Kaiser-Meyer-Olkin Measure of Sampling Adequacy是用于比较观测相关系数值与偏相关系数值的一个指标，其值愈逼近1，表明对这些变量进行因子分析的效果愈好.

The value of KMO should be. greater than 0.5 if the sample is adequate;

Factor analysis is a technique that requires a large sample size. Factor analysis is based on the correlation matrix of the variables involved, and correlations usually need a large sample size before they stabilize. Tabachnick and Fidell (2001, page 588) cite Comrey and Lee's (1992) advise regarding sample size: 50 cases is very poor, 100 is poor, 200 is fair, 300 is good, 500 is very good, and 1000 or more is excellent. As a rule of thumb, a bare minimum of 10 observations per variable is necessary to avoid computational difficulties

The Problem

First, the researcher may get a message saying that the input covariance or correlation matrix being analyzed is "not positive definite." Generalized least squares (GLS) estimation requires that the covariance or correlation matrix analyzed must be positive definite, and maximum likelihood (ML) estimation will also perform poorly in such situations. If the matrix to be analyzed is found to be not positive definite, many programs will simply issue an error message and quit.

Second, the message may refer to the asymptotic covariance matrix. This is not the covariance matrix being analyzed, but rather a weight matrix to be used with asymptotically distribution-free / weighted least squares (ADF/WLS) estimation.

Third, the researcher may get a message saying that its estimate of Sigma (), the model-implied covariance matrix, is not positive definite. LISREL, for example, will simply quit if it issues this message.

Fourth, the program may indicate that some parameter matrix within the model is not positive definite. This attribute is only relevant to parameter matrices that are variance/covariance matrices. In the language of the LISREL program, these include the matrices Theta-delta, Theta-epsilon, Phi () and Psi. Here, however, this "error message" can result from correct specification of the model, so the only problem is convincing the program to stop worrying about it.

Not Positive Definite"--What Does It Mean?

Strictly speaking, a matrix is "positive definite" if all of its eigenvalues are positive. Eigenvalues are the elements of a vector, e, which results from the decomposition of a square matrix S as:

S = e'Me

To an extent, however, we can discuss positive definiteness in terms of the sign of the "determinant" of the matrix. The determinant is a scalar function of the matrix. In the case of symmetric matrices, such as covariance or correlation matrices, positive definiteness wil only hold if the matrix and every "principal submatrix" has a positive determinant. ("Principal submatrices" are formed by removing row-column pairs from the original symmetric matrix.) A matrix which fails this test is "not positive definite." If the determinant of the matrix is exactly zero, then the matrix is "singular."

Why does this matter? Well, for one thing, using GLS estimation methods involves inverting the input matrix. Any text on matrix algebra will show that inverting a matrix involves dividing by the matrix determinant. So if the matrix is singular, then inverting the matrix involves dividing by zero, which is undefined. Using ML estimation involves inverting Sigma, but since the aim to maximize the similarity between the input matrix and Sigma, the prognosis is not good if the input matrix is not positive definite. Now, some programs include the option of proceeding with analysis even if the input matrix is not positive definite--with Amos, for example, this is done by invoking the $nonpositive command--but it is unwise to proceed without an understanding of the reason why the matrix is not positive definite. If the problem relates to the asymptotic weight matrix, the researcher may not be able to proceed with ADF/WLS estimation, unless the problem can be resolved.

In addition, one interpretation of the determinant of a covariance or correlation matrix is as a measure of "generalized variance." Since negative variances are undefined, and since zero variances apply only to constants, it is troubling when a covariance or correlation matrix fails to have a positive determinant.

Another reason to care comes from mathematical statistics. Sample covariance matrices are supposed to be positive definite. For that matter, so should Pearson and polychoric correlation matrices. That is because the population matrices they are supposedly approximating *are* positive definite, except under certain conditions. So the failure of a matrix to be positive definite may indicate a problem with the input matrix.

What Can We Do About It?

http://www2.gsu.edu/~mkteer/npdmatri.html