Structural Equation Modeling

Complete implementation of Structural Equation Modeling techniques for analyzing correlation, covariance, and moment matrices (structured means, models with intercepts). Simple or complex factor or path models can be specified via a simple path-language (which can also be created via dialogs or step-by-step wizards in the Statistica application). The program will compute, using constrained optimization techniques, the appropriate standard errors for standardized models, and for models fitted to correlation matrices. The results options include a comprehensive set of diagnostic statistics including the standard fit indices as well as noncentrality-based indices of fit, reflecting the most recent developments in the area of structural equation modeling. You can fit models to multiple samples (groups) and specify for each group fixed, free, or constrained (to be equal across groups) parameters. When analyzing moment matrices, these facilities allow you to test complex hypotheses for structured means in different groups. STATISTICA also includes powerful Monte Carlo simulation options: you can generate data files for predefined models, based on normal or skewed distributions. Bootstrap estimates can be computed, as well as distributions for various diagnostic statistics, parameter estimates, etc. over the Monte Carlo trials. Numerous flexible graphing options are available to visualize the results (e.g., distributions of parameters) from Monte Carlo runs.

General

Element Name Description
Detail of computed results reported Specifies the level of computed results reported. If Minimal results is requested, then only the final parameter estimates, fit indices, and residual covariance (correlation) matrix are reported; if All results is requested, various additional descriptive statistics, iteration history, and parameter correlations are also reported.
Analysis syntax Analysis syntax string for Structural Equation Modeling. You can specify here the complete syntax, as, for example, copied from a Statistica analysis.
Data to analyze Specifies whether to analyze the covariance matrix (or matrices for multiple-group problems), correlation matrix, or moments matrix (for modeling means).
Discrepancy function Specifies the discrepancy function (function to optimize); the resulting parameter estimates will be maximum likelihood estimates, GLS estimates, OLS estimates, etc.
Manifest exogenous Specifies how to account for exogenous manifest variables. Most models do not have manifest variables that are exogenous. Refer to the Electronic Manual for details.
Initial values, use Select the method to find initial values for free parameters; the default method uses .5 for all free parameters, except variances and covariances (or correlations) of manifest exogenous variables. These parameters are initialized at the values obtained from the sample data.
Standardization Choose to generate either a standardized solution (where latent variables all have unit variance) by one of two methods, or an unstandardized solution. Note: Select the New option to estimate a standardized solution via constrained estimation. This approach produces a solution where all latent variables, both independent and dependent, have variances of 1. Unlike the old method, however, standard errors are available with this option. Combining this option with the Data to Analyze - Correlations option (see above) allows you to estimate a completely standardized path model, where all variables, manifest and latent, have unit variances, and standard errors can be estimated for all parameters.
Number of groups Specifies the number of groups specified in the model (syntax).

Line Search Method

Element Name Description
Line search method Choose a basic line search method. Once the step direction has been chosen, the minimization problem is basically reduced from a problem in n unknowns to a problem in 1 unknown, i.e., the length of the step. There are three methods for choosing the length of the step. Their technical aspects are discussed in Unconstrained Minimization Techniques in the Electronic Manual.
Cubic LS alpha Enter a value in the Cubic LS Alpha field to control how large a reduction in the discrepancy function has to be made before a step is considered acceptable when the Cubic Interpolation line search method is used. The default value, .0001, allows virtually any improvement to be considered acceptable.
Golden section tau Enter a value in the Golden section tau field to control how wide a range to which the Golden Section line search is limited.
Golden search precision Enter a value in the Golden search precision field to control the precision of estimation in a Golden Section line search.
Max. no. of stephalves Enter a value in the Max. no. of stephalves field to set the maximum number of stephalves allowed on a single iteration if the Simple stephalving line search method is used.
Stephalve fraction Enter a value in the Stephalve fraction field to set the fraction the current step is multiplied by when Simple stephalving is used as the line search method.

Convergence and Iteration Parameters

Element Name Description
Max number of iterations Enter the maximum number of iterations allowed in the Maximum number of iterations field. When this number of iterations is reached, the iterative process will automatically terminate, and a message will be issued to indicate that the maximum number of iterations was exceeded. Note that the minimum number of iterations is zero in which case the discrepancy function and estimated covariance matrix will be computed.
Maximum step length Enter the maximum length of the step vector that will be allowed in the Maximum step length field. See Unconstrained Minimization Techniques in the Electronic Manual for a discussion of this parameter.
Steepest descent iterations Enter a number of steepest descent iterations to proceed the standard iterations in the No. of steepest descent iterations field. See Solving Iteration Problems in the Electronic Manual for suggestions on how to use this parameter to solve some problems encountered during iteration.
Max. residual cosine criterion Enter a value for the maximum residual cosine in the Maximum residual cosine criterion field. This criterion becomes small when parameter values have stabilized. The default value of .0001 works well across a wide variety of situations. For a technical definition of this criterion and a discussion of its merits, see Unconstrained Minimization Techniques in the Electronic Manual.
Rel. function change criterion Enter a value for the relative function change in the Relative function change criterion field. This criterion becomes small when the discrepancy function being minimized (see Statistical Estimation Theory in the Electronic Manual for a discussion) is no longer changing.
Step tolerance Enter a tolerance value in the Step tolerance field. Statistica uses this value when a parameter is temporarily eliminated from the iterative process. The tolerance value is one minus the squared multiple correlation of a parameter with the other parameters. If a parameter becomes highly redundant with other parameters during iteration, the approximate Hessian employed during Gauss-Newton iteration becomes unstable. This parameter controls when a parameter is temporarily removed from the iterative process. Very low values of this parameter mean that a parameter will never be removed.

Monte Carlo Analysis

Element Name Description
Monte Carlo analyses Performs a Monte-Carlo experiment (does not simply estimate the parameters of the model); a Monte Carlo experiment involves executing a specified number of replications of a statistical procedure. Each replication involves a complete analysis of a random sample created with certain characteristics, so Monte Carlo experiments can be very time consuming.
Random number seed 1 Enter an integer between 1 and 2,147,483,647 in the Random number seed 1 field. This seed is used to generate random numbers.
Random number seed 2 Enter a second integer between 1 and 2,147,483,647 in the Random number seed 2 field. This seed is used in Contaminated Normal generation only; see the Electronic Manual for details.
Number of replications Specifies the number of Monte Carlo replications.
Sample sizes Specifies the sample size for the Monte-Carlo samples (common to all groups, in multiple-group analyses). Sample sizes should be greater than the number of variables in the analysis, but they can be different for each group.
Make data for replication Specifies the replication number for which you want to generate data (if All results is requested).