The CALIS Procedure

specifies a confidence interval () for the Browne and Cudeck (1993) expected cross-validation index (ECVI). The default value is , which corresponds to a 90% confidence interval for the ECVI.

specifies the -level criterion for detecting leverage observations (or leverage points) in case-level (observation-level) residual diagnostics. The default ALPHALEV= value is 0.01. An observation is a leverage observation if the p-value of its squared Mahalanobis distance (M-distance) for its predictor variables (including observed and latent variables) is smaller than the specified -level, where the p-value is computed according to an appropriate theoretical chi-square distribution. The larger the ALPHALEV= value, the more liberal the criterion for detecting leverage observations.

In addition to displaying the leverage observations as defined by the ALPHALEV= criterion, PROC CALIS also displays the next 5 observations with the largest leverage M-distances for reference. However, the total number of observations in the displayed output cannot exceed 30 or the number of original observations, whichever is smaller.

This option is relevant only when residual analysis is requested with the RESIDUAL option and with raw data input.

specifies the -level criterion for detecting outliers in case-level (observation-level) residual diagnostics. The default ALPHAOUT= value is 0.01. An observation is an outlier if the p-value of its squared residual M-distance is smaller than the specified -level, where the p-value is computed according to an appropriate theoretical chi-square distribution. The larger the ALPHAOUT= value, the more liberal the criterion for detecting outliers.

In addition to displaying the outliers as defined by the ALPHAOUT= criterion, PROC CALIS also displays the next 5 observations with the largest residual M-distances for reference. However, the total number of observations in the displayed output in the displayed output cannot exceed 30 or the number of original observations, whichever is smaller.

This option is relevant only when residual analysis is requested with the RESIDUAL option and with raw data input.

specifies a confidence interval () for the Steiger and Lind (1980) root mean square error of approximation (RMSEA) coefficient (see Browne and Du Toit 1992). The default value is , which corresponds to a 90% confidence interval for the RMSEA.

specifies an absolute singularity criterion r (r > 0), for the inversion of the information matrix, which is needed to compute the covariance matrix. The default value for r or ASING= is the square root of the smallest positive double precision value.

When inverting the information matrix, the following singularity criterion is used for the diagonal pivot of the matrix:

where VSING and MSING are the specified values in the VSINGULAR= and MSINGULAR= options, respectively, and is the jth diagonal element of the information matrix. Note that in many cases a normalized matrix is decomposed (where ), and the singularity criteria are modified correspondingly.

specifies the formula for asymptotic covariances used in the weight matrix for WLS and DWLS estimation. The ASYCOV option is effective only if METHOD= WLS or METHOD=DWLS and no INWGT= input data set is specified. The following formulas are implemented:

By default, AYSCOV=BIASED is used for covariance analyses and ASYCOV=CORR is used for correlation analyses. Therefore, in almost all cases you do not need to set the ASYCOV= option once you specify the covariance or correlation analysis by the COV or CORR option.

inputs the SAS-data-set that contains the fit information of the baseline model of your choice. This customized baseline model replaces the default uncorrelatedness model for computing several fit indices of your target model. Typically, you create the BASEFIT= data set by using the OUTFIT= option in a previous PROC CALIS fitting of your customized baseline model. Using the BASEFIT= option assumes that you fit your customized baseline model and your target model with the same data, number of groups (for multiple-group analysis), and estimation method. Typically, your baseline model should be more restrictive (or have fewer parameters) than your target model.

For example, the following statements use the compound symmetry model (COVPATTERN=COMPSYM) as the customized baseline model for the subsequent factor model with two factors:

proc calis data=abc outfit=outf method=gls covpattern=compsym;
   var=x1-x10;
run;

proc calis data=abc method=gls basefit=outf;
   factor n=2;
   var=x1-x10;
run;

The fit information of the customized baseline model is saved as an OUTFIT= data set called outf, which is then read as a BASEFIT= data set in the next PROC CALIS run for fitting the target factor model. Notice that in this example the baseline model and the target factor model use the same data set, abc, and the same GLS estimation method.

Alternatively, you can use the BASEFUNC= option to input the function value and the degrees of freedom of your customized baseline model. See the BASEFUNC= option for details. The BASEFIT= option is ignored if you also specify the BASEFUNC= option.

Notice that the fit information in the BASEFIT= data set does not affect the computation of all fit indices. Mainly, it affects the incremental fit indices, because these indices are defined as the incremental fit of a target model over a baseline model. Among all absolute and parsimonious fit indices, only the parsimonious goodness-of-fit (PGFI) index (Mulaik et al., 1989) is affected by the input fit information provided in the BASEFIT= data set.

If you specify METHOD=LSDWLS, LSFIML, LSGLS, or LSML for your target model, the fit information in the BASEFIT= data set is assumed to have been obtained from the DWLS, FIML, GLS, or ML estimation of your customized baseline model. Hence, the fit information in the BASEFIT= data set applies only to the second estimation of your target model. The unweighted least squares (ULS) estimation of the target model still uses the uncorrelatedness model as the baseline model for computing fit indices.

If you use a BASEFIT= data set to input the fit information of your customized baseline model in a multiple-group analysis, then the baseline model function values, chi-squares, and degrees of freedom for individual groups are not known and hence not displayed in the multiple-group fit summary table. The Bentler-Bonett normed fit index (NFI) is also not displayed in the multiple-group fit summary table, because the baseline model function values for individual groups are not saved in the BASEFIT= data set.

inputs the fit function value r and the degrees of freedom i of the baseline model of your choice. This customized baseline model replaces the default uncorrelatedness model for computing several fit indices of your target model. To use this option, you must first fit your customized baseline model and then use this option to input the baseline model fit information when you fit your target model.

Using the BASEFUNC= option assumes that you fit your customized baseline model and your target model with the same data, number of groups (for multiple-group analysis), and estimation method. Typically, your baseline model should be more restrictive (or have fewer parameters) than your target model.

For example, assume that after fitting a customized baseline model you find that the function value of the baseline model is 20.54 and the model degrees of freedom are 15. The following code inputs the function value and the degrees of freedom of the customized baseline model by using the BASEFUNC= option:

proc calis data=abc basefunc(df=15)=20.54;
   path 
     f1 ===> x1-x5   = 1.,
     f2 ===> x6-x10  = 1.,
     f1 ===> f2;
run;

You can use the following equivalent syntax to provide the same baseline model fit information:

basefunc(df=15)=20.54
basefunc(15)=20.54 
basefunc=20.54(df=15)
basefunc=20.54(15)

It is emphasized here that you should input the fit function value, but not the model fit chi-square value, of the baseline model in the BASEFUNC= option. For all estimation methods except the full information maximum likelihood (FIML) method in PROC CALIS, the model fit chi-square values are some multiples of the fit function values. See the section Estimation Criteria for the definitions of the various fit functions that are assumed by the BASEFUNC= option.

Alternatively, it might be easier to use the BASEFIT= option to specify the SAS data set that contains the baseline model fit information. Such a SAS data set is typically created by using the OUTFIT= option in the PROC CALIS fitting of your customized baseline model. See the BASEFIT= option for details. However, the BASEFIT= option is ignored if you also specify the BASEFUNC= option.

Notice that the specified values in the BASEFUNC= option do not affect the computation of all fit indices. Mainly, they affect the incremental fit indices, because these indices are defined as the incremental fit of a target model over a baseline model. Among all absolute and parsimonious fit indices, only the parsimonious goodness-of-fit (PGFI) index (Mulaik et al., 1989) is affected by the values provided in the BASEFUNC= option.

If you specify METHOD=LSDWLS, LSFIML, LSGLS, or LSML for your target model, the fit information provided by the BASEFUNC= option is assumed to have been obtained from the DWLS, FIML, GLS, or ML estimation of your customized baseline model. Hence, the fit information provided by the BASEFUNC= option applies only to the second estimation of your target model. The unweighted least squares (ULS) estimation of the target model still uses the uncorrelatedness model as the baseline model for computing fit indices.

If you use the BASEFUNC= option to input the fit information of your customized baseline model in a multiple-group analysis, then the baseline model function values, chi-squares, and degrees of freedom for individual groups are not known and hence not displayed in the multiple-group fit summary table. The Bentler-Bonett NFI is also not displayed in the multiple-group fit summary table, because the baseline model function values for individual groups are not provided with the BASEFUNC= option.

computes univariate skewness and kurtosis by formulas uncorrected for bias.

See the section Measures of Multivariate Kurtosis for more information.

specifies a correction factor c for the chi-square statistics for model fit. You can specify a name for a built-in correction factor or a value between 0 and 1 as the CHICORRECT= value. The model fit chi-square statistic is computed as:

where N is the total number of observations, k is the number of independent groups, and F is the optimized function value. Application of these correction factors requires appropriate specification of the covariance structural model suitable for the chi-square correction. For example, using CHICORRECT=UNCORR assumes that you are fitting a covariance structure with free parameters on the diagonal elements and fixed zeros off-diagonal elements of the covariance matrix. Because all the built-in correction factors assume multivariate normality in their derivations, the appropriateness of applying these built-in chi-square corrections to estimation methods other than METHOD=ML is not known.

Valid names for the CHICORRECT= value are as follows:

defines the criterion value p for indicating a close fit. The smaller the better fit. The default value for close fit is .05.

analyzes the correlation matrix, instead of the default covariance matrix. See the COVARIANCE option for more details.

analyzes the covariance matrix. Because this is also the default analysis in PROC CALIS, you can simply omit this option when you analyze covariance rather than correlation matrices. If the DATA= input data set is a TYPE=CORR data set (containing a correlation matrix and standard deviations), the default COV option means that the covariance matrix is computed and analyzed.

Unlike many other SAS/STAT procedures (for example, the FACTOR procedure) that analyze correlation matrices by default, PROC CALIS uses a different default because statistical theories of structural equation modeling or covariance structure analysis are mostly developed for covariance matrices. You must use the CORR option if correlation matrices are analyzed.

specifies one of the built-in covariance structures for the data. The purpose of this option is to fit some commonly-used direct covariance structures efficiently without the explicit use of the MSTRUCT model specifications. With this option, the covariance structures are defined internally in PROC CALIS. The following names for the built-in covariance structures are supported:

When you specify the covariance structure model by means of the COVPATTERN= option, you can define the set of variables in the analysis by the VAR statement (either within the scope of the PROC CALIS statement or the GROUP statements). If the VAR statement is not used, PROC CALIS uses all numerical variables in the data sets.

Except for the EQCOVMAT pattern, all other built-in covariance patterns are primarily designed for single-group analysis. However, you can still use these covariance pattern options for multiple-group situations. For example, consider the following three-group analysis:

proc calis covpattern=compsym;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
run;

In this specification, all three groups are fitted by the compound symmetry pattern. However, there would be no constraints across these groups. PROC CALIS generates two distinct parameters for each group: _varparm_mdl1 and _covparm_mdl1 for Group 1, _varparm_mdl2 and _covparm_mdl2 for Group 2, and _varparm_mdl3 and _covparm_mdl3 for Group 3. Similarly, the _mdlxx suffix, where xx represents the model number, is applied to the parameters defined by the SATURATED, SPHERICITY (or SIGSQI), and UNCORR (or DIAG) covariance patterns in multiple-group situations. However, chi-square correction, whenever it is applicable to single-group analysis, is not applied to such multiple-group analyses.

You can also apply the COVPATTERN= option partially to the groups in the analysis. For example, the following statements apply the spheric pattern to Group 1 and Group 2 only:

proc calis covpattern=sphericity;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
   model 3 / group=3;
     path    x1 ===> y3;
run;

Group 3 is fitted by Model 3, which is specified explicitly by a PATH model with distinct covariance structures.

If the EQCOVMAT pattern is specified instead, as shown in the following statements, the equality of covariance matrices still holds for Groups 1 and 2:

proc calis covpattern=eqcovmat;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
   model 3 / group=3;
     path    x1 ===> y3;
run;

However, Group 3 has it own covariances structures as specified in Model 3. In this case, the chi-square correction due to Box (1949) is not applied because the null hypothesis is no longer testing the equality of covariance matrices among the groups in the analysis.

Use the MEANPATTERN= option if you also want to analyze some built-in mean structures along with the covariance structures.

specifies a nonnegative threshold r, which determines whether the eigenvalues of the information matrix are considered to be zero. If the inverse of the information matrix is found to be singular (depending on the VSINGULAR=, MSINGULAR=, ASINGULAR=, or SINGULAR= option), a generalized inverse is computed using the eigenvalue decomposition of the singular matrix. Those eigenvalues smaller than r are considered to be zero. If a generalized inverse is computed and you do not specify the NOPRINT option, the distribution of eigenvalues is displayed.

specifies an input data set that can be an ordinary SAS data set or a specially structured TYPE=CORR, TYPE=COV, TYPE=UCORR, TYPE=UCOV, TYPE=SSCP, or TYPE=FACTOR SAS data set, as described in the section Input Data Sets. If the DATA= option is omitted, the most recently created SAS data set is used.

changes the initial values of all variance parameters by the relationship:

where is the new initial value and is the original initial value. The initial values of all variance parameters should always be nonnegative to generate positive definite predicted model matrices in the first iteration. By using values of , for example, r = 2, r = 10, and so on, you can increase these initial values to produce predicted model matrices with high positive eigenvalues in the first iteration. The DEMPHAS= option is effective independent of the way the initial values are set; that is, it changes the initial values set in the model specification as well as those set by an INMODEL= data set and those automatically generated for the FACTOR, LINEQS, LISMOD, PATH, or RAM models. It also affects the initial values set by the START= option, which uses, by default, DEMPHAS=100 if a covariance matrix is analyzed and DEMPHAS=10 for a correlation matrix.

reduces the degrees of freedom of the model fit test by i. In general, the number of degrees of freedom is the total number of nonredundant elements in all moment matrices minus the number of parameters, t. Because negative values of i are allowed, you can also increase the number of degrees of freedom by using this option.

makes the effective number of observations n + 1. You can also use the NOBS= option to specify the number of observations.

computes and displays total, direct, and indirect effects for the unstandardized and standardized estimation results. Standard errors for the effects are also computed. Note that this displayed output is not automatically included in the output generated by the PALL option.

Note also that in some situations computations of total effects and their partitioning are not appropriate. While total and indirect effects must converge in recursive models (models with no cyclic paths among variables), they do not always converge in nonrecursive models. When total or indirect effects do not converge, it is not appropriate to partition the effects. Therefore, before partitioning the total effects, the convergence criterion must be met. To check the convergence of the effects, PROC CALIS computes and displays the “stability coefficient of reciprocal causation”— that is, the largest modulus of the eigenvalues of the matrix, which is the square matrix that contains the path coefficients of all endogenous variables in the model. Stability coefficients less than one provide a necessary and sufficient condition for the convergence of the total and the indirect effects. Otherwise, PROC CALIS does not show results for total effects and their partitioning. See the section Stability Coefficient of Reciprocal Causation for more information about the computation of the stability coefficient.

displays the extended path estimates such as the variances, covariances, means, and intercepts in the table that contains the ordinary path effect (coefficient) estimates. This option applies to the PATH model only.

specifies the relative function convergence criterion. The optimization process is terminated when the relative difference of the function values of two consecutive iterations is smaller than the specified value of r; that is,

where FSIZE can be defined by the FSIZE= option in the NLOPTIONS statement. The default value is , where FDIGITS either can be specified in the NLOPTIONS statement or is set by default to , where is the machine precision.

instructs that the algorithm to compute the approximate covariance matrix of parameter estimates used for computing the approximate standard errors and modification indices when the information matrix is singular. If the number of parameters t used in the model you analyze is smaller than the value of i, the time-expensive Moore-Penrose (G4) inverse of the singular information matrix is computed by eigenvalue decomposition. Otherwise, an inexpensive pseudo (G1) inverse is computed by sweeping. By default, i = 60.

See the section Estimation Criteria for more details.

specifies the relative gradient convergence criterion.

Termination of all techniques (except the CONGRA technique) requires the following normalized predicted function reduction to be smaller than r. That is,

where FSIZE can be defined by the FSIZE= option in the NLOPTIONS statement. For the CONGRA technique (where a reliable Hessian estimate is not available),

is used. The default value is .

specifies an input data set that contains initial estimates for the parameters used in the optimization process and can also contain boundary and general linear constraints on the parameters. Typical applications of this option are to specify an OUTEST= data set from a previous PROC CALIS analysis. The initial estimates are taken from the values of the PARMS observation in the INEST= data set.

specifies an input data set that contains information about the analysis model. A typical use of the INMODEL= option is when you run an analysis with its model specifications saved as an OUTMODEL= data set from a previous PROC CALIS run. Instead of specifying the main or subsidiary model specification statements in the new run, you use the INMODEL= option to input the model specification saved from the previous run.

Sometimes, you might create an INMODEL= data set from modifying an existing OUTMODEL= data set. However, editing and modifying OUTMODEL= data sets requires good understanding of the formats and contents of the OUTMODEL= data sets. This process could be error-prone for novice users. For details about the format of INMODEL= or OUTMODEL= data sets, see the sectionInput Data Sets.

It is important to realize that INMODEL= or OUTMODEL= data sets contain only the information about the specification of the model. These data sets do not store any information about the bounds on parameters, linear and nonlinear parametric constraints, and programming statements for computing dependent parameters. If required, these types of information must be provided in the corresponding statement specifications (for example, BOUNDS, LINCON, and so on) in addition to the INMODEL = data set.

An OUTMODEL= data set might also contain default parameters added automatically by PROC CALIS from a previous run (for example, observations with _TYPE_=ADDPCOV, ADDMEAN, or ADDPVAR). When reading the OUTMODEL= model specification as an INMODEL= data set in a new run, PROC CALIS ignores these added parameters so that the model being read is exactly like the previous PROC CALIS specification (that is, before default parameters were added automatically). After interpreting the specification in the INMODEL= data set, PROC CALIS will then add default parameters appropriate to the new run. The purpose of doing this is to avoid inadvertent parameter constraints in the new run, where another set of automatic default parameters might have the same generated names as those of the generated parameter names in the INMODEL= data set.

If you want the default parameters in the INMODEL= data set to be read as a part of model specification, you must also specify the READADDPARM option. However, using the READADDPARM option should be rare.

For highly nonlinear objective functions, such as the EXP function, the default initial radius of the trust-region algorithms (TRUREG, DBLDOG, and LEVMAR) or the default step length of the line-search algorithms can produce arithmetic overflows. If an arithmetic overflow occurs, specify decreasing values of 0 < r < 1 such as INSTEP=1E–1, INSTEP=1E–2, INSTEP=1E–4, and so on, until the iteration starts successfully.

For more details, see the section Computational Problems.

specifies an input data set that contains the weight matrix used in generalized least squares (GLS), weighted least squares (WLS, ADF), or diagonally weighted least squares (DWLS) estimation, if you do not specify the INV option at the same time. The weight matrix must be positive definite because its inverse must be defined in the computation of the objective function. If the weight matrix defined by an INWGT= data set is not positive definite, it can be ridged using the WRIDGE= option. See the section Estimation Criteria for more information. If you specify the INWGT(INV)= option, the INWGT= data set contains the inverse of the weight matrix, rather than the weight matrix itself. Specifying the INWGT(INV)= option is equivalent to specifying the INWGT= and INWGTINV options simultaneously. With the INWGT(INV)= specification, the input matrix is not required to be positive definite. See the INWGTINV option for more details. If no INWGT= data set is specified, default settings for the weight matrices are used in the estimation process. The INWGT= data set is described in the section Input Data Sets. Typically, this input data set is an OUTWGT= data set from a previous PROC CALIS analysis.

specifies that the INWGT= data set contains the inverse of the weight matrix, rather than the weight matrix itself. This option is effective only with an input weight matrix specified in the INWGT= data set and with the generalized least squares (GLS), weighted least squares (WLS or ADF), or diagonally weighted least squares (DWLS) estimation. With this option, the input matrix provided in the INWGT= data set is not required to be positive definite. Also, the ridging requested by the WRIDGE= option is ignored when you specify the INWGTINV option.

computes and displays univariate kurtosis and skewness, various coefficients of multivariate kurtosis, and the numbers of observations that contribute most to the normalized multivariate kurtosis. See the section Measures of Multivariate Kurtosis for more information. Using the KURTOSIS option implies the SIMPLE display option. This information is computed only if the DATA= data set is a raw data set, and it is displayed by default if the PRINT option is specified. The multivariate least squares kappa and the multivariate mean kappa are displayed only if you specify METHOD=WLS and the weight matrix is computed from an input raw data set. All measures of skewness and kurtosis are corrected for the mean. Using the BIASKUR option displays the biased values of univariate skewness and kurtosis.

specifies the line-search method for the CONGRA, QUANEW, and NEWRAP optimization techniques. See Fletcher (1980) for an introduction to line-search techniques. The value of i can be any integer between 1 and 8, inclusively; the default is i=2.

specifies the degree of accuracy that should be obtained by the line-search algorithms LIS=2 and LIS=3. Usually an imprecise line search is inexpensive and successful. For more difficult optimization problems, a more precise and more expensive line search might be necessary (Fletcher, 1980, p. 22). The second (default for NEWRAP, QUANEW, and CONGRA) and third line-search methods approach exact line search for small LSPRECISION= values. If you have numerical problems, you should decrease the LSPRECISION= value to obtain a more precise line search. The default LSPRECISION= values are displayed in the following table.

For more details, see Fletcher (1980, pp. 25–29).

specifies the maximum number i of function calls in the optimization process. The default values are displayed in the following table.

The default is used if you specify MAXFUNC=0. The optimization can be terminated only after completing a full iteration. Therefore, the number of function calls that is actually performed can exceed the number that is specified by the MAXFUNC= option.

specifies the maximum number i of iterations in the optimization process. Except for the iteratively reweighted least squares (IRLS) algorithm for the robust estimation of model parameters, the default values are displayed in the following table.

The default maximum number of iterations for IRLS is 5000. The default value is used if you specify MAXITER=0 or if you omit the MAXITER option.

The optional second value n is valid only for OMETHOD=QUANEW with nonlinear constraints. It specifies an upper bound n for the number of iterations of an algorithm and reduces the violation of nonlinear constraints at a starting point. The default is n = 20. For example, specifying

maxiter= . 0

means that you do not want to exceed the default number of iterations during the main optimization process and that you want to suppress the feasible point algorithm for nonlinear constraints.

specifies the maximum number of missing patterns to display in the output, where n is between 1 and 9,999. The default MAXMISSPAT= value is 10 or the number of missing patterns in the data, whichever is smaller. The number of missing patterns to display cannot exceed this MAXMISSPAT= value. This option is relevant only when there are incomplete observations (with some missing values in the analysis variables) in the input raw data set and when you use METHOD=FIML or METHOD=LSFIML for estimation.

Because the number of missing patterns could be quite large, PROC CALIS displays a limited number of the most frequent missing patterns in the output. The MAXMISSPAT= and the TMISSPAT= options are used in determining the number of missing patterns to display. The missing patterns are ordered according to the data proportions they account for, from the largest to the smallest. PROC CALIS displays a minimum number of the highest-frequency missing patterns. This minimum number is the smallest among five, the actual number of missing patterns, and the MAXMISSPAT= value. Then, PROC CALIS displays the subsequent high-frequency missing patterns if the data proportion accounted for by each of these patterns is at least as large as the proportion threshold set by the TMISSPAT= value (default at 0.05) until the total number of missing patterns displayed reaches the maximum set by the MAXMISSPAT= option.

specifies one of the built-in mean structures for the data. The purpose of this option is to fit some commonly-used direct mean structures efficiently without the explicit use of the MSTRUCT model specifications. With this option, the mean structures are defined internally in PROC CALIS. The following names for the built-in mean structures are supported:

When you specify the mean structure model by means of the MEANPATTERN= option, you can define the set of variables in the analysis by the VAR statement (either within the scope of the PROC CALIS statement or the GROUP statements). If the VAR statement is not used, PROC CALIS uses all numerical variables in the data sets.

Except for the EQMEANVEC pattern, all other built-in mean patterns are primarily designed for single-group analysis. However, you can still use these mean pattern options for multiple-group situations. For example, consider the following three-group analysis:

proc calis meanpattern=uniform;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
run;

In this specification, all three groups are fitted by the uniform mean pattern. However, there would be no constraints across these groups. PROC CALIS generates a distinct mean parameter for each group: _meanparm_mdl1 for Group 1, _meanparm_mdl2 for Group 2, and _meanparm_mdl3 for Group 3. Similarly, the _mdlxx suffix, where xx represents the model number, is applied to the parameters defined by the SATURATED mean pattern in multiple-group situations.

You can also apply the MEANPATTERN= option partially to the groups in the analysis. For example, the following statements apply the ZERO mean pattern to Group 1 and Group 2 only:

proc calis meanpattern=zero;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
   model 3 / group=3;
     path    x1 ===> y3;
     means x1 = mean_x1;
run;

Group 3 is fitted by Model 3, which is specified explicitly by a PATH model with a distinct mean parameter mean_x1.

If the EQMEANVEC pattern is specified instead, as shown in the following statements, the equality of mean vectors still holds for Groups 1 and 2:

proc calis meanpattern=eqmeanvec;
   group 1 / data=set1;
   group 2 / data=set2;
   group 3 / data=set3;
   model 3 / group=3;
     path    x1 ===> y3;
     means x1 = mean_x1;
run;

However, Group 3 has it own mean structures as specified in Model 3.

Use the COVPATTERN= option if you also want to analyze some built-in covariance structures along with the mean structures. If you use the MEANPATTERN= option but do not specify the COVPATTERN= option, a saturated covariance structure model (that is, COVPATTERN=SATURATED) is assumed by default.

invokes the analysis of mean structures. By default, no mean structures are analyzed. You can specify the MEANSTR option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it propagates to all models. When this option is specified in the MODEL statement, it applies only to the local model. Except for the COSAN model, the MEANSTR option adds default mean parameters to the model. For the COSAN model, the MEANSTR option adds null mean vectors to the model. Instead of using the MEANSTR option to analyze the mean structures, you can specify the mean and the intercept parameters explicitly in the model by some model specification statements. That is, you can specify the intercepts in the LINEQS statement, the intercepts and means in the PATH or the MEAN statement, the _MEAN_ matrix in the MATRIX statement, or the mean structure formula in the COSAN statement. The explicit mean structure parameter specifications are useful when you need to constrain the mean parameters or to create your own references of the parameters.

specifies the method of parameter estimation. The default is METHOD=ML. Valid values for name are as follows:

computes and displays Lagrange multiplier (LM) test indices for constant parameter constraints, equality parameter constraints, and active boundary constraints, as well as univariate and multivariate Wald test indices. The modification indices are not computed in the case of unweighted or diagonally weighted least squares estimation.

The Lagrange multiplier test (Bentler, 1986; Lee, 1985; Buse, 1982) provides an estimate of the reduction that results from dropping the constraint. For constant parameter constraints and active boundary constraints, the approximate change of the parameter value is displayed also. You can use this value to obtain an initial value if the parameter is allowed to vary in a modified model. See the section Modification Indices for more information.

Relying solely on the LM tests to modify your model can lead to unreliable models that capitalize purely on sampling errors. See MacCallum, Roznowski, and Necowitz (1992) for the use of LM tests.

specifies a relative singularity criterion r (r > 0) for the inversion of the information matrix, which is needed to compute the covariance matrix. If you do not specify the SINGULAR= option, the default value for r or MSING= is 1E–12; otherwise, the default value is 1E–4 SING, where SING is the specified SINGULAR= value.

When inverting the information matrix, the following singularity criterion is used for the diagonal pivot of the matrix:

where ASING and VSING are the specified values of the ASINGULAR= and VSINGULAR= options, respectively, and is the jth diagonal element of the information matrix. Note that in many cases a normalized matrix is decomposed (where ), and the singularity criteria are modified correspondingly.

turns off the automatic adjustment of degrees of freedom when there are active constraints in the analysis. When the adjustment is in effect, most fit statistics and the associated probability levels will be affected. This option should be used when you believe that the active constraints observed in the current sample will have little chance to occur in repeated sampling. See the section Adjustment of Degrees of Freedom for more discussion on the issue.

specifies the number of observations. If the DATA= input data set is a raw data set, nobs is defined by default to be the number of observations in the raw data set. The NOBS= and EDF= options override this default definition. You can use the RDF= option to modify the nobs specification. If the DATA= input data set contains a covariance, correlation, or scalar product matrix, you can specify the number of observations either by using the NOBS=, EDF=, and RDF= options in the PROC CALIS statement or by including a _TYPE_='N' observation in the DATA= input data set.

disables the display of index types in the fit summary table.

deactivates the inherited MEANSTR option for the analysis of mean structures. You can specify the NOMEANSTR option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it does not have any apparent effect because by default the mean structures are not analyzed. When this option is specified in the MODEL statement, it deactivates the inherited MEANSTR option from the PROC CALIS statement. In other words, this option is mainly used for resetting the default behavior in the local model that is specified within the scope of a particular MODEL statement. If you specify both the MEANSTR and NOMEANSTR options in the same statement, the NOMEANSTR option is ignored.

Caution: This option does not remove the mean structure specifications from the model. It only deactivates the MEANSTR option inherited from the PROC CALIS statement. The mean structures of the model are analyzed as long as there are mean structure specifications in the model (for example, when you specify the means or intercepts in any of the main or subsidiary model specification statements).

suppresses the display of the analytic results of the missing patterns. This option is relevant only when there are incomplete observations (with some missing values in the analysis variables) in the input raw data set and when you use METHOD=FIML or METHOD=LSFIML for estimation.

suppresses the computation of modification indices. The NOMOD option is useful in connection with the PALL option because it saves computing time.

prints the model results in the order they appear in the input specifications. This is the default printing behavior. In contrast, the ORDERSPEC option arranges the model results by the types of parameters. You can specify the NOORDERSPEC option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it does not have any apparent effect because by default the model results display in the same order as that in the input specifications. When this option is specified in the MODEL statement, it deactivates the inherited ORDERSPEC option from the PROC CALIS statement. In other words, this option is mainly used for resetting the default behavior in the local model that is specified within the scope of a particular MODEL statement. If you specify both the ORDERSPEC and NOORDERSPEC options in the same statement, the NOORDERSPEC option is ignored.

suppresses the printing of parameter names in the model results. The default is to print the parameter names. You can specify the NOPARMNAME option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it propagates to all models. When this option is specified in the MODEL statement, it applies only to the local model.

suppresses the displayed output. Note that this option temporarily disables the Output Delivery System (ODS). See Chapter 20: Using the Output Delivery System, for more information.

suppresses the printing of standardized results. The default is to print the standardized results.

suppresses the printing of the standard error estimates. Standard errors are not computed for unweighted least squares (ULS) or diagonally weighted least squares (DWLS) estimation. In general, standard errors are computed even if the STDERR display option is not used (for file output). You can specify the NOSTDERR option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it propagates to all models. When this option is specified in the MODEL statement, it applies only to the local model.

specifies the optimization method or technique. Because there is no single nonlinear optimization algorithm available that is clearly superior (in terms of stability, speed, and memory) for all applications, different types of optimization methods or techniques are provided in the CALIS procedure. The optimization method or technique is specified by using one of the following names in the OMETHOD= option:

For fewer than 500 parameters (), OMETHOD=NRRIDG (Newton-Raphson Ridge) is the default optimization technique for the FIML estimation, and OMETHOD=LEVMAR (Levenberg-Marquardt) is the default optimization technique for the all other estimation methods. For , OMETHOD=QUANEW (quasi-Newton) is the default method, and for , OMETHOD=CONGRA (conjugate gradient) is the default method. Each optimization method or technique can be modified in various ways. See the section Use of Optimization Techniques for more details.

prints the model and group results in the order of the model or group numbers, starting from the smallest number. It also arrange some model results by the parameter types. In effect, this option turns on the ORDERGROUPS, ORDERMODELS, and ORDERSPEC options. The ORDERALL is not a default option. By default, the printing of the results follow the order of the input specifications.

prints the group results in the order of the group numbers, starting from the smallest number. The default behavior, however, is to print the group results in the order they appear in the input specifications.

prints the model results in the order of the model numbers, starting from the smallest number. The default behavior, however, is to print the model results in the order they appear in the input specifications.

arranges some model results by the types of parameters. The default behavior, however, is to print the results in the order they appear in the input specifications. You can specify the ORDERSPEC option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it propagates to all models. When this option is specified in the MODEL statement, it applies only to the local model.

creates an output data set that contains the parameter estimates, their gradient, Hessian matrix, and boundary and linear constraints. For METHOD=ML, METHOD=GLS, and METHOD=WLS, the OUTEST= data set also contains the information matrix, the approximate covariance matrix of the parameter estimates ((generalized) inverse of information matrix), and approximate standard errors. If linear or nonlinear equality or active inequality constraints are present, the Lagrange multiplier estimates of the active constraints, the projected Hessian, and the Hessian of the Lagrange function are written to the data set.

See the section OUTEST= Data Set for a description of the OUTEST= data set. If you want to create a SAS data set in a permanent library, you must specify a two-level name. For more information about permanent libraries and SAS data sets, see SAS Language Reference: Concepts.

creates an output data set that contains the values of the fit indices. See the section OUTFIT= Data Set for details.

creates an output data set that contains the model information for the analysis, the parameter estimates, and their standard errors. An OUTMODEL= data set can be used as an input INMODEL= data set in a subsequent analysis by PROC CALIS. If you want to create a SAS data set in a permanent library, you must specify a two-level name. For more information about permanent libraries and SAS data sets, see SAS Language Reference: Concepts.

creates an output data set that contains the BY group variables, the analyzed covariance or correlation matrices, and the predicted and residual covariance or correlation matrices of the analysis. You can specify the correlation or covariance matrix in an OUTSTAT= data set as an input DATA= data set in a subsequent analysis by PROC CALIS. See the section OUTSTAT= Data Set for a description of the OUTSTAT= data set. If the model contains latent variables, this data set also contains the predicted covariances between latent and manifest variables and the latent variable score regression coefficients (see the PLATCOV option ). If the FACTOR statement is used, the OUTSTAT= data set also contains the rotated and unrotated factor loadings, the unique variances, the matrix of factor correlations, the transformation matrix of the rotation, and the matrix of standardized factor loadings.

You can use the latent variable score regression coefficients with PROC SCORE to compute factor scores.

If you want to create a SAS data set in a permanent library, you must specify a two-level name. For more information about permanent libraries and SAS data sets, see SAS Language Reference: Concepts.

creates an output data set that contains the elements of the weight matrix or the its inverse used in the estimation process. The inverse of the weight matrix is output only when you specify an INWGT= data set with the INWGT= and INWGTINV options (or the INWGT(INV)= option alone) in the same analysis. As a result, the entries in the INWGT= and OUTWGT= data sets are consistent. In other situations where the weight matrix is computed by the procedure or obtained from the OUTWGT= data set without the INWGTINV option, the weight matrix is output in the OUTWGT= data set. Furthermore, if the weight matrix is computed by the procedure, the OUTWGT= data set contains the elements of the weight matrix on which the WRIDGE= and the WPENALTY= options are applied.

You cannot create an OUTWGT= data set with an unweighted least squares or maximum likelihood estimation. The weight matrix is defined only in the GLS, WLS (ADF), or DWLS fit function. An OUTWGT= data set can be used as an input INWGT= data set in a subsequent analysis by PROC CALIS. See the section OUTWGT= Data Set for the description of the OUTWGT= data set. If you want to create a SAS data set in a permanent library, you must specify a two-level name. For more information about permanent libraries and SAS data sets, see SAS Language Reference: Concepts.

displays all optional output except the output generated by the PCOVES and PDETERM options.

Caution: The PALL option includes the very expensive computation of the modification indices. If you do not really need modification indices, you can save computing time by specifying the NOMOD option in addition to the PALL option.

prints the parameter names in the model results. This is the default printing behavior. In contrast, the NOPARMNAME option suppresses the printing of the parameter names in the model results. You can specify the PARMNAME option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it does not have any apparent effect because by default model results show the parameter names. When this option is specified in the MODEL statement, it deactivates the inherited NOPARMNAME option from the PROC CALIS statement. In other words, this option is mainly used for resetting the default behavior in the local model that is specified within the scope of a particular MODEL statement. If you specify both the PARMNAME and NOPARMNAME options in the same statement, the PARMNAME option is ignored.

displays the covariance or correlation matrix that is analyzed and the predicted model covariance or correlation matrix.

displays the following:

The covariance matrix of the parameter estimates is not computed for estimation methods ULS and DWLS. This displayed output is not included in the output generated by the PALL option.

displays three coefficients of determination: the determination of all equations (DETAE), the determination of the structural equations (DETSE), and the determination of the manifest variable equations (DETMV). These determination coefficients are intended to be global means of the squared multiple correlations for different subsets of model equations and variables. The coefficients are displayed only when you specify a FACTOR, LINEQS, LISMOD, PATH, or RAM model, but they are displayed for all five estimation methods: ULS, GLS, ML, WLS, and DWLS.

You can use the STRUCTEQ statement to define which equations are structural equations. If you do not use the STRUCTEQ statement, PROC CALIS uses its own default definition to identify structural equations.

The term “structural equation” is not defined in a unique way. The LISREL program defines the structural equations by the user-defined BETA matrix. In PROC CALIS, the default definition of a structural equation is an equation that has a dependent left-side variable that appears at least once on the right side of another equation, or an equation that has at least one right-side variable that appears at the left side of another equation. Therefore, PROC CALIS sometimes identifies more equations as structural equations than the LISREL program does.

displays the parameter estimates. In some cases, this includes displaying the standard errors and t values.

displays the model specification with initial estimates and the vector of initial values.

displays the following:

The estimated covariances between latent and manifest variables and the latent variable score regression coefficients are written to the OUTSTAT= data set. You can use the score coefficients with PROC SCORE to compute factor scores.

specifies the ODS Graphics plots. When you specify only one plot-request, you can omit the parentheses around the plot-request. For example:

PLOTS=ALL PLOTS=RESIDUALS PLOTS=(PP RESBYPRED QQ)

ODS Graphics must be enabled before plots can be requested. For example:

ods graphics on;
proc calis plots;
   path y <=== x,
        y <=== z;
run;
ods graphics off;

For more information about enabling and disabling ODS Graphics, see the section Enabling and Disabling ODS Graphics in Chapter 21: Statistical Graphics Using ODS.

You can specify the following plot-requests:

displays parameter estimates, approximate standard errors, and t values in matrix form if you specify the analysis model using the RAM or LINEQS statement.

adds the options KURTOSIS, RESIDUAL, PLATCOV, and TOTEFF to the default output.

excludes the output produced by the PINITIAL, SIMPLE, and STDERR options from the default output.

displays the fit assessment table only.

displays the weight matrix used in the estimation. The weight matrix is displayed after the WRIDGE= and the WPENALTY= options are applied to it. However, if you specify an INWGT= data set by the INWGT= and INWGTINV options (or the INWGT(INV)= option alone) in the same analysis, this option displays the elements of the inverse of the weight matrix.

is an alias for the INSTEP= option for Levenberg-Marquardt minimization.

specifies a positive integer as a seed value for the pseudo-random number generator to generate initial values for the parameter estimates for which no other initial value assignments in the model definitions are made. Except for the parameters in the diagonal locations of the central matrices in the model, the initial values are set to random numbers in the range . The values for parameters in the diagonals of the central matrices are random numbers multiplied by 10 or 100. See the section Initial Estimates for more information.

makes the effective number of observations the actual number of observations minus the RDF= value. The degree of freedom for the intercept should not be included in the RDF= option. If you use PROC CALIS to compute a regression model, you can specify RDF= number-of-regressor-variables to get approximate standard errors equal to those computed by PROC REG.

inputs the generated default parameters (for example, observations with _TYPE_=ADDPCOV, ADDMEAN, or ADDPVAR) in the INMODEL= data set as if they were part of the original model specification. Typically, these default parameters in the INMODEL= data set were generated automatically by PROC CALIS in a previous analysis and stored in an OUTMODEL= data set, which is then used as the INMODEL= data set in a new run of PROC CALIS. By default, PROC CALIS does not input the observations for default parameters in the INMODEL= data set. In most applications, you do not need to specify this option because PROC CALIS is able to generate a new set of default parameters that are appropriate to the new situation after it reads in the INMODEL= data set. Undistinguished uses of the READADDPARM option might lead to unintended constraints on the default parameters.

displays the raw and normalized residual covariance matrix, the rank order of the largest residuals, and a bar chart of the residual tallies. If mean structures are modeled, mean residuals are also displayed and ranked.

For raw data input, this option also displays tables for case-level (observation-level) residual analysis, including outlier and leverage detections and departures of residuals from the theoretical residual distributions. To set the criterion for detecting outliers, use the ALPHAOUT= option. To set the criterion for leverage observations, use the ALPHALEV= option. Case-level residual analysis is not available when you specify METHOD=FIML.

For the covariance and mean residuals, three types of normalized or standardized residual matrices can be chosen with the RESIDUAL= specification.

When ODS Graphics plots of covariance and mean residuals are also requested, the bar charts of residual tallies are suppressed. They are replaced with high quality graphical histograms showing residual distributions. If you still want to display the bar charts in this situation, use the RESIDUAL(TALLY) or RESIDUAL(TALLY)= option.

The RESIDUAL option is also enabled by the PRINT option. See the section Assessment of Fit for more details about the definitions of residuals.

defines a ridge factor r for the diagonal of the covariance or correlation matrix that is analyzed. The matrix is transformed to:

If you do not specify r in the RIDGE option, PROC CALIS tries to ridge the covariance or correlation matrix so that the smallest eigenvalue is about . Because the weight matrix in the GLS method is the same as the observed covariance or correlation matrix, the RIDGE= option also applies to the weight matrix for the GLS estimation, unless you input the weight matrix by the INWGT= option.

Caution: The covariance or correlation matrix in the OUTSTAT= output data set does not contain the ridged diagonal.

specifies the maximum number i of iterations for the iteratively reweighted least squares (IRLS) method to compute the robust mean and covariance matrices with the two-stage robust estimation. This option is relevant only with the use of the ROBUST= option and with raw data input. The default value is 5,000.

You can also specify this option in the GROUP statement so that different groups can use different ROBITER= values. Notice that the ROBITER= option does not specify the maximum number of iterations for the IRLS algorithm used in the direct robust estimation or in the second stage of the two-stage robust estimation. You can specify the MAXITER= option for this purpose.

sets the tuning parameter r () for the robust estimation method that you specify using the ROBUST= option. The ROBPHI= value controls the criterion for downweighting observations. This value indicates approximately the proportion of observations that would receive weights less than 1 (that is, would be downweighted) according to certain theoretical distributions. The larger the ROBPHI= value, the more observations are downweighted (that is, with weights less than 1). The default value is 0.05.

You can also specify this option in the GROUP statement so that different groups can use different ROBPHI= values for the tuning parameters.

invokes the robust estimation method that downweights the outliers in estimation. You can use the ROBUST option only in conjunction with the ML method (METHOD=ML). More accurately, the robust estimation is done by using the iteratively reweighted least squares (IRLS) method under the normal distribution assumption. The model fit of robust estimation is evaluated with the ML discrepancy function.

You must provide raw data input for the robust estimation method to work. With the robust method, the Huber weights are applied to the observations so that outliers are downweighted during estimation. See the section Robust Estimation for details.

You can request the three different types of robust methods by using one of the following names:

For details about these robust methods, see the section Robust Estimation.

To control the proportion of the observations that are downweighted during the robust estimation, you can specify the value of the tuning parameter , which is between 0 and 1, by using the ROBPHI= option. Approximately, of observations would receive weights less than 1 according to certain theoretical distributions. By default, the value of the tuning parameter is set to 0.05 for all robust methods in PROC CALIS.

By default, the robust method uses a maximum of 5,000 iterations to obtain parameter convergence through the IRLS algorithm. You can override this default maximum number of iterations by specifying the ROBITER= option. The default relative parameter convergence criterion for the robust method is 1E–8. See the XCONV= option for the mathematical definition of this criterion and for information about overriding the default convergence criterion.

Because all robust methods reweight the observations iteratively, the observed variable means are always implicitly updated with the robust weights. Therefore, in a sense all robust methods intrinsically analyze the mean structures of models. If you do not specify the MEANSTR option or any mean parameters for your models, PROC CALIS assumes appropriate saturated mean structures for the models. However, when you are computing fit statistics, these saturated mean structures are ignored as if they were never modeled. If you do specify the MEANSTR option or any mean parameters for your models, these mean structures are taken into account in computing fit statistics.

In this release, robust estimation with the IRLS method is not supported when you specify the BOUNDS, LINCON, or NLINCON statement. However, you can still set parameter constraints by using the same parameter names or by specifying the PARAMETERS statement and the SAS programming statements. See the section Setting Constraints on Parameters for techniques to set up implicit parameter constraints by using the PARAMETERS statement and SAS programming statements.

is an alias for the INSTEP= option for line-search algorithms.

displays means, standard deviations, skewness, and univariate kurtosis if available. This information is displayed when you specify the PRINT option. If the KURTOSIS option is specified, the SIMPLE option is set by default.

specifies the singularity criterion r (0 < r < 1) used, for example, for matrix inversion. The default value is the square root of the relative machine precision or, equivalently, the square root of the largest double precision value that, when added to 1, results in 1.

specifies the probability limit used for computing the stepwise multivariate Wald test. The process stops when the univariate probability is smaller than r. The default value is r=0.05.

is an alias for the LSPRECISION= option.

specifies initial estimates for parameters as multiples of the r value. In all CALIS models, you can supply initial estimates individually as parenthesized values after each parameter name. Unspecified initial estimates are usually computed by various reasonable initial estimation methods in PROC CALIS. If none of the initialization methods is able to compute all the unspecified initial estimates, then the remaining unspecified initial estimates are set to r, 10 , or 100 . For variance parameters, 100 is used for covariance structure analyses and 10 is used for correlation structure analyses. For other types of parameters, r is used. The default value is r = 0.5. If the DEMPHAS= option is used, the initial values of the variance parameters are multiplied by the value specified in the DEMPHAS= option. See the section Initial Estimates for more information.

displays approximate standard errors if estimation methods other than unweighted least squares (ULS) or diagonally weighted least squares (DWLS) are used (and the NOSTDERR option is not specified). In contrast, the NOSTDERR option suppresses the printing of the standard error estimates. If you specify neither the STDERR nor the NOSTDERR option, the standard errors are computed for the OUTMODEL= data set. This information is displayed by default when you specify the PRINT option.

You can specify the STDERR option in both the PROC CALIS and the MODEL statements. When this option is specified in the PROC CALIS statement, it does not have any apparent effect because by default the model results display the standard error estimates (for estimation methods other than ULS and DWLS). When this option is specified in the MODEL statement, it deactivates the inherited NOSTDERR or NOSE option from the PROC CALIS statement. In other words, this option is mainly used for resetting the default behavior in the local model that is specified within the scope of a particular MODEL statement. If you specify both the STDERR and NOSTDERR options in the same statement, the STDERR option is ignored.

specifies the proportion threshold for the missing patterns to display in the output, where n is between 0 and 1. The default TMISSPAT= value is 0.05. This option is relevant only when there are incomplete observations (with some missing values in the analysis variables) in the input raw data set and when you use METHOD=FIML or METHOD=LSFIML for estimation.

Because the number of missing patterns could be quite large, PROC CALIS displays a limited number of the most frequent missing patterns in the output. Together with the MAXMISSPAT= option, this option controls the number of missing patterns to display in the output. See the MAXMISSPAT= option for a detailed description about how the number of missing patterns to display is determined.

specifies the update method for the quasi-Newton or conjugate-gradient optimization technique.

For OMETHOD=CONGRA, the following updates can be used:

For OMETHOD=DBLDOG, the following updates (Fletcher, 1987) can be used:

For OMETHOD=QUANEW, the following updates (Fletcher, 1987) can be used:

specifies the divisor used in the calculation of covariances and standard deviations. The default value is VARDEF=N for the METHOD=FIML, and VARDEF=DF for other estimation methods. The values and associated divisors are displayed in the following table, where k is the number of partial variables specified in the PARTIAL statement. When a WEIGHT statement is used, is the value of the WEIGHT variable in the jth observation, and the summation is performed only over observations with positive weight.

specifies a relative singularity criterion r (r > 0) for the inversion of the information matrix, which is needed to compute the covariance matrix. If you do not specify the SINGULAR= option, the default value for r or VSING= is 1E–8; otherwise, the default value is SING, which is the specified SINGULAR= value.

When inverting the information matrix, the following singularity criterion is used for the diagonal pivot of the matrix:

where ASING and MSING are the specified values of the ASINGULAR= and MSINGULAR= options, respectively, and is the jth diagonal element of the information matrix. Note that in many cases a normalized matrix is decomposed (where ), and the singularity criteria are modified correspondingly.

specifies the penalty weight for the WLS and DWLS fit of the diagonal elements of a correlation matrix (constant 1s). The criterion for weighted least squares estimation of a correlation structure is

where r is the penalty weight specified by the WPENALTY=r option and the are the elements of the inverse of the reduced weight matrix that contains only the nonzero rows and columns of the full weight matrix . The second term is a penalty term to fit the diagonal elements of the correlation matrix. The default value is 100. The reciprocal of this value replaces the asymptotic variance corresponding to the diagonal elements of a correlation matrix in the weight matrix , and it is effective only with the ASYCOV=CORR option, which is the default for correlation analyses. The often used value r = 1 seems to be too small in many cases to fit the diagonal elements of a correlation matrix properly. The default WPENALTY= value emphasizes the importance of the fit of the diagonal elements in the correlation matrix. You can decrease or increase the value of r if you want to decrease or increase the importance of the diagonal elements fit. This option is effective only with the WLS or DWLS estimation method and the analysis of a correlation matrix.

See the section Estimation Criteria for more details.

Caution: If you input the weight matrix by the INWGT= option, the WPENALTY= option is ignored.

defines a ridge factor r for the diagonal of the weight matrix used in GLS, WLS, or DWLS estimation. The weight matrix is transformed to

The WRIDGE= option is applied on the weight matrix before the following actions occur:

Caution: If you input the weight matrix by the INWGT= option, the OUTWGT= data set will contain the same weight matrix without the ridging requested by the WRIDGE= option. This ensures that the entries in the INWGT= and OUTWGT= data sets are consistent. The WRIDGE= option is ignored if you input the inverse of the weight matrix by the INWGT= and INWGTINV options (or the INWGT(INV)= option alone).

specifies the relative parameter convergence criterion. Termination requires a small relative parameter (x) change in subsequent iterations, that is,

The default value for r is 1E–8 for robust estimation (see the ROBUST option) with the iteratively reweighted least squares method, and it is 0 for other estimation methods. The default value for XSIZE is 0. You can change this default value by specifying the XSIZE= option in the NLOPTIONS statement.