KMO: Find the Kaiser, Meyer, Olkin Measure of Sampling Adequacy

Description

Henry 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 `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.

Usage

KMO(r)

Arguments

A correlation matrix or a data matrix (correlations will be found)

Value

- MSA
{The overall Measure of Sampling Adequacy}
MSAiThe measure of sampling adequacy for each item
item{Image}{The Image correlation matrix (Q)}

Details

Let $S^2 = diag(R^{-1})^{-1}$ and $Q = SR^{-1}S$. Then Q is said to the be the anti-image intercorrelation matrix. Let $sumr2 = \sum{R^2}$ and $sumq2 = \sum{Q^2}$ for all off diagonal elements of R and Q, then $SMA=sumr2)/(sumr2 + sumq2)$. Although originally MSA was 1 - sumq2/sumr2 (Kaiser, 1970), this was modified in Kaiser and Rice, (1974) to be $SMA=sumr2)/(sumr2 + sumq2)$. This is the formula used by Dziuban and Shirkey (1974) and by SPSS.

References

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.

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.

Examples

Run this code

KMO(Thurstone)
KMO(Harman.political)   #compare to the results in Dziuban and Shirkey (1974)

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