High KMO but low communality in factor analysis

Question

I'm performing a factor analysis and I have for a variable a Kaiser-Meyer-Olkin (KMO) measurement of .710 and a communality of .136. I recall that we are recommended to delete variables with a low KMO statistic (<=0.5) or with a low communality. In this case, I am not sure how to deal with this particular variable.

Answer 1

First, quite high KMO value for a variable does not necessarily refute or contradict its low communality. The individual KMO says how much the variable is free from partial correlations. FA assumes that latent common factors load more than just pairs of variables, and so a variable correlating high with only one specific another variable isn't a good candidate for FA. KMO is computed before the analysis. On the other hand, the communality says how much the variable is loaded by all the common factors extracted in the analysis done (and so it depends on the number of factors and on the method of extraction). A variable may occur loaded weakly, which means that it poorly correlates with any of the other input variables at all. Or, sometimes, number of factors fitted is too low to "appreciate" its correlations. And that variable may be "good" from the KMO point of view.

Second, KMO and communality are things considered in the scope of true factor analysis and not PCA. FA is modeling, PCA is summarizing. You may use PCA as a substitute for FA because not infrequently it gives quite similar results. But PCA does not build communalities like FA does. There is no theoretical or logical reason to watch after communality, and also after KMO, when doing PCA, as PCA does not hunt after pairwise correlations to explain them.

Third, one should remember that FA or PCA is sometimes done on covariances, not correlations. If the raw (non-rescaled) communality of .136 corresponds to a variable with variance much lower than 1, than it is high communality! KMO value, on the other hand, is typically computed from correlations and therefore isn't affected by variance magnitude.

Forth, very skewed data - not acceptible in FA - may probably add to a "discrepancy" between a KMO and a communality (I haven't explored that possibility closer).

Fifth, Heywood case - impossibly high extraction communality for one variable -, may be a cause of too low communalities for some other variables.

Finally, I hope that you understand it, so saying it just in case, that KMO value and communality value cannot be compared together directly by magnitude. They just have very different formulas.

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In the comment to this answer, @gung specifically inquires

Do you mean that one manifest variable is largely uncorrelated with the rest of the manifest variables? So, in that case, the KMO might be high for that one variable, but the communality would be low?

Yes. That is possible. KMO for a variable i, also called its MSA ("measure of sampling adequacy") is the proportion $MSA_i=\frac {\sum r_{ij}^2}{\sum r_{ij}^2+\sum q_{ij}^2}$, where $r_{ij}$ is correlation b/w variable i and each other variable j and $q_{ij}$ is their partial correlation controlling for all the rest manifest variables. So, if all $q$s are very small then even small $r$s yield high MSA. But a variable with small $r$s won't receive high communality, the sum of its squared loadings, because it doesn't bear enough common factor(s) inside to correlate with other variables.