Let \(S^2 = diag(R^{-1})^{-1} \) and \(Q = SR^{-1}S\). Then Q is said to be the anti-image intercorrelation matrix. Let \(sumr^2 = \sum{R^2}\) and \(sumq^2 = \sum{Q^2}\) for all off diagonal elements of R and Q, then \(SMA=sumr^2/(sumr^2 + sumq^2)\). Although originally MSA was 1 - sumq^2/sumr^2 (Kaiser, 1970), this was modified in Kaiser and Rice, (1974) to be \(SMA=sumr^2/(sumr^2 + sumq^2)\). This is the formula used by Dziuban and Shirkey (1974) and by SPSS.
In his delightfully flamboyant style, Kaiser (1975)
suggested that KMO > .9 were marvelous, in the .80s, mertitourious, in the .70s, middling, in the .60s, medicore, in the 50s, miserable, and less than .5, unacceptable.
An alternative measure of whether the matrix is factorable is the Bartlett test cortest.bartlett
which tests the degree that the matrix deviates from an identity matrix.
Note that except for the reversal of signs, the anti-image correlation matrix is the same as that returned by partial.r
.