Please see an old text in this field.

https://files.eric.ed.gov/fulltext/ED328591.pdf

Abolfazl Ghoodjani, thank you very much for this document. In my question I should have added that the information I was recently receiving indicated that each item had only one anti-image correlation that needed to be taken into account, and, furthermore, that the items' correlations to be taken into account were displayed along the diagonal of the matrix.

The authors in the document you've attached do refer to anti-image correlations that lie on the diagonal, and the authors recommend that these be high. They don't indicate how high, but they are happy with these correlations ranging from .77 to .93 in their own data.

The authors also recommend that a large proportion of the correlations below the diagonal in the anti-image matrix should be low (something I'd not seen mention of before). From the document you sent me, presumably this means that only a small percentage if these correlations should be > .20.

The only thing that concerns me (apart from a bit of vagueness in the document) is whether, in the 30 years since it was written/presented, researchers might have abandoned some strategies mentioned there and might have either adopted different strategies or relied more heavily on other metrics, e.g., the KMO index and Bartlett's test of sphericity.

Thanks, again. I very much appreciate your help.

Robert, is anti-image the same thing as reverse coded? If so does that help?

Best wishes, David Booth

David Eugene Booth, thanks for trying to help. My answer is, no, I'm pretty sure they're not the same thing. For what it's worth, in the same set of options that SPSS lets researchers select the KMO index and Bartlett's test (that's within the Descriptives option), it's possible to request anti-image correlations, and it was those correlations I was referring to.

One of my big questions is now whether many researchers look at those correlations when they conduct either PCA or EFA.

Thanks, again, for stepping in.

Thanks for the reply. My next guess is can you tell what is being correlated in an example. I'm somewhat confused about all of this. Best wishes and good luck, David

BTW it just occurred to me that sometimes contacting SPSS and asking what is being computed by these methods actually works. I would try that next. Best wishes, David Booth

David Eugene Booth, thanks for your extra suggestions. I can't fathom from any "basics" what is actually being correlated, and I've already looked up several sites on the web that refer to what is being computed with respect to anti-image correlations, but I have to confess that the kind of thing I find there doesn't float my boat (which, I confess, is attributable to my statistical ignorance).

As I indicated above, I'd probably be happy to know what other (good) researchers do when conducting PCAs and EFAs: Do they take the anti-image correlations into account? I'm accustomed to seeing mention of only the KMO index and Bartlett's test at the start of those analyses in order to get a sense of the "factorability" of the data.

I'd still contact SPSS.. IF they compute something they should know why they computed it. BEST WISHES, DAVID BOOTH

David Eugene Booth, OK, thanks, I might take up your suggestion, but do you know who / where to contact at SPSS? I imagine it's a pretty "impenetrable" organisation for the kind of thing that's concerning me. Maybe I'm wrong about that.

Wow You might try looking through this link for something that looks like it might help: spss help - Bing

Back in the day Maria Norusis (sp?) at SPSS and John Sall (SAS) would actually take calls. Those days appear to be over. Another reason to prefer R is the package maintainers list their contact info. I was able to contact a couple of them when I had problems. My inside guy for both Maria and John has been retired longer than I have. I'll ask him if he still has SPSS contacts, If I can get any info I'll pass it to you. Good luck and Best wishes, David

Hi Robert Trevethan ,

directly from SPSS website:

https://www.ibm.com/docs/en/spss-statistics/26.0.0?topic=analysis-factor-descriptives

Roberto Molteni, thanks for this. The bits of the jig saw puzzle are falling into place to some extent, though I'm getting inconsistent information, so maybe it would be better to say that the bits of the puzzle are scattering themselves all over the place.

In the material you've just recommended, there is the following statement:

"The anti-image correlation matrix contains the negatives of the partial correlation coefficients, and the anti-image covariance matrix contains the negatives of the partial covariances. In a good factor model, most of the off-diagonal elements will be small." (Bolding added)

In other places, I've found that only the anti-image correlations on the diagonal need to be taken into account, and/or that it's those correlations below the diagonal that should be taken into account, and that the anti-image covariances can be totally ignored.

On top of that, for what it's worth, I'm currently working with someone whose 15 items of data have only three interitem correlations > .30 and there is an unimpressive KMO index (.64), but the anti-image correlations (both on the diagonal and either side of the diagonal) seem to satisfy the "requirements" for being good.

It's difficult knowing what to believe. I just hope that some other RG members are able to obtain a degree of reassurance through all this.

Thanks, again, for your input.

Roberto, Super how did you know this? I have been most of the afternoon trying to find this. Best wishes, David Booth

Hi Robert Trevethan ,

the link may help https://www.stata.com/manuals13/mvglossary.pdf

and dear prof. David Eugene Booth , it comes from the past when I was more into the subject. Now it's more of a pleasure, having a nice virtual chat with all of you.

Best regards,

roberto

Abolfazl Ghoodjani,

David Eugene Booth

Roberto Molteni

Thank you, all, for trying to help me. I decided to check two sets of analyses against the information about anti-image matrices that surfaced from your input. One set of analyses was based on 15 items that had only four interitem correlations > .30 (only one of which was > .50). The other set of analyses was based on 33 items among which almost all interitem correlations were > .30, but few were > .60.

Despite the data in the second set of analyses being much more suited to PCA (and EFA), the anti-image matrices indicated that both sets of data carried an acceptable level of interitem association - at least based on entries in the diagonal and entries under, or on both sides of, the diagonal.

Having said that, almost all of the anti-image correlations on the diagonal of the first set of data hovered around .50 but almost all of those correlations were > .90 in the second set of data. Nevertheless, both data sets met the criterion of anti-image correlations being > .50.

Interestingly, the 11 items that we finally removed from consideration in the second, good, set of analyses (we got down to only 22 items) all satisfied the criteria that I'd seen pertaining to the anti-image matrices.  So, "bad" items can easily satisfy anti-image correlation criteria.

I could be wrong, but the conclusions I have drawn from that exercise are that the anti-image correlation and covariance matrices are not very useful for deciding whether data are suited to PCA/EFA, and they have little (or no) bearing on whether items are likely to be retained in iterative processes. Furthermore, I don't feel I've been "neglectful" in not using these matrices in the past. That's a relief.

Thanks, again. I couldn't have moved ahead without your support.