The CALIS Procedure

The most important factor in choosing an appropriate algorithm is the nature of the interrelationships among variables in the model. Depending on the pattern of interrelationships, the CALIS procedure provides three basic algorithms: the process-flow algorithm, grouped-flow algorithm, and GRIP algorithm. You can request these algorithms by specifying the ARRANGE= option in the PATHDIAGRAM statement. This section describes these algorithms and the situations in which they work well. The section also describes how the CALIS procedure selects the "best" algorithm by default (by specifying the ARRANGE=AUTOMATIC option).

No layout algorithm for path diagrams works perfectly in all situations. Chances are that some paths in your model would violate the ideal pattern that each of the basic layout algorithms assumes. In some cases, even if the violations are minor—for example, they occur in just one or two paths—they are sufficient to throw off the layout of the diagram.

There are two ways for you to alleviate the problems that are caused by these minor violations. One way is to proactively identify the paths in your model that violate the ideal pattern that the intended layout algorithm assumes. If these minor violations are ignored during the layout process, then the layout algorithm can work well for the majority of patterns in the path diagram. To accomplish this, the CALIS procedure provides the DESTROYER= option for you to specify these minor violations. This section presents examples to illustrate the use of this option.

The other way to alleviate these problems is to use the ODS Graphics Editor to improve the graphical quality of the path diagram that PROC CALIS produces. The ODS Graphics Editor provides several functionalities to improve your path diagrams manually. The section Editing of Path Diagrams with the ODS Graphics Editor describes these functionalities.

To illustrate the use of the DESTROYER= option, consider a higher-order factor model that is specified by the following statements:

proc calis;
   path
      FactorG ===> Factor1-Factor3,
      Factor1 ===> y1-y3  ,
      Factor2 ===> y4-y6  ,
      Factor3 ===> y7-y9  ,
      FactorG ===> y2 y7  ;
run;

In this model, Factor1, Factor2, Factor3, and FactorG are all latent factors. All other variables are observed variables. Which would be the best layout algorithm to draw the path diagram for this model?

Does this model exhibit an ideal process-flow pattern? In this model, FactorG is the only exogenous non-error variable and is considered to be at level 1. Factor1, Factor2, Factor3, y2, and y7 all have directional paths from FactorG, and therefore they are considered to be at level 2. However, because y2 and y7 also have directional paths from Factor1 and Factor3, respectively, they can also be considered to be at level 3. Hence, this model does not exhibit an ideal process-flow pattern.

Does this model exhibit an ideal grouped-flow pattern? If you consider only the factors in the model, FactorG is at level 1 and all other latent factors are at level 2 without ambiguities. Hence, this model exhibits an ideal process-flow pattern in factors or an ideal grouped-flow pattern in all non-error variables. You expect the grouped-flow algorithm to be used to draw the path diagram for this model if you specify the following statements:

proc calis;
   path
      FactorG ===> Factor1-Factor3,
      Factor1 ===> y1-y3  ,
      Factor2 ===> y4-y6  ,
      Factor3 ===> y7-y9  ,
      FactorG ===> y2 y7  = destroyer1 destroyer2;
   pathdiagram diagram=initial notitle;
run;

Figure 29.14 shows the output diagram. Although the path diagram still shows the hierarchical ordering of the latent factors, the two paths, "FactorG ===> y2" and "FactorG ===> y7," weaken the display of the factor-variable clusters for Factor1 and Factor3. Therefore, these two paths can be called destroyer paths, or simply destroyers. For this reason, the effect parameters of these two paths are labeled as "destroyer1" and "destroyer2" in the PATH statement to show their disruptive characteristics.

What if you ignore these destroyer paths in the layout process of the diagram? To do this, you can use the DESTROYER= option in the PATHDIAGRAM statement, as follows:

   pathdiagram diagram=initial notitle destroyer=[FactorG ===> y2 y7];

Because the two destroyer paths are ignored when PROC CALIS determines the layout algorithm, an ideal process-flow pattern is recognized, and the process-flow algorithm is used to lay out all variables in the model. Then, all paths, including the destroyer paths, are simply put back into the picture to produce the final path diagram, as shown in Figure 29.15.

The diagram in Figure 29.15 is preferable to the one in Figure 29.14 because it clearly shows the three levels of variables in the higher-order factor model.

Destroyer paths do not necessarily occur only in ideal process-flow or grouped-flow patterns. Even when the GRIP algorithm is used for a general pattern of variable interrelationships, you might be able to identify potential destroyers in the path diagram. Consider the model that is specified by the following statements:

proc calis;
   path
      Factor1 ===> y1 y2,
      Factor2 ===> y3 y4,
      Factor3 ===> y5 y6,
      Factor1 ===> Factor2 Factor3,
      Factor2 ===> Factor3,
      y2      ===> y5    = destroyer;
   pathdiagram diagram=initial notitle;
run;

This model exhibits neither an ideal process-flow pattern nor an ideal grouped-flow pattern. As a result, PROC CALIS uses the GRIP algorithm automatically to draw the path diagram. Figure 29.16 shows the output diagram.

Because of the presence of the destroyer path "y2 ===> y5," the two factor clusters for Factor1 and Factor3 are drawn closer to each other in the path diagram, thus destroying their distinctive identities that are associated with the related measured variables.

You can use the DESTROYER= option to fix this problem, as shown in the following modified PATHDIAGRAM statement:

   pathdiagram diagram=initial notitle destroyer=[y2 ===> y5];

Figure 29.17 now shows a more preferable path diagram. The factor clusters in this diagram are more distinctive than those in Figure 29.16, in which the algorithm does not handle the destroyer path.

You can edit the output path diagram in PROC CALIS by using the ODS Graphics Editor. The ODS Graphics Editor provides the following useful functionalities for editing path diagrams:

To use the ODS Graphics Editor, you must first enable the creation of editable graphs. For information about the enabling procedures, see the section Enabling and Disabling ODS Graphics in Chapter 21: Statistical Graphics Using ODS. For complete documentation about the ODS Graphics Editor, see the SAS 9.4 ODS Graphics Editor: User's Guide. For information about editing nodes and links in output path diagrams, see Chapter 8 of the SAS 9.4 ODS Graphics Editor: User's Guide. Note that different terminology is used in the SAS 9.4 ODS Graphics Editor: User's Guide. Variables and paths in the current context are referred to as nodes and links in the User’s Guide. For introductory examples of path diagram editing, see Yung (2014).

Taking the conventions and clarity of graphical representations into consideration, the CALIS procedure employs a set of default settings for producing path diagrams. These default settings define the style or scheme for representing the path diagram elements graphically. For example, PROC CALIS displays the error variances as double-headed paths that are attached to the associated variables. In this way, there is no need to display the error variables so that more space is available to present other important graphical elements in path diagrams.

However, researchers do not always agree on the best style or scheme for representing the path diagram elements. Even the same researcher might want to use different representation style or scheme for his or her path diagrams in different situations. In this regard, you can override most of the default settings to customize path diagrams to meet your own requirements. Table 29.6 summarizes the default graphical and nongraphical settings of the path diagram elements. Related options are listed next to these default settings.

One often-used customization in structural equation modeling is to display only the structural component (or structural model) of the full model, which consists of the measurement and the structural components. Traditionally, the structural component refers to the latent factors and their interrelationships, whereas the measurement component refers to observed indicator variables and their relationships with the latent factors.

For example, consider the full model that is displayed as a path diagram in Figure 29.18. The latent factors are Factor1, Factor2, and Factor3. The remaining variables, y1–y9, are observed variables.

The structural component of this full model includes the three latent factors, their directional relationships (represented by the three directed paths), and their variance parameters (represented by the three double-headed arrows that are attached to the factors).

Showing only the structural component is useful when your research is focused mainly on the interrelationships among the latent factors. The measurement component, which usually contains a relatively large number of observed variables for reflecting or defining the latent factors, is of secondary interest. Eliminating the measurement component leads to a clearer path diagram to represent your model.

To request a path diagram that shows the structural component, you can specify the STRUCTURAL option in the PATHDIAGRAM statement, as in the following statements:

proc calis;
   path
      Factor1 ===> y1-y3 ,
      Factor2 ===> y4-y6 ,
      Factor3 ===> y7-y9 ,
      Factor1 ===> Factor2 Factor3,
      Factor2 ===> Factor3;
   pathdiagram diagram=initial structural;
run;

Figure 29.19 shows the path diagram for the structural model. For simplicity in presentation, the DIAGRAM=INITIAL option is used to show the initial model specifications only.

The preceding specification produces the path diagram for the structural component, in addition to that for the full model. To output the path diagram for the structural component only, use the STRUCTURAL(ONLY) option in the PATHDIAGRAM statement, as follows:

pathdiagram diagram=initial structural(only);

Instead of eliminating the measurement component from the path diagram, PROC CALIS provides the EMPHSTRUCT option to highlight the structural component in the full model. For example, the following PATHDIAGRAM statement produces the path diagram in Figure 29.19:

pathdiagram diagram=initial emphstruct;

Figure 29.20 shows a complete model, with emphasis on the structural component. The path diagram labels only the latent factors. Observed variables are represented by small rectangles and are not labeled.

Certainly, the EMPHSTRUCT option is not limited to the initial path diagram, such as the one shown in Figure 29.20. For the unstandardized and standardized solutions, the EMPHSTRUCT option puts similar emphasis on the structural variables. In addition, the diagram displays only the parameter estimates of the structural components. Figure 29.21 shows the path diagram of an unstandardized solution that emphasizes the structural components.

The traditional definition of "structural model" or "structural component" might be too restrictive, in the sense that only latent factors and their interrelationships are included. In contemporary structural equation modeling, sometimes observed variables can take the role of "factors" in the structural component. For example, observed variables that have negligible measurement errors and serve as exogenous or predictor variables in a system of functional equations naturally belong to the structural component. For this reason or other reasons, researchers might want to expand the definition of "structural component" to include some designated observed variables. The STRUCTADD= option enables you to do that.

For example, the following statements specify a model that has two latent factors, Factor1 and Factor2:

proc calis;
   path
      Factor1 ===> y1-y3,
      Factor2 ===> y4-y6,
      y7      ===> Factor1 Factor2;
   pathdiagram diagram=initial notitle structadd=[y7]
               label=[y7="y7 is Exogenous"] ;
   pathdiagram diagram=initial notitle structadd=[y7] struct(only);
               label=[y7="y7 is Exogenous"] ;
   pathdiagram diagram=initial notitle structadd=[y7] emphstruct;
               label=[y7="y7 is Exogenous"] ;
run;

By default, only the latent factors Factor1 and Factor2 are treated as structural variables. However, because the observed variable y7 is exogenous and is a predictor of the two latent variables, it is not unreasonable to also treat it as a structural variable when you are producing path diagrams. Hence, all three PATHDIAGRAM statements use the STRUCTADD= option to include y7 in the set of structural variables. For illustration purposes, variable y7 is labeled as "y7 is Exogenous" in the path diagrams. Although all three PATHDIAGRAM statements produce diagrams for the initial solution, they each display the structural component in a different way.

The first PATHDIAGRAM statement produces a diagram for the full model in Figure 29.22. The structural component is not specifically emphasized. Because y7 is treated as a structural variable, it is larger than the other observed variables, and its size is comparable to that of the default structural variables Factor1 and Factor2.

The second PATHDIAGRAM statement produces a diagram that emphasizes the structural component of the full model in Figure 29.23. Again, because y7 is treated as a structural variable, it is much larger than the other observed variables, and it is emphasized in the same way as the default structural variables Factor1 and Factor2.

The third PATHDIAGRAM statement produces a diagram for the structural component in Figure 29.24. Because y7 is treated as a structural variable, it is also shown in this path diagram, and its size is comparable to that of Factor1 and Factor2.

If you do not include y7 as a structural variable in this PATHDIAGRAM statement, Figure 29.24 would display Factor1 and Factor2 as isolated variables that do not have any directional paths that connect them, making it an uninteresting path diagram display.