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Structural Equation Models

You can use the CALIS procedure for analysis of covariance structures, fitting systems of linear structural equations, and path analysis. These terms are more or less interchangeable, but they emphasize different aspects of the analysis. The analysis of covariance structures refers to the formulation of a model for the variances and covariances among a set of variables and the fitting of the model to an observed covariance matrix. In linear structural equations, the model is formulated as a system of equations relating several random variables with assumptions about the variances and covariances of the random variables. In path analysis, the model is formulated as a path diagram, in which arrows connecting variables represent variances, covariances, and regression (or path) coefficients. Path models and linear structural equation models can be converted to models of the covariance matrix and can, therefore, be fitted by the methods of covariance structure analysis. All of these methods support the use of hypothetical latent variables and measurement errors in the models.

Below are highlights of the capabilities of the CALIS procedure:

CALIS Procedure


The CALIS procedure fits structural equation models, which express relationships among a system of variables that can be either observed variables (manifest variables) or unobserved hypothetical variables (latent variables). PROC CALIS enables you to do the following:

  • estimate parameters and test hypotheses for constrained and unconstrained problems in the following:
    • multiple and multivariate linear regression
    • linear measurement-error models
    • path analysis and causal modeling
    • simultaneous equation models with reciprocal causation
    • exploratory and confirmatory factor analysis of any order
    • canonical correlation
    • a wide variety of other (non)linear latent variable models
  • estimate parameters by using the following criteria:
    • unweighted least squares
    • generalized least squares
    • weighted least squares
    • diagonally weighted least squares
    • maximum likelihood
    • full information maximum likelihood
    • maximum likelihood with Satorra-Bentler scaled model fit chi-square statistic and sandwich-type standard error estimation
    • robust estimation with maximum likelihood model evaluation
  • specify models using the following modeling languages:
    • FACTOR—supports the input of factor-variable relations
    • LINEQS—uses equations to describe variable relationships
    • LISMOD—utilizes LISREL model matrices for defining models
    • MSTRUCT—supports direct parameterization in the mean and covariance matrices
    • PATH—provides an intuitive causal path specification interface
    • RAM—utilizes the formulation of the reticular action model
    • REFMODEL—provides a quick way for model referencing and respecification
  • choose among the following optimization algorithms:
    • Levenberg-Marquardt algorithm (More, 1978)
    • trust-region algorithm (Gay 1983)
    • Newton-Raphson algorithm with line search
    • ridge-stabilized Newton-Raphson algorithm
    • various quasi-Newton and dual quasi-Newton algorithms: Broyden-Fletcher-Goldfarb-Shanno and Davidon-Fletcher-Powell, including a sequential quadratic programming algorithm for processing nonlinear equality and inequality constraints
    • various conjugate gradient algorithms: automatic restart algorithm of Powell (1977), Fletcher-Reeves, Polak-Ribiere, and conjugate descent algorithm of Fletcher (1980)
  • use the following methods to automatically generate initial values for the optimization process:
    • two-stage least squares estimation
    • instrumental variable factor analysis
    • approximate factor analysis
    • ordinary least squares estimation
    • McDonald's (McDonald and Hartmann 1992) method
  • formulate general equality and inequality constraints by using programming statements
  • specify free unnamed parameters in all models
  • analyze linear dependencies in the information matrix (approximate Hessian matrix) that might be helpful in detecting unidentified models
  • perform multiple-group analysis. Groups can also be fitted by multiple models simultaneously.
  • specify linear and nonlinear equality and inequality constraints on the parameters with several different statements, depending on the type of input
  • compute Lagrange multiplier test indices for simple constant and equality parameter constraints and for active boundary constraints
  • produce a SAS data set that contains information about the optimal parameter estimates (parameter estimates, gradient, Hessian, projected Hessian and Hessian of Lagrange function for constrained optimization, the information matrix, and standard errors)
  • produce a SAS data set that contains residuals and, for exploratory factor analysis, the rotated and unrotated factor loadings
  • perform analysis of multiple samples with equal sample size by analyzing a moment supermatrix that contains the individual moment matrices as block diagonal submatrices
  • perform residual analysis at the case level or observation level
  • perform direct robust estimation based on residual weighting or two-stage robust estimation based on analyzing the robust mean and covariance matrices
  • input the model fit information of the customized baseline model of your choice. PROC CALIS then computes various fit indices (mainly the incremental fit indices) based on your customized model fit rather than the fit of the default uncorrelatedness model.
  • compute weighted covariances or correlations
  • create a SAS data set that corresponds to any output table
  • perform BY group processing, which enables you to obtain separate analyses on grouped observations
  • automatically create the following types of graphs:
    • distribution of residuals in the moment matrices
    • case-level residual diagnostics
    • path diagrams
For further details, see CALIS Procedure