Principal Component Analysis - Using 4 components to form single index

[Klaus Klausen]

Hi,

to gauge the psychological trait of narcissism among CEOs, I employ a 14-item measurement. The majority of papers applying this index combine these variables into a unified narcissism measurement using Principal Component Analysis. Despite the concerns many people have about PCA's overvalue in forming such indices, I'm considering it, potentially as a robustness test.

In one of the most cited paper, the authors detail a method to develop a CEO narcissism score:

The next step is to develop one overall CEO narcissism score, taking into account the balanced view and the broadness of the concept. For each of the four principal components, the factor loadings are multiplied by each individual CEO observation. The four standardized scores are added to form one CEO narcissism score. This CEO narcissism score ranges from 0 to 20, with a mean of 7.1 (SD = 2.1).

As I am not sure if the following commands replicate this approach, I would greatly appreciate your comments on that.

pca $CNS, components(4)  blanks(.2) //Extract four components
estat kmo //Check feasibility of data for pca
rotate, promax blanks(.2) //use oblique rotation
predict CNS, score //Form the index based on the four extracted components

If I'm not completely off track here, according to #5 in this thread, my approach should use the first components to form my overall narcissism index CNS, right?

Thank you!

[George Ford]

If you have 4 components, then I'd think you'd need them to be separate variables, as they are measuring different aspects of narcissism. Are more than one Eigenvalue greater than 1? It would be more interesting, to me at least, if all the responses collapsed to a single factor. If not, then understanding why they do not should be considered. What is each measuring, and how might those differences affect the outcome?

Also, make sure you use the polychoric correlation matrix if the responses are not continuous but on a Likert-type scale (or tetrachoric if all are dichotomous).

And, consider whether a Bartlett prediction is preferred.

[Klaus Klausen]

Thank you.

Here are the unrotaded results from the PCA. Indeed, I have 6 components with an Eigenvalue > 1:

Principal components/correlation                 Number of obs    =      6,765
                                                 Number of comp.  =          4
                                                 Trace            =         14
    Rotation: (unrotated = principal)            Rho              =     0.4791

    --------------------------------------------------------------------------
       Component |   Eigenvalue   Difference         Proportion   Cumulative
    -------------+------------------------------------------------------------
           Comp1 |      2.10949     .0961184             0.1507       0.1507
           Comp2 |      2.01337      .638738             0.1438       0.2945
           Comp3 |      1.37463      .165252             0.0982       0.3927
           Comp4 |      1.20938      .122447             0.0864       0.4791
           Comp5 |      1.08693       .01282             0.0776       0.5567
           Comp6 |      1.07411      .100407             0.0767       0.6334
           Comp7 |      .973702      .147541             0.0696       0.7030
           Comp8 |      .826161     .0179238             0.0590       0.7620
           Comp9 |      .808237      .127173             0.0577       0.8197
          Comp10 |      .681064     .0583256             0.0486       0.8684
          Comp11 |      .622739       .10783             0.0445       0.9128
          Comp12 |      .514908     .0634065             0.0368       0.9496
          Comp13 |      .451502      .197711             0.0323       0.9819
          Comp14 |      .253791            .             0.0181       1.0000
    --------------------------------------------------------------------------

Narcissism is recognized as a multidimensional personality trait encompassing a range of characteristics. The commonly acknowledged four key dimensions include (1) Superiority/Arrogance, (2) Exploitativeness/Entitlement, (3) Self-absorption/Self-admiration, and (4) Leadership/Authority. Consequently, it is unsurprising that the extraction of multiple components with Eigenvalues greater than 1 occurs, I guess.
Indeed, the classification might also be the reason why most papers restrict to 4 components which each component referring to one of these dimensions.

My intuition would also be to include the 4 components on their own in the regression which somewhat contradicts with what the rest of the existing papers are doing. What the authors I am referring to in #1 might have done is extracting 4 components using PCA, standardize each of the components and than add them up.
Just to verify: With if all the responses collapsed to a single factor you mean that all 14 variables load on the same component? So responses refer to my 14 items/variables?

[Nick Cox]

I don't really understand what you're trying to do or (not necessarily identical) what people in this field are trying to do. So it's dangerous for me and dangerous to you to comment at all, but some small points.

There is nothing in the quotation about rotation.

We don't have the original reference to check for ourselves, always assuming it is not behind a paywall.

I am used to a sales pitch that the first PC is the best single summary and to a different but consistent sales pitch that you may need more than one PC as a part of a composite summary. What I don't get is the idea that mushing four PCs together is somehow an improvement on either approach. How do you do that defensibly? Shouldn't you weight by the eigenvalues? I don't know.

Digression: The fact that this is in some literature doesn't convince me as such. In my own discipline of geography PCA and factor analysis (often conflated) were for a few years (early 1970s) just about the most discussed statistical method, or methods, but journals were full of agitated and angry papers explaining that the other people just didn't understand what they were doing with the method. That's a different field, but to me a tell-tale sign of confusion is when people mix terminology and talk about factor loadings in a PCA context.

To end positively: this sounds right to me!

My intuition would also be to include the 4 components on their own in the regression

[George Ford]

I've used FA with some success but am no expert, so keep that in mind.

The FA is telling you is that the 14 factors are not measuring one thing, but six things. Combining them seems problematic. As Nick points out, it's not clear how you would combine them.

I'd argue that your analysis suggests the way it's been used in the literature is incorrect (and I've seen papers make the same criticism). There is no single type of narcissism.

You might look at the loadings and see if you can track the components to the 4 types (are there questions that clearly point to Arrogance, Authority, and so forth). Then use those 4 factors. You can always test for equal effects from the regression results to see if they are sensibly combined (though I'd think you may need to make sure they are all scaled the same if the DV is not logged).

If the questions are designed to track the four types, then you can do a FA on just the relevant questions for each type and get a single variable for each.

[Klaus Klausen]

Thank you both for the helpful comments