The CALIS Procedure |

PATH Statement |

- PATH path < , path ...> ;

and

A

<---, <--, <-, or <

A

--->, -->, ->, or >

The PATH statement specifies the paths in your structural equation model. You can specify at most one PATH statement in a model within the scope of either the PROC CALIS statement or a MODEL statement. To complete the PATH model specifications, you might need to add some subsidiary model specification statements such as the PVAR, PCOV, and the MEAN statements. The following is the syntax for the PATH modeling language:

- PATH path < , path ...> ;
- PVAR partial-variance-parameters ;
- PCOV partial-covariance-parameters ;
- MEAN mean-parameters ;

Paths in structural equation modeling represent the functional relationships among observed and latent variables. You can specify the paths in your model by using the PATH statement. Paths in the PATH statement are separated by commas. Notice that paths from the errors or disturbances are not necessary in the PATH statement. Essentially, the roles of error or disturbance terms in the PATH model are represented by the associated error variances of the endogenous variables in the model.

The PVAR statement specifies the parameters for the variances or error (partial) variances. The PCOV statement specifies the parameters for the covariances or error (partial) covariances. The MEAN statement specifies the parameters for the means or intercepts. For details about these subsidiary model specification statements, see the syntax of the individual statements.

In each path entry of the PATH statement, you specify two lists of variables: *var_list* and *var_list2*. Depending on the direction of the *arrow* specification, one group of variables contains the outcome variables and the other group contains the predictor variables. Optionally, you can specify the *parameter-spec* at the end of each path entry. You can specify the following five types of the parameters for the path entries:

unnamed free parameters

initial values

fixed values

free parameters with names provided

free parameters with names and initial values provided

For example, in the following statement you specify a model with five paths:

PATH V1 <--- F1 , V2 <--- F1 = (0.5), V3 <--- F1 = 1., V4 <--- F1 = b1, V5 <--- F1 = b2 (.4);

The first path entry specifies a path from F1 to V1. The effect of F1 (or the path coefficient) is an unnamed free parameter. For this path effect parameter, PROC CALIS generates a parameter name with the _Parm prefix and appended with a unique integer (for example, _Parm1). The second path entry specifies a path from F1 to V2. The effect of F1 is also an unnamed free parameter with an initial estimate of 0.5. PROC CALIS also generates a parameter name for effect parameter. The third path entry specifies a path from F1 to V3. The effect of F1 is also a fixed value of 1.0. This value stays the same in the model estimation. The fourth path entry specifies a path from F1 to V4. The effect of F1 is a free parameter named b1. The fifth path entry specifies a path from F1 to V5. The effect of F1 is a free parameter named b2, with an initial value of 0.4.

You can specify multiple variables in the *var_list* and *var_list2* lists. For example, the following statement specifies five paths from F1 to V1–V5:

PATH F1 ---> V1-V5;

All the five effects of F1 on the five variables are unnamed free parameters. If both *var_list* and *var_list2* lists contain multiple variables, you must be careful about the order of the variables when you also specify parameters at the end of the path entry. For example, the following statement specifies the paths from the predictor variables x1–x2 to the outcome variables y1–y3:

PATH y1-y3 <--- x1-x2 = a1-a6;

The PATH statement specifies six paths in the path entry. These six paths have effect parameters a1–a6. This specification is equivalent to the following specification:

PATH y1 <--- x1 = a1; y1 <--- x2 = a2; y2 <--- x1 = a3; y2 <--- x2 = a4; y3 <--- x1 = a5; y3 <--- x2 = a6;

The following statement shows another example of multiple-path specification:

PATH x1-x2 ---> y1-y3 = b1-b6;

This specification is equivalent to the following specification with separate path specifications:

PATH x1 ---> y1 = b1; x1 ---> y2 = b2; x2 ---> y3 = b3; x2 ---> y1 = b4; x2 ---> y2 = b5; x2 ---> y3 = b6;

You can also specify parameter with mixed types in any path entry, as shown in the following specification:

PATH F1 ---> y1-y3 = 1. b1(.5) (.3), F2 ---> y4-y6 = 1. b2 b3(.7);

This specification is equivalent to the following expanded version:

PATH F1 ---> y1 = 1., F1 ---> y2 = b1(.5), F1 ---> y3 = (.3), F2 ---> y4 = 1., F2 ---> y5 = b2, F2 ---> y6 = b3(.7);

Notice that in the original specification with multiple-path entries, 0.5 is interpreted as the initial value for the parameter b1, but not as the initial estimate for the path from F1 to y3. In general, an initial value that follows a parameter name is associated with the free parameter.

If you indeed want to specify that b1 is a free parameter *without* an initial estimate and 0.5 is the initial estimate for the path from F1 to y3 (while keeping all other specification the same), you can use a null initial value specification, as shown in the following statement:

PATH F1 ---> y1-y3 = 1. b1() (.5) , F2 ---> y4-y6 = 1. b2 b3(.7);

This way 0.5 becomes the initial value for the path from F1 to y3. Because a parameter list with mixed types might be confusing, you can break down the specifications into separate path entries to remove ambiguities. For example, you can use the following specification equivalently:

PATH F1 ---> y1 = 1., F1 ---> y2 = b1, F1 ---> y3 = (.5) , F2 ---> y4-y6 = 1. b2 b3(.7);

The equal signs in the path entries are optional when the parameter lists do not start with a parameter name. For example, the preceding specification is the same as the following specification:

PATH F1 ---> y1 1., F1 ---> y2 = b1, F1 ---> y3 (.5) , F2 ---> y4-y6 1. b2 b3(.7);

Notice that in the second path entry, you must retain the equal sign because b1 is a parameter name. Omitting the equal sign makes the specification erroneous because b1 is treated as a variable. This might cause serious estimation problems. Omitting the equal signs might be cosmetically appealing in specifying fixed values or initial values (for example, the first and the third path entries). However, the gain of doing that is not much as compared to the clarity of specification that results from using the equal signs consistently.

If you provide fewer parameters than the number of paths in a path entry, all the remaining parameters are treated as unnamed free parameters. For example, the following specification specifies the free parameter beta to the first path and assigns unnamed free parameters to the remaining four paths:

PATH F1 ---> y1 z1 z2 z3 z4 = beta;

This specification is equivalent to the following specification:

PATH F1 ---> y1 = beta, F1 ---> z1 z2 z3 z4;

If you intend to fill up all values with the last parameter specification in the list, you can use the continuation syntax ** [...]**,

PATH F1 ---> y1 z1 z2 z3 z4 = beta gamma [...];

This specification is equivalent to the following specification:

PATH F1 ---> y1 z1 z2 z3 z4 = beta 4*gamma;

The repetition factor ** 4*** means that gamma repeats 4 times.

However, you must be careful not to provide too many parameters. For example, the following specification results in an error:

PATH SES_Factor ---> y1 z1 z2 z3 z4 = beta gamma1-gamma6;

Because there are only five paths in the specification, parameters gamma5 and gamma6 are excessive.

It is important to understand the default parameters in the PATH model. First, if you know which parameters are default free parameters, you can make your specification more efficient by omitting the specifications of those parameters that can be set by default. For example, because all variances and covariances among exogenous variables (excluding error terms) are free parameters by default, you do not need to specify them with the PCOV and PVAR statements if these variances and covariances are not constrained. Second, if you know which parameters are default fixed zero parameters, you can specify your model accurately. For example, because all error covariances in the PATH model are fixed zeros by default, you must use the PCOV statement to specify the partial (error) covariances among the endogenous variables if you want to fit a model with correlated errors. See the section Default Parameters in the PATH Model for details about the default parameters of the PATH model.

If you define a new model by using a reference (old) model in the REFMODEL statement, you might want to modify some path specifications from the PATH statement of the reference model before transferring the specifications to the new model. To change a particular path specification from the reference model, you can simply respecify the same path with the desired parameter specification in the PATH statement of the new model. To delete a particular path and its associated parameter from the reference model, you can specify the desired path with a missing value specification in the PATH statement of the new model.

- PATH path < , path ...> ;
- PVAR partial-variance-parameters ;
- PCOV partial-covariance-parameters ;
- MEAN mean-parameters ;

The new model is formed by integrating with the old model in the following ways:

- Duplication:
If you do not specify in the new model a parameter location that exists in the old model, the old parameter specification is duplicated in the new model.

- Addition:
If you specify in the new model a parameter location that does not exist in the old model, the new parameter specification is used in the new model.

- Deletion:
If you specify in the new model a parameter location that also exists in the old model and the new parameter is denoted by the missing value '.', the old parameter specification is not copied into the new model.

- Replacement:
If you specify in the new model a parameter location that also exists in the old model and the new parameter is not denoted by the missing value '.', the new parameter specification replaces the old one in the new model.

For example, consider the following specification of a two-group analysis:

proc calis; group 1 / data=d1; group 2 / data=d2; model 1 / group=1; path V1 <--- F1 = 1., V2 <--- F1 = load1, V3 <--- F1 = load2, F1 <--- V4 = b1, F1 <--- V5 = b2, F1 <--- V6 = b3; pvar E1-E3 = ve1-ve3, F1 = vd1, V5-V6 = phi4-phi6; pcov V1 V2 = cve12; model 2 / group=2; refmodel 1; path V3 <--- F1 = load1, pcov V1 V2 = ., V2 V3 = cve23; run;

You specify Model 2 by referring to Model 1 in the REFMODEL statement. Model 2 is the new model that refers to the old model, Model 1. This example illustrates the four types of model integration rules for the new model:

Duplication: All parameter specifications, except for the partial covariance between V1 and V2 and the V3 <--- F1 path in the old model, are duplicated in the new model.

Addition: The parameter cve23 for the partial covariance between V2 and V3 is added in the new model because there is no corresponding specification in the old model.

Deletion: The specification of partial covariance between V1 and V2 in the old model is not copied into the new model, as indicated by the missing value '.' specified in the new model.

Replacement: The new path V3 <--- F1 replaces the same path in the old model with parameter load1 for the path coefficient. Thus, in the new model paths V3 <--- F1 and v2 <--- F1 are now constrained to have the same path coefficient parameter load1.

The motivation of the extended path modeling language is to express all the features in the path diagram by the *paths* in the PATH statement. The PATH statement discussed so far specifies only the single-headed paths in the path diagram. However, the extended path modeling language includes also the double-headed paths that represent the variances or covariances in the path diagram. With the extended path modeling language, you can specify the variances, covariances, means, and intercepts in the PATH statement, instead of the MEAN, PCOV, and PVAR statements.

- PATH
*var_list**two-head-arrow**var_list2*<=*parameter-spec*><, ...> ;

where a *two-head-arrow* represents one of the following:

<-->, <->, or <>

This syntax enables you to specify covariances between the variables in the *var_list* list and the variables in the *var_list2* list. Consider the following example:

PATH v1 <--> v2, v3 v4 <--> v5 v6 v7 = cv1-cv6;

The first *path* entry specifies the covariance between v1 and v2 as an unnamed free parameter. PROC CALIS generates a name for this parameter. The second *path* entry specifies six covariances with parameters named cv1–cv6. This multiple-covariance specification is equivalent to the following elementwise covariance specification:

PATH v3 <--> v5 = cv1, v3 <--> v6 = cv2, v3 <--> v7 = cv3, v4 <--> v5 = cv4, v4 <--> v6 = cv5, v4 <--> v7 = cv6;

Note that the order of variables in the list is important for determining the assignment of the parameters in the *parameter-spec* list.

If the same variable appears in both of the *var_list* and *var_list2* lists, the "covariance" specification becomes a variance specification for that variable. For example, the following statement specifies two variances:

PATH v1 <--> v1 = 1.0, v2 <--> v2 v3 = var2 cv23;

The first *path* entry specifies the variance of v1 as a fixed value of 1.0. The second *path* entry specifies the variance of v2 as a free parameter named var2, and then the covariance between v2 and v3 as a free parameter named cv23.

It might result in an error if you attempt to use this syntax to specify the variance and covariances among a set of variables. For example, suppose you intend to specify the variances and covariances among v1–v3 as unnamed free parameters by the following statement:

PATH v1-v3 <--> v1-v3 ;

This specification expands to the following elementwise specification:

PATH v1 <--> v1 , v1 <--> v2 , v1 <--> v3 , v2 <--> v1 , v2 <--> v2 , v2 <--> v3 , v3 <--> v1 , v3 <--> v2 , v3 <--> v3 ;

There are nine variance or covariance specifications, but all of the covariances are specified twice. This is treated as a duplication error. The correct way is to specify only the nonredundant covariances, as shown in the following elementwise specification:

PATH v1 <--> v1 , v2 <--> v1 , v2 <--> v2 , v3 <--> v1 , v3 <--> v2 , v3 <--> v3 ;

However, the elementwise specification is quite tedious when the number of variables is large. Fortunately, there is another syntax to deal with this situation. This syntax is discussed in the section Path Syntax for Specifying Variances and Covariances.

- PATH
*two-head-arrow**var_list*<=*parameter-spec*><, ...> ;

This syntax enables you to specify variances among the variables in the *var_list* list. Consider the following example:

PATH <--> v1 = (0.8), <--> v2-v4 ;

The first *path* entry specifies the variance of v1 as an unnamed free parameter with an initial estimate of 0.8. The second *path* entry specifies the variances of v2–v4 as unnamed free parameters. No initial values are given for these three variances. PROC CALIS generates names for all these variance parameters. You can specify these variances equivalently by the elementwise covariance specification syntax, as shown in the following, but former syntax is much more efficient.

PATH v1 <--> v1 = (0.8), v2 <--> v2 , v3 <--> v3 , v4 <--> v4 ;

- PATH
*two-head-arrow*[*var_list*]<=*parameter-spec*><, ...> ;

This syntax enables you to specify all the variances and covariances among the variables in the *var_list* list. For example,the following statement specifies all the variances and covariances among v2–v4:

PATH <--> [v2-v4] = 1.0 cv32 cv33(0.5) cv42 .7 cv44;

This specification is equivalent to the following elementwise specification:

PATH v2 <--> v2 = 1.0, v3 <--> v2 = cv32 , v3 <--> v3 = cv33(0.5), v4 <--> v2 = cv42, v4 <--> v3 = .7, v4 <--> v2 = cv44;

- PATH
*two-head-arrow*(*var_list*)<=*parameter-spec*><, ...> ;

This syntax enables you to specify all the nonredundant covariances among the variables in the *var_list*. For example, the following statement specifies all the nonredundant covariances between v2–v4:

PATH <--> (v2-v5) = cv1-cv6;

This specification is equivalent to the following elementwise specification:

PATH v3 <--> v2 = cv1 , v4 <--> v2 = cv2 , v4 <--> v3 = cv3 , v5 <--> v2 = cv4 , v5 <--> v3 = cv5 , v5 <--> v4 = cv6 ;

- PATH
**1***right-arrow**var_list*<=*parameter-spec*><, ...> ;

where a *right-arrow* is one of the following:

--->, -->, ->, or >

This syntax enables you to specify the means or intercepts of the variables in the *var_list* list as paths from the constant **1**. Consider the following example:

PATH 1 ---> v1 = alpha, 1 ---> v2-v4 = 3*kappa;

The first *path* entry specifies the mean or intercepts of v1 as a free parameter named alpha. The second *path* entry specifies the means or intercepts of v2–v4 as constrained parameters. All these means or intercepts are named kappa so that they have the same estimate.

Whether the mean or intercept is specified depends on whether the variable is endogenous or exogenous. The intercept is specified if the variable is endogenous in the model. Otherwise, the mean of the variable is specified. Fortunately, any variable in the model can have either a mean or intercept (but not both) to specify. Therefore, the "shared" syntax for the means and intercepts specification does not cause any conflicts.

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