Structural Equation Modeling - Covariance Matrices vs. Correlation Matrices

In the Data to Analyze box of the Analysis Parameters dialog box, select which type of data to analyze. For standard path analyses or structural equation models, you choose either Covariances or Correlations. Use the following information to help you decide whether you should use the covariance or correlation matrix.

Covariances. When you select this option,STATISTICA will analyze the Covariance Matrix of the input variables, regardless of what kind of data is input. So, for example, if you input a correlation matrix file with standard deviations, SEPATH will calculate the covariance matrix and analyze it. If your input file is a covariance matrix file, SEPATH will analyze it directly. If your file contains raw data, SEPATH will analyze it and calculate the covariance matrix for you.  

The statistical distribution of the elements of a covariance matrix is not the same as that of a correlation matrix. This is obvious if you consider the diagonal elements of a covariance matrix, which are the variances of the variables. These are random variables - they vary from sample to sample. However, the diagonal elements of a correlation matrix are not random variables - they are always 1. The methods employed by previous structural modeling programs are based on the assumption that a covariance matrix is being analyzed. The sampling distribution theory they employ is not applicable to a correlation matrix, except in special circumstances.

Recent research has emphasized that it is possible (indeed likely) that you will get some wrong results if you analyze a correlation matrix as if it were a covariance matrix. However, a number of currently distributed structural modeling programs will analyze a correlation matrix as though it were a covariance matrix. The fact that such programs yield incorrect results has been described in the literature (see, for example, Cudeck, 1989). In order to provide compatibility with these other programs, SEPATH will analyze a correlation matrix as if it were a covariance matrix, but it can, unlike most other programs, directly and automatically provide correct analysis of a correlation matrix as well.

Correlations. When you select this option, SEPATH will calculate the correlation matrix from the input data and analyze it. SEPATH analyzes the correlation matrix correctly, using constrained estimation theory developed by Michael Browne (see Browne, 1982; Mels, 1989; Browne & Mels, 1992), and implemented first in the computer program RAMONA. As a result, SEPATH gives the correct standard errors, estimates, and test statistics when a correlation matrix is analyzed directly. When combined with the Standardization - New option SEPATH can estimate a completely standardized path model, where all variables are standardized to have unit variance, and standard errors can be estimated as well.