The CALIS Procedure
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Overview
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Getting Started
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Syntax
Classes of Statements in PROC CALIS Single-Group Analysis Syntax Multiple-Group Multiple-Model Analysis Syntax PROC CALIS Statement BOUNDS Statement BY Statement COSAN Statement COV Statement DETERM Statement EFFPART Statement FACTOR Statement FITINDEX Statement FREQ Statement GROUP Statement LINCON Statement LINEQS Statement LISMOD Statement LMTESTS Statement MATRIX Statement MEAN Statement MODEL Statement MSTRUCT Statement NLINCON Statement NLOPTIONS Statement OUTFILES Statement PARAMETERS Statement PARTIAL Statement PATH Statement PCOV Statement PVAR Statement RAM Statement REFMODEL Statement RENAMEPARM Statement SAS Programming Statements SIMTESTS Statement STD Statement STRUCTEQ Statement TESTFUNC Statement VAR Statement VARIANCE Statement VARNAMES Statement WEIGHT Statement
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Details
Input Data Sets Output Data Sets The COSAN Model The FACTOR Model The LINEQS Model The LISMOD Model and Submodels The MSTRUCT Model The PATH Model The RAM Model Naming Variables and Parameters Setting Constraints on Parameters Automatic Variable Selection Estimation Criteria Relationships among Estimation Criteria Gradient, Hessian, Information Matrix, and Approximate Standard Errors Counting the Degrees of Freedom Assessment of Fit Total, Direct, and Indirect Effects Standardized Solutions Modification Indices Missing Values and the Analysis of Missing Patterns Measures of Multivariate Kurtosis Initial Estimates Use of Optimization Techniques Computational Problems Displayed Output ODS Table Names ODS Graphics
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Examples
Estimating Covariances and Correlations Estimating Covariances and Means Simultaneously Testing Uncorrelatedness of Variables Testing Covariance Patterns Testing Some Standard Covariance Pattern Hypotheses Linear Regression Model Multivariate Regression Models Measurement Error Models Testing Specific Measurement Error Models Measurement Error Models with Multiple Predictors Measurement Error Models Specified As Linear Equations Confirmatory Factor Models Confirmatory Factor Models: Some Variations The Full Information Maximum Likelihood Method Comparing the ML and FIML Estimation Path Analysis: Stability of Alienation Simultaneous Equations with Mean Structures and Reciprocal Paths Fitting Direct Covariance Structures Confirmatory Factor Analysis: Cognitive Abilities Testing Equality of Two Covariance Matrices Using a Multiple-Group Analysis Testing Equality of Covariance and Mean Matrices between Independent Groups Illustrating Various General Modeling Languages Testing Competing Path Models for the Career Aspiration Data Fitting a Latent Growth Curve Model Higher-Order and Hierarchical Factor Models Linear Relations among Factor Loadings Multiple-Group Model for Purchasing Behavior Fitting the RAM and EQS Models by the COSAN Modeling Language Second-Order Confirmatory Factor Analysis Linear Relations among Factor Loadings: COSAN Model Specification Ordinal Relations among Factor Loadings Longitudinal Factor Analysis
- References
| Measures of Multivariate Kurtosis |
In many applications, the manifest variables are not even approximately multivariate normal. If this happens to be the case with your data set, the default generalized least squares and maximum likelihood estimation methods are not appropriate, and you should compute the parameter estimates and their standard errors by an asymptotically distribution-free method, such as the WLS estimation method. If your manifest variables are multivariate normal, then they have a zero relative multivariate kurtosis, and all marginal distributions have zero kurtosis (Browne; 1982). If your DATA= data set contains raw data, PROC CALIS computes univariate skewness and kurtosis and a set of multivariate kurtosis values. By default, the values of univariate skewness and kurtosis are corrected for bias (as in PROC UNIVARIATE), but using the BIASKUR option enables you to compute the uncorrected values also. The values are displayed when you specify the PROC CALIS statement option KURTOSIS.
In the following formulas, 
denotes the sample size and 
denotes the number of variables.
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corrected variance for variable
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uncorrected univariate skewness for variable
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corrected univariate skewness for variable
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uncorrected univariate kurtosis for variable
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corrected univariate kurtosis for variable
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Mardia’s multivariate kurtosis
where
is the biased sample covariance matrix with as the divisor. -
relative multivariate kurtosis
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normalized multivariate kurtosis
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Mardia based kappa
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mean scaled univariate kurtosis
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adjusted mean scaled univariate kurtosis
with
If variable 
is normally distributed, the uncorrected univariate kurtosis 
is equal to 0. If 
has an -variate normal distribution, Mardia’s multivariate kurtosis 
is equal to 0. A variable is called leptokurtic if it has a positive value of and is called platykurtic if it has a negative value of . The values of 
, 
, and 
should not be smaller than the following lower bound (Bentler; 1985):
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PROC CALIS displays a message if , , or falls below the lower bound.
If weighted least squares estimates (METHOD=WLS or METHOD=ADF) are specified and the weight matrix is computed from an input raw data set, the CALIS procedure computes two more measures of multivariate kurtosis.
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multivariate mean kappa
where
and
is the number of elements in the vector 
(Bentler; 1985). -
multivariate least squares kappa
where
is the vector of the elements in the denominator of 
(Bentler; 1985) and 
is the vector of the , which is defined as: 
The occurrence of significant nonzero values of Mardia’s multivariate kurtosis and significant amounts of some of the univariate kurtosis values indicate that your variables are not multivariate normal distributed. Violating the multivariate normality assumption in (default) generalized least squares and maximum likelihood estimation usually leads to the wrong approximate standard errors and incorrect fit statistics based on the 
value. In general, the parameter estimates are more stable against violation of the normal distribution assumption. For more details, refer to Browne (1974, 1982, 1984).