3 15 Managers here give feedback and evaluate employees fairly, r

OFF4 16 Manager's here make layoff decisions fairly, regardless of f

OFF5 17 Managers interpret human resource policies (such as sick lea

OFF6 18 Managers here give assignments based on the skills and abili

Perception of Organizational Inclusion

OIF1 19 Management here encourages the formation of employee network

OIF2 20 There is a mentoring program in use here that identifies and

OIF3 21 The old boys' network is alive and well here.

OIF3R 22 Reverse of - The old boys' network is alive and well here.

OIF4 23 The company spends enough money and time on diversity awaren

Personal Value for Diversity

PDVF1 24 Knowing more about cultural norms of diverse groups would he

PDVF2 25 I think that diverse viewpoints add value.

PDVF3 26 I believe diversity is a strategic business issue.

Personal Comfort with Diversity

PCF1 27 I feel at ease with people from backgrounds other than my ow

PCF2 28 I am afraid to disagree with members of other groups for fea

PCF2R 29 Reverse of - I am afraid to disagree with members of other g

PCF3 30 Diversity issues keep some work teams here from performing t

PCF3R 31 Reverse of - Diversity issues keep some work teams here from

_

See if an EFA gives the same factors. Be sure to investigate an oblique-factors solution.

Exercise 6: Some output you should get

There are 4 eigenvalues >= 1, which is as it should be given the knowledge of the items.

But the scree plot doesn’t really indicate 4 factors.

Exercise 6 output

Exercise 7: CFA on same data

Perform a CFA of the Caldwellnm data using the results from the EFA to guide your choice of loadings.

Use the CaldwellObliqueCFAStarter.amw file as a starting point. It should look like the following . . .

Note that in this file, the residual variances are set equal to 1, rather than the residual regression arrows. This was done on orders from the dark side.

Exercise 7: Your result should look something like the following

Note the correlations between error terms. These may represent specific similarities between items in working or content. The “across factor” loadings were suggested by modification indices.

Entering Summary Data for Amos

Since the analyses in Amos are based on summary statistics - the variances and covariances between the variables - only the variances and covariances need be entered.

They can be entered

1) as variances and covariances along with means or

2) as correlations along with means and standard deviations.

They can be entered into SPSS, into Excel, into Access, or even Word.

Example from SPSS (Data from AMOS 4.0 manual)

Example from Excel (Data from Barrick, et. al. (2002) JAP.

Rules For Names of columns in the data file.

A. First column’s name is rowtype_ (Underscore is important!!)

B. Second column’s name is varname_ (Again, the underscore!!)

C. 3^rd and subsequent columns.

The names of these columns are the names of the variables.

II. Rows of the data file

A. Row 1: Contains the letter, n, in column 1.

Contains nothing in column 2.

Contains sample size in subsequent columns.

B. Row 2 through K+1, where K is the number of variables:

Column 1 contains either “corr” without the quotes or “cov” dependent on whether the entries are correlations or covariances.

Column 2 contains the variable names, in same order as listed across the top.

Columns 3 through K+1 contain correlations or covariances, depending on what you have, until the diagonal of the matrix.

C. Row K+2

Contains the word, stddev, in column 1, nothing in column 2, and standard deviations in columns 3 through K+2.

D. Row K+3

Contains the word, mean, in column 1, nothing in column 2, and means in columns 3 through K+3.

Analyzing Correlations

If you want to analyze correlations, and enter 1 for each standard deviation and 0 for each mean.