Displayed Output

The CALIS Procedure

Displayed Output

The output displayed by PROC CALIS depends on the statement used to specify the model. Since an analysis requested by the LINEQS or RAM statement implies the analysis of a structural equation model, more statistics can be computed and displayed than for a covariance structure analysis following the generalized COSAN model requested by the COSAN statement. The displayed output resulting from use of the FACTOR statement includes all the COSAN displayed output as well as more statistics displayed only when you specify the FACTOR statement. Since the displayed output using the RAM statement differs only in its form from that generated by the LINEQS statement, in this section distinctions are made between COSAN and LINEQS output only.

The unweighted least-squares and diagonally weighted least-squares estimation methods do not provide a sufficient statistical basis to provide the following output (neither displayed nor written to an OUTEST= data set):

most of the fit indices
approximate standard errors
normalized or asymptotically standardized residuals
modification indices
information matrix
covariance matrix of parameter estimates

The notation S = (s_ij) is used for the analyzed covariance or correlation matrix, C = (c_ij) for the predicted model matrix, W for the weight matrix (for example, W = I for ULS, W = S for GLS, W = C for ML estimates), X for the vector of optimal parameter estimates, n for the number of manifest variables, t for the number of parameter estimates, and N for the sample size.

The output of PROC CALIS includes the following:

COSAN and LINEQS: List of the matrices and their properties specified by the generalized COSAN model if you specify at least the PSHORT option.
LINEQS: List of manifest variables that are not used in the specified model and that are automatically omitted from the analysis. Note that there is no automatic variable reduction with the COSAN or FACTOR statement. If necessary, you should use the VAR statement in these cases.
LINEQS: List of the endogenous and exogenous variables specified by the LINEQS, STD, and COV statements if you specify at least the PSHORT option.
COSAN: Initial values of the parameter matrices indicating positions of constants and parameters. The output, or at least the default output, is displayed if you specify the PINITIAL option.
LINEQS: The set of structural equations containing the initial values and indicating constants and parameters, and output of the initial error variances and covariances. The output, or at least the default output, is displayed if you specify the PINITIAL option.
COSAN and LINEQS: The weight matrix W is displayed if GLS, WLS, or DWLS estimation is used and you specify the PWEIGHT or PALL option.
COSAN and LINEQS: General information about the estimation problem: number of observations (N), number of manifest variables (n), amount of independent information in the data matrix (information, n(n+1)/2), number of terms and matrices in the specified generalized COSAN model, and number of parameters to be estimated (parameters, t). If there are no exogenous manifest variables, the difference between the amount of independent information (n(n+1)/2) and the number of requested estimates (t) is equal to the degrees of freedom (df). A necessary condition for a model to be identified is that the degrees of freedom are nonnegative. The output, or at least the default output, is displayed if you specify the SIMPLE option.
COSAN and LINEQS: Mean and Std Dev (standard deviation) of each variable if you specify the SIMPLE option, as well as skewness and kurtosis if the DATA= data set is a raw data set and you specify the KURTOSIS option.
COSAN and LINEQS: Various coefficients of multivariate kurtosis and the numbers of observations that contribute most to the normalized multivariate kurtosis if the DATA= data set is a raw data set and the KURTOSIS option, or you specify at least the PRINT option. See the section "Measures of Multivariate Kurtosis" for more information.
COSAN and LINEQS: Covariance or correlation matrix to be analyzed and the value of its determinant if you specify the output option PCORR or PALL. A 0 determinant indicates a singular data matrix. In this case, the generalized least-squares estimates with default weight matrix S and maximum likelihood estimates cannot be computed.
LINEQS: If exogenous manifest variables in the linear structural equation model are specified, then there is a one-to-one relationship between the given covariances and corresponding estimates in the central model matrix $\Phi$ or P. The output indicates which manifest variables are recognized as exogenous, that is, for which variables the entries in the central model matrix are set to fixed parameters. The output, or at least the default output, is displayed if you specify the PINITIAL option.
COSAN and LINEQS: Vector of parameter names, initial values, and corresponding matrix locations, also indicating dependent parameter names used in your program statements that are not allocated to matrix locations and have no influence on the fit function. The output, or at least the default output, is displayed if you specify the PINITIAL option.
COSAN and LINEQS: The pattern of variable and constant elements of the predicted moment matrix that is predetermined by the analysis model is displayed if there are significant differences between constant elements in the predicted model matrix and the data matrix and you specify at least the PSHORT option. It is also displayed if you specify the PREDET option. The output indicates the differences between constant values in the predicted model matrix and the data matrix that is analyzed.
COSAN and LINEQS: Special features of the optimization technique chosen if you specify at least the PSHORT option.
COSAN and LINEQS: Optimization history if at least the PSHORT option is specified. For more details, see the section "Use of Optimization Techniques".
COSAN and LINEQS: Specific output requested by options in the NLOPTIONS statement; for example, parameter estimates, gradient, gradient of Lagrange function, constraints, Lagrange multipliers, projected gradient, Hessian, projected Hessian, Hessian of Lagrange function, Jacobian of nonlinear constraints.
COSAN and LINEQS: The predicted model matrix and its determinant, if you specify the output option PCORR or PALL.
COSAN and LINEQS: Residual and normalized residual matrix if you specify the RESIDUAL, or at least the PRINT option. The variance standardized or asymptotically standardized residual matrix can be displayed also. The average residual and the average off-diagonal residual are also displayed. See the section "Assessment of Fit" for more details.
COSAN and LINEQS: Rank order of the largest normalized residuals if you specify the RESIDUAL, or at least the PRINT option.
COSAN and LINEQS: Bar chart of the normalized residuals if you specify the RESIDUAL, or at least the PRINT option.
COSAN and LINEQS: Value of the fit function F. See the section "Estimation Criteria" for more details. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Goodness-of-fit index (GFI), adjusted goodness-of-fit index (AGFI), and root mean square residual (RMR) (J $\ddot{o}$ reskog and S $\ddot{o}$ rbom 1985). See the section "Assessment of Fit" for more details. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Parsimonious goodness-of-fit index (PGFI) of Mulaik et al. (1989). See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Overall $\chi^2$ , df, and Prob>Chi**2 if the METHOD= option is not ULS or DWLS. The $\chi^2$ measure is the optimum function value F multiplied by (N - 1) if a CORR or COV matrix is analyzed or multiplied by N if a UCORR or UCOV matrix is analyzed; $\chi^2$ measures the likelihood ratio test statistic for the null hypothesis that the predicted matrix C has the specified model structure against the alternative that C is unconstrained. The notation Prob>Chi**2 means "the probability under the null hypothesis of obtaining a greater $\chi^2$ statistic than that observed." This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If METHOD= is not ULS or DWLS, the $\chi^2_0$ value of the independence model and the corresponding degrees of freedom can be used (in large samples) to evaluate the gain of explanation by fitting the specific model (Bentler 1989). See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If METHOD= is not ULS or DWLS, the value of the Steiger & Lind (1980) root mean squared error of approximation (RMSEA) coefficient and the lower and upper limits of the confidence interval. The size of the confidence interval is defined by the option ALPHARMS= $\alpha$ , $0 \leq \alpha \leq 1$ . The default is $\alpha=0.1$ ,which corresponds to a 90% confidence interval. See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If the value of the METHOD= option is not ULS or DWLS, the value of the probability of close fit (Browne and Cudeck 1993). See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If the value of the METHOD= option is not ULS or DWLS, the value of the Browne & Cudeck (1993) expected cross validation (ECVI) index and the lower and upper limits of the confidence interval. The size of the confidence interval is defined by the option ALPHAECV= $\alpha$ , $0 \leq \alpha \leq 1$ . The default is $\alpha=0.1$ ,which corresponds to a 90% confidence interval. See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If the value of the METHOD= option is not ULS or DWLS, Bentler's (1989) Comparative Fit Index. See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If you specify METHOD=ML or METHOD=GLS, the $\chi^2$ value and corresponding probability adjusted by the relative kurtosis coefficient $\eta_2$ , which should be a close approximation of the $\chi^2$ value for elliptically distributed data (Browne 1982). See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: The Normal Theory Reweighted LS $\chi^2$ Value is displayed if METHOD= ML. Instead of the function value F_ML, the reweighted goodness-of-fit function F_GWLS is used. See the section "Assessment of Fit" for more detail.
COSAN and LINEQS: Akaike's Information Criterion if the value of the METHOD= option is not ULS or DWLS. See the section "Assessment of Fit". This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Bozdogan's (1987) Consistent Information Criterion, CAIC. See the section "Assessment of Fit". This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Schwarz's Bayesian Criterion (SBC) if the value of the METHOD= option is not ULS or DWLS (Schwarz 1978). See the section "Assessment of Fit". This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: If the value of the METHOD= option is not ULS or DWLS, the following fit indices based on the overall value are displayed:
- McDonald's (McDonald and Hartmann 1992) measure of centrality
- Parsimonious index of James, Mulaik, and Brett (1982)
- Z-Test of Wilson and Hilferty (1931)
- Bentler and Bonett's (1980) nonnormed coefficient
- Bentler and Bonett's (1980) normed coefficient
- Bollen's (1986) normed index $\rho_1$
- Bollen's (1989a) nonnormed index $\Delta_2$
See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Hoelter's (1983) Critical N Index is displayed (Bollen 1989b, p. 277). See the section "Assessment of Fit" for more detail. This output can be suppressed only by the NOPRINT option.
COSAN and LINEQS: Equations of linear dependencies among the parameters used in the model specification if the information matrix is recognized as singular at the final solution.
COSAN: Model matrices containing the parameter estimates. Except for ULS or DWLS estimates, the approximate standard errors and t values are also displayed. This output is displayed if you specify the PESTIM option or at least the PSHORT option.
LINEQS: Linear equations containing the parameter estimates. Except for ULS and DWLS estimates, the approximate standard errors and t values are also displayed. This output is displayed if you specify the PESTIM option, or at least the PSHORT option.
LINEQS: Variances and covariances of the exogenous variables. This output is displayed if you specify the PESTIM option, or at least the PSHORT.
LINEQS: Linear equations containing the standardized parameter estimates. This output is displayed if you specify the PESTIM option, or at least the PSHORT option.
LINEQS: Table of correlations among the exogenous variables. This output is displayed if you specify the PESTIM option, or at least the PSHORT option.
LINEQS: Correlations among the exogenous variables. This output is displayed if you specify the PESTIM option, or at least the PSHORT option.
LINEQS: Squared Multiple Correlations table, which displays the error variances of the endogenous variables. These are the diagonal elements of the predicted model matrix. Also displayed is the Total Variance and the R² values corresponding to all endogenous variables. See the section "Assessment of Fit" for more detail. This output is displayed if you specify the PESTIM option, or at least the PSHORT option.
LINEQS: If you specify the PDETERM or the PALL option, the total determination of all equations (DETAE), the total determination of the structural equations (DETSE), and the total determination of the manifest variables (DETMV) are displayed. See the section "Assessment of Fit" for more details. If one of the determinants in the formulas is 0, the corresponding coefficient is displayed as a missing value. If there are structural equations, PROC CALIS also displays the Stability Coefficient of Reciprocal Causation, that is, the largest eigenvalue of the BB' matrix, where B is the causal coefficient matrix of the structural equations.
LINEQS: The matrix of estimated covariances among the latent variables if you specify the PLATCOV option, or at least the PRINT option.
LINEQS: The matrix of estimated covariances between latent and manifest variables used in the model if you specify the PLATCOV option, or at least the PRINT option.
LINEQS and FACTOR: The matrix FSR of latent variable scores regression coefficients if you specify the PLATCOV option, or at least the PRINT option. The FSR matrix is a generalization of Lawley and Maxwell's (1971, p.109) factor scores regression matrix,
FSR = C_yx C_xx^-1

where C_xx is the n ×n predicted model matrix (predicted covariances among manifest variables) and C_yx is the n_lat ×n matrix of the predicted covariances between latent and manifest variables. You can multiply the manifest observations by this matrix to estimate the scores of the latent variables used in your model.
LINEQS: The matrix TEF of total effects if you specify the TOTEFF option, or at least the PRINT option. For the LINEQS model, the matrix of total effects is
${TEF} = (I- {\beta})^{-1} {\gamma}- (O:I)$

(For the LISREL model, refer to J $\ddot{o}$ reskog and S $\ddot{o}$ rbom 1985) The matrix of indirect effects is displayed also.
FACTOR: The matrix of rotated factor loadings and the orthogonal transformation matrix if you specify the ROTATE= and PESTIM options, or at least the PSHORT options.
FACTOR: Standardized (rotated) factor loadings, variance estimates of endogenous variables, R² values, correlations among factors, and factor scores regression matrix, if you specify the PESTIM option, or at least the PSHORT option. The determination of manifest variables is displayed only if you specify the PDETERM option.
COSAN and LINEQS: Univariate Lagrange multiplier and Wald test indices are displayed in matrix form if you specify the MODIFICATION (or MOD) or the PALL option. Those matrix locations that correspond to constants in the model in general contain three values: the value of the Lagrange multiplier, the corresponding probability (df=1), and the estimated change of the parameter value should the constant be changed to a parameter. If allowing the constant to be an estimated parameter would result in a singular information matrix, the string 'sing' is displayed instead of the Lagrange multiplier index. Those matrix locations that correspond to parameter estimates in the model contain the Wald test index and the name of the parameter in the model. See the section "Modification Indices" for more detail.
COSAN and LINEQS: Univariate Lagrange multiplier test indices for releasing equality constraints if you specify the MODIFICATION (or MOD) or the PALL option. See the section "Modification Indices" for more detail.
COSAN and LINEQS: Univariate Lagrange multiplier test indices for releasing active boundary constraints specified by the BOUNDS statement if you specify the MODIFICATION (or MOD) or the PALL option. See the section "Modification Indices" for more detail.
COSAN and LINEQS: If the MODIFICATION (or MOD) or the PALL option is specified, the stepwise multivariate Wald test for constraining estimated parameters to zero constants is performed as long as the univariate probability is larger than the value specified in the PMW= option (default PMW=0.05). See the section "Modification Indices" for more details.

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