The CALIS Procedure
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Overview
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Getting Started
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SyntaxClasses of Statements in PROC CALISSingle-Group Analysis SyntaxMultiple-Group Multiple-Model Analysis SyntaxPROC CALIS StatementBOUNDS StatementBY StatementCOSAN StatementCOV StatementDETERM StatementEFFPART StatementFACTOR StatementFITINDEX StatementFREQ StatementGROUP StatementLINCON StatementLINEQS StatementLISMOD StatementLMTESTS StatementMATRIX StatementMEAN StatementMODEL StatementMSTRUCT StatementNLINCON StatementNLOPTIONS StatementOUTFILES StatementPARAMETERS StatementPARTIAL StatementPATH StatementPATHDIAGRAM StatementPCOV StatementPVAR StatementRAM StatementREFMODEL StatementRENAMEPARM StatementSAS Programming StatementsSIMTESTS StatementSTD StatementSTRUCTEQ StatementTESTFUNC StatementVAR StatementVARIANCE StatementVARNAMES StatementWEIGHT Statement
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DetailsInput Data SetsOutput Data SetsDefault Analysis Type and Default ParameterizationThe COSAN ModelThe FACTOR ModelThe LINEQS ModelThe LISMOD Model and SubmodelsThe MSTRUCT ModelThe PATH ModelThe RAM ModelNaming Variables and ParametersSetting Constraints on ParametersAutomatic Variable SelectionPath Diagrams: Layout Algorithms, Default Settings, and CustomizationEstimation CriteriaRelationships among Estimation CriteriaGradient, Hessian, Information Matrix, and Approximate Standard ErrorsCounting the Degrees of FreedomAssessment of FitCase-Level Residuals, Outliers, Leverage Observations, and Residual DiagnosticsLatent Variable ScoresTotal, Direct, and Indirect EffectsStandardized SolutionsModification IndicesMissing Values and the Analysis of Missing PatternsMeasures of Multivariate KurtosisInitial EstimatesUse of Optimization TechniquesComputational ProblemsDisplayed OutputODS Table NamesODS Graphics
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ExamplesEstimating Covariances and CorrelationsEstimating Covariances and Means SimultaneouslyTesting Uncorrelatedness of VariablesTesting Covariance PatternsTesting Some Standard Covariance Pattern HypothesesLinear Regression ModelMultivariate Regression ModelsMeasurement Error ModelsTesting Specific Measurement Error ModelsMeasurement Error Models with Multiple PredictorsMeasurement Error Models Specified As Linear EquationsConfirmatory Factor ModelsConfirmatory Factor Models: Some VariationsResidual Diagnostics and Robust EstimationThe Full Information Maximum Likelihood MethodComparing the ML and FIML EstimationPath Analysis: Stability of AlienationSimultaneous Equations with Mean Structures and Reciprocal PathsFitting Direct Covariance StructuresConfirmatory Factor Analysis: Cognitive AbilitiesTesting Equality of Two Covariance Matrices Using a Multiple-Group AnalysisTesting Equality of Covariance and Mean Matrices between Independent GroupsIllustrating Various General Modeling LanguagesTesting Competing Path Models for the Career Aspiration DataFitting a Latent Growth Curve ModelHigher-Order and Hierarchical Factor ModelsLinear Relations among Factor LoadingsMultiple-Group Model for Purchasing BehaviorFitting the RAM and EQS Models by the COSAN Modeling LanguageSecond-Order Confirmatory Factor AnalysisLinear Relations among Factor Loadings: COSAN Model SpecificationOrdinal Relations among Factor LoadingsLongitudinal Factor Analysis
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Example 29.3 Testing Uncorrelatedness of Variables
This example uses the sales data in Example 29.1 and tests the uncorrelatedness of the variables in the model by using the MSTRUCT model specification. With the multivariate
normality assumption, this is also the test of independence of the variables. The MATRIX
statement defines the parameters in the model.
The uncorrelatedness model assumes that the correlations or covariances among the four variables are zero. Therefore, only the four diagonal elements of the covariance matrix, which represent the variances of the variables, are free parameters in the covariance structure model. To specify these parameters, use the MATRIX statement with the MSTRUCT model specification:
proc calis data=sales; mstruct var=q1-q4; matrix _cov_ [1,1], [2,2], [3,3], [4,4]; run;
Example 29.1 specifies exactly the same MSTRUCT statement for the four variables. The difference here is the addition of the MATRIX statement.
Without a MATRIX statement, the MSTRUCT model assumes that all nonredundant elements in the covariance matrix are model parameters.
This assumption is not the case in the current specification. The MATRIX statement specification for the covariance matrix
(denoted by the _cov_ keyword) specifies four free parameters on the diagonal of the covariance matrix: [1,1], [2,2], [3,3], and [4,4]. All other unspecified elements in the covariance matrix are fixed zeros by default.
The uncorrelatedness model is displayed in the output for the initial model specification. Output 29.3.1 shows that all off-diagonal elements of the covariance matrix are fixed zeros while the diagonal elements are missing and
labeled with _Parm1–_Parm4. PROC CALIS generates these parameter names automatically and estimates these four parameters in the analysis.
Output 29.3.1: Initial Uncorrelatedness Model for the Sales Data
Output 29.3.2 shows the model fit chi-square test of the uncorrelatedness model. The chi-square is 6.528 (df = 6, p = 0.3667), which is not significant. This means that you fail to reject the uncorrelatedness model. In other words, the data is consistent with the uncorrelatedness model (zero covariances or correlations among the quarterly sales).
Output 29.3.2: Fit Summary of the Uncorrelatedness Model for the Sales Data
Output 29.3.3 shows the estimates of the covariance matrix under the uncorrelatedness model, together with standard error estimates and t values. All off-diagonal elements are fixed zeros in the estimation results.
Output 29.3.3: Estimates of Variance under the Uncorrelatedness Model for the Sales Data
This example shows how to specify free parameters in the MSTRUCT model by using the MATRIX statement. To specify the covariance matrix, use the _COV_ keyword in the MATRIX statement. To specify the parameters in the mean structures, you need use an additional MATRIX statement with the _MEAN_ keyword.
Two important notes regarding the MSTRUCT model specification are now in order:
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When you use the MSTRUCT statement without any MATRIX statements, all elements in the covariance matrix are free parameters in the model (for example, see Example 29.1). However, if the MATRIX statement includes at least one free or fixed parameter in the covariance matrix, PROC CALIS assumes that all other unspecified elements in the covariance matrix are fixed zeros (such as the current example).
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Using parameter names in the MATRIX statement specification is optional. In the context of the current example, naming the parameters is optional because there is no need to refer to them anywhere in the specification. PROC CALIS automatically generates unique names for these parameters. Alternatively, you can specify your own parameter names in the MATRIX statement. Naming parameters is not only useful for references, but is also indispensable when you need to constrain model parameters by referring to their names. See Example 29.4 to use parameter names to define a covariance pattern.