TECNOLOGÍAS PROPIETARIAS DE ALTO RENDIMJENTO

ements needed to set up command files using the EQS syntax and notation. An input file in EQS is made up of various commands. The beginning of each command is signaled by a forward slash (/) and its end by a semicolon (;). A command is typically followed by several keywords, or subcommands, and can go on for several lines. To begin the introduction to EQS, a list of commands that arguably are most often used in practice when conducting SEM analyses are presented next (for further details see Bentler, 2004). Each command is illustrated using the factor analysis model originally dis- played in Fig. 6 in Chapter 1. For ease of discussion, the same figure is dis- played again in Fig. 7. In addition, and to keep all command files visually separate from the regular text, all command lines are capitalized through- out the book.

Title Command

One of the first things needed to create an EQS input file is a title command. This command simply describes in plain English the type of model exam- ined (e.g., a confirmatory factor analysis model or a path analytic model) and perhaps some of its specifics (e.g., a study of college students or middle managers). The title command is initiated by the keyword /TITLE. On the line immediately following, and for as many lines as needed, an explanatory title is provided. For example, suppose that the model of concern is the fac- tor analysis model displayed in Fig. 7. The title command could then be listed as

/TITLE

EQS INPUT FILE FOR A FACTOR ANALYSIS MODEL OF THREE INTERRELATED CONSTRUCTS EACH MEASURED BY THREE INDICATORS;

We emphasize that although a title command can be kept to a single or a couple of descriptive lines, like in this example, the more details are pro-

vided in it the better one will recall the purpose of the modeling, especially when revisiting the command file later.

Data Command

The data command lists details about the data to be analyzed. This is initi- ated by using the keyword /SPECIFICATIONS. On the next line, the exact number of variables in the data set are provided using the keyword (sub- command) VARIABLES= , followed by that number. Then information about the number of observations in the data set is given using the keyword CASES= , followed by sample size. The data to be used in the study may be placed directly in the input file or in a separate data file. If the data are avail- able as a covariance matrix, the keyword MATRIX=COV can be used (al- though it is not strictly needed as it is the default option). If the data are in

FIG. 7. Example factor analysis model. F1 = Parental dominance; F2 = Child intelli- gence; F3 = Achievement motivation.

raw form (one column per variable, in as many lines as sample size), then the keyword MATRIX=RAW is used and the name of the file enclosed in apostrophes (including its location, i.e., path to it) are provided with the keyword DATA_FILE= (e.g., DATA_FILE=‘C:\DATA\DATAFILE’). If the data are in the form of a covariance matrix and to be placed directly in the input file (e.g., the covariance matrixSin Chap. 1), the command /MATRIX is used later in the input file followed by the matrix itself. (Although the covariance matrix typically would appear at the very end of the input file for conve- nience reasons—see final EQS input file below—for continuity reasons we mention its inclusion into the command file at this point.) If a matrix of vari- able interrelationships other than the covariance matrix is to be analyzed, this information is provided using the keyword ANALYSIS= , followed by MOMENTS if the mean structure is to be analyzed (i.e., variable means along with covariance matrix, as in Chap. 6), or by CORRELATION if the correlation matrix is to be analyzed.

The default method of estimation in EQS is maximum likelihood (ML). If a method other than ML is to be used for estimation purposes, it is stated af- ter the keyword METHOD=, which is followed by its abbreviation in the program language (e.g., GLS or LS for unweighted least squares, and ROBUST for the robust ML method; e.g., Bentler, 2004). Although EQS pro- vides the option of selecting from among several estimation procedures, as indicated in Chapter 1, only the use of the ML method will be exemplified in this introductory text. Utilizing the factor analysis model in Fig. 7 as an illus- tration, the data command line can then be listed as (in case the model is fit- ted to data from 245 subjects; we note that the last 3 subcommands of the command /SPECIFICATIONS are defaults and do not need to be explicitly stated in the input file):

/SPECIFICATIONS VARIABLES=9; CASES=245; METHOD=ML; MATRIX=COV; ANALYSIS=COV; /MATRIX 1.01 .32 1.50 .43 .40 1.22 .38 .25 .33 1.13 .30 .20 .30 .7 1.06 .33 .22 .38 .72 .69 1.12 .20 .08 .07 .20 .27 .20 1.30 .33 .19 .22 .09 .22 .12 .69 1.07 .52 .27 .36 .33 .37 .29 .50 .62 1.16

It should be noted that because each of the subcommands ends with a semico- lon, they can all be stated also in a single line; the only command in which no semicolon is needed to mark its end is /MATRIX (in addition to the data lines). Model-Definition Commands

The next commands needed in the EQS input file, following the flowchart presented earlier, deal with model description. To accomplish this aim, one can take a close look at the path diagram of the model to be fitted and provide to the software information about its parameters. Setting up the model defi- nition commands involves: (a) writing out the equations relating each de- pendent variable to its explanatory variables; (b) determining the status of all variances of independent variables (whether free or fixed, and at what value in the latter case); and (c) determining the status of the covariances for all in- dependent variables. These activities are achieved by using the commands /EQUATIONS, /VARIANCES, and /COVARIANCES, respectively.

Each free or constrained parameter in the model is denoted in EQS by an asterisk. Following closely the path diagram in Fig. 7 results in 21 asterisks appearing in the model-definition equations and commands, which num- ber as discussed in Chap. 1 equals that of free model parameters. The com- mand /EQUATIONS initiates a listing of the model equations, which in the current example of Fig. 7 is as follows

/EQUATIONS V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F2 + E4; V5 = *F2 + E5; V6 = *F2 + E6; V7 = *F3 + E7; V8 = *F3 + E8; V9 = *F3 + E9;

We stress that model parameters (i.e., the ninel’s in Equations 1 in Chap. 1) have been explicitly represented by asterisks in the listing of model equa- tions. In addition, if needed, one can also assign a special start value to any model parameter by writing that value immediately before the asterisk in the corresponding equation (e.g., V9 = .9*F3 + E9 assigns a start value of .9 to the loading of V9 upon the third latent variable). This does not change the status of the parameter in question (e.g., from free to fixed), but only signals to the software that this value will be the one this parameter will re- ceive at the initial step of the numerical iteration process. Alternatively, fixed factor loadings are represented by their value placed immediately be-

fore the latent variable they belong to (i.e., they are not followed by an aster- isk, unlike free parameters).

The command /VARIANCES is used to inform the software about the sta- tus of the independent variable variances (recall Rule 1 in Chap. 1). Accord- ing to Fig. 7, there are nine residual variances and three factor variances, which represent all variances of independent variables in this model. The factor variances, however, will be fixed to 1 following Rule 6 in order to in- sure that the latent variable metrics are set. Hence, we add the following lines to the input file:

/VARIANCES

F1 TO F3 = 1; E1 TO E9 = *;

Note that one can use the TO convention in order to save tedious writing of all independent variables in the model, which becomes a particularly handy feature with large models having many error and/or latent variable variances.

Finally, information about independent variable covariances—that is, the three factor covariances in Fig. 7—must be communicated to the pro- gram. This is accomplished with the command /COVARIANCES:

/COVARIANCES

F2,F1=*; F3,F1=*; F3,F2=*;

Using the TO convention, the last line can be shortened to F1 TO F3 = *; . Once the commands dealing with model definition have been com- pleted, it is important to ensure that Rules 5 and 6 have not been contra- dicted in the input file. Thus, for the model in Fig. 7, a final check should make sure that each of the three factor variances is indeed fixed at 1 (Rule 6) and in particular that no variance or covariance of dependent variables as well as no covariance of a dependent and an independent variable have been declared model parameters (Rule 5). Lastly, in the example of Fig. 7, counting the number of asterisks one finds 21 model parameters declared in the input file—just as many as there are asterisks in the path diagram. Ob- viously, if the two numbers differ, some model parameters have either been left out or incorrectly declared as such (for the case of no parameter constraints).

In conclusion of this section, we note that the example considered does not include any requests for particular information from the final solution (see last part of earlier flow chart). This is because no additional output in- formation beyond that provided by the default settings of EQS was needed. Later in this book we will include such requests, however, which either re- late to the execution of the iteration process or ask the program to list infor- mation that it otherwise does not routinely provide.

Complete EQS Command File for the Model in Fig. 7

Based on the above discussion, the following complete EQS command file emerges for the confirmatory factor analysis model of concern in this sec- tion. The use of the command /END signals the end of the EQS input file for this model.

/TITLE

EQS INPUT FILE FOR A FACTOR ANALYSIS MODEL OF THREE INTERRELATED CONSTRUCTS EACH MEASURED BY THREE INDICATORS;

/SPECIFICATIONS

VARIABLES=9; CASES=245; METHOD=ML; MATRIX=COV; ANALYSIS=COV; /EQUATIONS V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F2 + E4; V5 = *F2 + E5; V6 = *F2 + E6; V7 = *F3 + E7; V8 = *F3 + E8; V9 = *F3 + E9; /VARIANCES F1 TO F3 = 1; E1 TO E9 = *; /COVARIANCES F1,F2=*; F1,F3=*; F2,F3=*; /MATRIX 1.01 .32 1.50 .43 .40 1.22 .38 .25 .33 1.13 .30 .20 .30 .7 1.06 .33 .22 .38 .72 .69 1.12 .20 .08 .07 .20 .27 .20 1.30 .33 .19 .22 .09 .22 .12 .69 1.07 .52 .27 .36 .33 .37 .29 .50 .62 1.16 /END; A Useful Abbreviation

A particularly helpful feature of EQS (as well as other software) is that each command and keyword/subcommand can be abbreviated to its first three

letters. This often saves a considerable amount of time for the researcher when setting up command files. In this way, the input file of the immedi- ately preceding subsection can be shortened as follows.

/TIT

EQS INPUT FILE FOR A FACTOR ANALYSIS MODEL OF THREE INTERRELATED CONSTRUCTS EACH MEASURED BY THREE INDICATORS;

/SPE

VAR=9; CAS=245; MET=ML; MAT=COV; ANA=COV; /EQU V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F2 + E4; V5 = *F2 + E5; V6 = *F2 + E6; V7 = *F3 + E7; V8 = *F3 + E8; V9 = *F3 + E9; /VAR F1 TO F3 = 1; E1 TO E9 = *; /COV F1,F2=*; F1,F3=*; F2,F3=*; /MAT 1.01 .32 1.50 .43 .40 1.22 .38 .25 .33 1.13 .30 .20 .30 .7 1.06 .33 .22 .38 .72 .69 1.12 .20 .08 .07 .20 .27 .20 1.30 .33 .19 .22 .09 .22 .12 .69 1.07 .52 .27 .36 .33 .37 .29 .50 .62 1.16 /END;

We note that the last three subcommands of the specifications command can be omitted as they are default options.

Imposing Parameter Restrictions

An issue that frequently arises in empirical research is testing substantively meaningful hypotheses. For example, suppose that when dealing with the model in Fig. 7 one were interested in examining the plausibility of the hy-

pothesis that the first three observed variables—the indicators of the Paren- tal domination factor—have equal factor loadings (i.e., represent a triplet of tau-equivalent tests). This assumption is tantamount to the three measures assessing the same construct, Parental dominance, in the same units of measurement. In order to introduce this constraint in the model under consideration, a new command handling parameter restrictions needs to be included in the input file. This command is /CONSTRAINTS and contains the following specification of the imposed parameter equalities:

/CONSTRAINTS

(V1,F1)=(V2,F1)=(V3,F1);

We note that within parentheses first comes the dependent and then inde- pendent variable to which the parameter pertains, in case it is a regression coefficient; this order is immaterial for variance and covariance parameters. For consistency reasons, we suggest that all constraints imposed in a model be included immediately after the /COVARIANCE command, which usually ends the model-definition part of an EQS input file.

INTRODUCTION TO THE LISREL NOTATION AND SYNTAX