kmo

Factor analysis can be used as a guide to how coherently a set of variables relate to a hypothesized underlying dimension that they are all being used to measure. External validity analysis assesses whether the scale that has been constructed performs as theoretically expected in correlation with other variables to which it is expected to be related.

There are some assumptions about the characteristics of factors that are extracted and defined that are unobserved common dimensions that may be listed to account for the correlations among observed variables. Sampling adequacy predicts if data are likely to factor well, based on correlation and partial correlation. It is used to assess which variables to drop from the model because they are too multicollinear.

It has been suggested that inv(R) should be a near-diagonal matrix in order to successfully fit a factor analysis model. To assess how close inv(R) is to a diagonal matrix, Kaiser (1970) proposed a measure of sampling adequacy, now called KMO (Kaiser-Meyer-Olkin) index. The common part, called the image of a variable, is defined as that part which is predictable by regressing each variable on all other variables.

The anti-image is the specific part of the variable that cannot be predicted. Examining the anti-image of the correlation matrix. That is the negative of the partial correlations, partialling out all other variables. There is a KMO statistic for each individual variable and their sum is the overall statistic. If it is not > 0.6 drop the indicator variables with the lowest individual statistic value until the overall one rises above 0.6: factors which is meritorious. The diagonal elements on the anti-image correlation matrix are the KMO individual statistics for each variable. A KMO index <= 0.5 indicates the correlation matrix is not suitable for factor analysis.

Input:
X - Input matrix can be a data matrix (size n-data x p-variables)

Output(s):
- Kaiser-Meyer-Olkin Index.
- Degree of Common Variance Report (shared by a set of variables
and thus assesses the degree to which they measure a common
underlying factor).
optional(s):
- Anti-image Covariance Matrix.
- Anti-image Correlation Matrix