Chapter 21 – Factor analysis

Exercises

Exercise 21.1

In this chapter we carried out an Exploratory Factor Analysis on the data in the file FAdata.sav Now load up the file FAdata.sav and have a go at carrying out a Principal Components Analysis instead. The exercise, apart from getting factor analysis practice, is to inspect the SPSS Output files and see what differences there are between the EFA and PCA results.

To carry out the PCA do the following:

In SPSS proceed exactly as for stage 1 of the EFA analysis except:

When you select the Extraction button select Principal Components Analysis from the top drop down box after Method.

Select Rotation and make sure None is selected. Since we do in fact know the number of factors we will select you could rotate at this point, but usually you would not know the number of factors required.

Stage 2 would involve exactly the same decision process as for EFA and so we will extract four factors.

For stage 3 again, the steps are exactly the same but ensure that Principal Components Analysis is still selected in the Extraction box.

Now examine the two sets of results for differences before moving on to the answers below.

Answers

The first thing to note is that many tables stay the same in PCA as they were in EFA, including the first two – the Correlation Matrix and the KMO and Bartlett’s test results. This is expected since they deal with exactly the same data in exactly the same way. However, in the Communalities table the values for the initial solution in PCA are all 1. Remember that PCA explains all the variance across items and not just shared variance. Hence, since a communality is the amount of variation in an item explained by all the identified factors then this amount as a proportion will be 100% or simply 1 when other amounts of variance are expressed as decimal parts of 1.

Next note that below the scree plot, instead of a Factor Matrix there is now a Component Matrix. In this latter table note that there are more loadings that have exceeded the criterion value we set of .3 and that in almost all cases the loadings here are larger than those in the EFA Factor Matrix. Again this is because PCA incorporates all sources of variance into the analysis.

Moving to the all important Pattern Matrix tables we find that the two tables generally agree with one another but with some exceptions (the EFA table is in the book, p.xxx, and the PCA table appears below. The arts oriented creativity Factor 1 loads very highly on the creative item, more than for the equivalent PCA component, and relatively highly  on visual and verbal creativity. The trivial loading for problem solving can be ignored and in the PCA table this item does not reach our criterion. However in the PCA table science creativity cross loads on components 1 and 2 and for component 1 the loading is not trivial being close to .4

Hence this item looks decidedly weak and ambiguous. The items included in factor 2 and component 2 are identical apart from emotionally aware which creeps into the PCA table with another trivial loading of .214 whereas, and as would be expected, this item loads much higher on component  3. Factor 3 (possibly related to emotional intelligence) and Component 3 also agree on included items except for the fact that creative is also included in Component 3 but again with a tiny loading just above the set criterion. The anger and impulsiveness related Factor 4 and Component 4 are again comparable apart from the inclusion of a small loading on creative for the latter.

Whilst inspecting and interpreting these tables it is worth remembering again that while the EFA table theoretically points to latent variables that are causal in creating the factors produced through EFA, the components in the PCA table are really constituents in an overall parsimonious description of the original items.

Pattern Matrixa

Component

1		2	3	4
SRverbal_creativity	.790	-.259
SRcreative	.712		.213	.232
SRvisual_crerativity	.585			.489
SRintelligent		-.868
SRproblem_solving		-.793
SRknowledge		-.689
SRscience_creativity	-.396	-.625
SRemotions_recognition_of_ot hers			.825
SRemotionally_aware		-.214	.748
SRemotionally_expressive			.743	.322
SRanger				.792
SRimpulsive				.690

Extraction Method: Principal Component Analysis. Rotation Method: Oblimin with Kaiser Normalization.

a. Rotation converged in 10 iterations.

Exercise 21.2

Have a go at this short quiz to test your understanding of factor analysis and identify any gaps in your knowledge.

---

Weblinks

Factor analysis

The Analysis Factor explains FA in relatively simple terms and a few other related issues like the difference between EFA and PCA.

Factor Analysis: A Short Introduction, Part 1 (theanalysisfactor.com)

An article on FA by the researchers who kindly gave permission to use their data for Chapter 21.

(PDF) Test Development (researchgate.net)

Site to download a copy of the MonteCarlo software, free at the time of writing and also safe.

Thank you for downloading Monte Carlo PCA for Parallel Analysis from CNET Download.com