KMO.RdHenry Kaiser (1970) introduced an Measure of Sampling Adequacy (MSA) of factor analytic data matrices. Kaiser and Rice (1974) then modified it. This is just a function of the squared elements of the `anti-image' matrix compared to the squares of the original correlations. The overall MSA as well as estimates for each item are found. The index is known as the Kaiser-Meyer-Olkin (KMO) index.
KMO(r)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.
MSA: The overall Measure of Sampling Adequacy
MSAi: The measure of sampling adequacy for each item
Image: The anti-image correlation matrix (Q)
H.~F. Kaiser. (1970) A second generation little jiffy. Psychometrika, 35(4):401–415.
H.~F. Kaiser and J.~Rice. (1974) Little jiffy, mark iv. Educational and Psychological Measurement, 34(1):111–117.
H.F. Kaiser. 1974) An index of factor simplicity. Psychometrika, 39 (1) 31-36.
Dziuban, Charles D. and Shirkey, Edwin C. (1974) When is a correlation matrix appropriate for factor analysis? Some decision rules. Psychological Bulletin, 81 (6) 358 - 361.
See Also as fa, cortest.bartlett, Harman.political, partial.r
KMO(Thurstone)
#> Kaiser-Meyer-Olkin factor adequacy
#> Call: KMO(r = Thurstone)
#> Overall MSA = 0.88
#> MSA for each item =
#> Sentences Vocabulary Sent.Completion First.Letters
#> 0.86 0.86 0.90 0.86
#> Four.Letter.Words Suffixes Letter.Series Pedigrees
#> 0.88 0.92 0.85 0.93
#> Letter.Group
#> 0.87
k.m <- KMO(Harman.political) #compare to the results in Dziuban and Shirkey (1974)
k.m
#> Kaiser-Meyer-Olkin factor adequacy
#> Call: KMO(r = Harman.political)
#> Overall MSA = 0.81
#> MSA for each item =
#> Lewis Roosevelt Party Voting Median Rental Homeownership
#> 0.73 0.76 0.84 0.87 0.53
#> Unemployment Mobility Education
#> 0.93 0.78 0.86
lowerMat(k.m$Image)
#> Lewis Rsvlt PrtyV MdnRn Hmwnr Unmpl Mblty Edctn
#> Lewis 1.00
#> Roosevelt -0.76 1.00
#> Party Voting 0.32 -0.52 1.00
#> Median Rental -0.15 0.08 -0.04 1.00
#> Homeownership -0.01 0.15 0.24 -0.23 1.00
#> Unemployment 0.22 -0.32 -0.13 0.16 -0.11 1.00
#> Mobility 0.10 -0.13 0.26 -0.22 0.69 0.03 1.00
#> Education 0.27 -0.08 0.29 -0.57 0.26 0.24 -0.04 1.00
lowerMat(partial.r(Harman.political)) #identical to image, except for sign
#> Lewis Rsvlt PrtyV MdnRn Hmwnr Unmpl Mblty Edctn
#> Lewis 1.00
#> Roosevelt 0.76 1.00
#> Party Voting -0.32 0.52 1.00
#> Median Rental 0.15 -0.08 0.04 1.00
#> Homeownership 0.01 -0.15 -0.24 0.23 1.00
#> Unemployment -0.22 0.32 0.13 -0.16 0.11 1.00
#> Mobility -0.10 0.13 -0.26 0.22 -0.69 -0.03 1.00
#> Education -0.27 0.08 -0.29 0.57 -0.26 -0.24 0.04 1.00