AN EXAMPLE CONFIRMATORY FACTOR ANALYSIS MODEL To demonstrate a confirmatory factor analysis model, consider the follow-

Confirmatory Factor Analysis

AN EXAMPLE CONFIRMATORY FACTOR ANALYSIS MODEL To demonstrate a confirmatory factor analysis model, consider the follow-

ing example model with three latent variables: Ability, Achievement Motiva- tion, and Aspiration. The model proposes that three variables are indicators of Ability, three variables are proxies of Motivation, and two variables are in- dicative of Aspiration. Here the primary interest lies in estimating the relationships among Ability, Motivation, and Aspiration. For the purposes of this chapter, assume data are available from a sample ofN= 250 second- year college students for which the normality assumption is plausible. The following observed variables are used in this study:

1. A general ability score (ABILITY1).

2. Grade point average obtained in last year of high school (ABILITY2).

3. Grade point average obtained in first year of college (ABILITY3). 4. Achievement motivation score 1 (MOTIVN1).

5. Achievement motivation score 2 (MOTIVN2). 6. Achievement motivation score 3 (MOTIVN3). 7. A general educational aspiration score (ASPIRN1). 8. A general vocational aspiration score (ASPIRN2).

The example confirmatory factor analysis model is presented in Fig. 10 and the observed covariance matrix in Table 1. The model is initially de- picted in EQS notation usingV1toV8for the observed variables,E1toE8for

the error terms associated with the observed variables, andF1toF3for the

To determine the parameters of the model in Fig. 10, which are designated by asterisks there, we follow the six rules outlined in Chap. 1. Accord- ing to Rule 1, all eight error-term variances are model parameters, and according to Rule 3 the eight factor loadings are also model parameters. In addition, the three construct variances are tentatively designated model parameters (but see the use of Rule 6 later in this paragraph). Following Rule 2, the three covariances between latent variables are also model parameters. Rule 4 is not applicable to this model with regard to latent variables be- cause no explanatory relationships are assumed among any of them. For Rule 5, observe that there are no two-way arrows connecting dependent variables, or a dependent and independent variable in the model in Fig. 10. Finally, Rule 6 requires that the scale of each latent variable be fixed. Be-

cause this study’s primary interest is in estimating the correlations between Ability, Motivation, and Aspiration (which are identical to their covariances if the variances of the latent variables are set equal to 1), the variances of the latent variables are fixed to unity. This decision makes the construct variances fixed parameters rather than free model parameters. Hence, the model in Fig. 10 has altogether 19 parameters (8 factor loadings + 3 factor covariances + 8 error variances = 19), which are symbolized by asterisks.

EQS, LISREL, and MplusCOMMAND FILES EQS Input File

The EQS input file is constructed following the guidelines outlined in Chap. 2. Accordingly, the file begins with a title command line followed by a specifi- cation line providing the number of variables in the model and sample size.

/TITLE

EXAMPLE CONFIRMATORY FACTOR ANALYSIS; /SPECIFICATIONS

CASES=250; VARIABLES=8;

TABLE 1

Covariance Matrix for Confirmatory Factor Analysis Example of Ability, Motivation, and Aspiration

Variable AB1 AB2 AB3 MOT1 MOT2 MOT3 ASP1 ASP2

AB1 .45 AB2 .32 .56 AB3 .27 .32 .45 MOT1 .17 .20 .19 .55 MOT2 .20 .21 .18 .30 .66 MOT3 .19 .25 .20 .30 .36 .61 ASP1 .08 .12 .09 .23 .27 .22 .58 ASP2 .11 .10 .07 .21 .25 .27 .39 .62

To facilitate interpretation of the output, labels are provided for all variables included in the model using the command line /LABELS.

/LABELS

V1=ABILITY1; V2=ABILITY2; V3=ABILITY3; V4=MOTIVN1; V5=MOTIVN2; V6=MOTIVN3; V7=ASPIRN1; V8=ASPIRN2; F1=ABILITY; F2=MOTIVATN; F3=ASPIRATN;

Next the model definition equations are stated followed by the remaining model parameters in the variance and covariance commands. The /LMTEST command requests the modification indices discussed in Chapter 1.

/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; /VARIANCES F1 TO F3=1; E1 TO E8=*; /COVARIANCES F1 TO F3=*; /LMTEST;

Finally, the data are provided along with the end of input file command. /MATRIX .45 .32 .56 .27 .32 .45 .17 .20 .19 .55 .20 .21 .18 .30 .66 .19 .25 .20 .30 .36 .61 .08 .12 .09 .23 .27 .22 .58 .11 .10 .07 .21 .25 .27 .39 .62 /END;

The complete EQS command file, using the appropriate abbreviations, looks now as follows:

/TIT

CONFIRMATORY FACTOR ANALYSIS MODEL; /SPE

CAS=250; VAR=8; /LAB

V1=ABILITY1; V2=ABILITY2; V3=ABILITY3; V4=MOTIVN1; V5=MOTIVN2; V6=MOTIVN3; V7=ASPIRN1; V8=ASPIRN2; F1=ABILITY; F2=MOTIVATN; F3=ASPIRATN;

/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; /VAR F1 TO F3=1; E1 TO E8=*; /COV F1 TO F3=*; /LMTEST; /MAT .45 .32 .56 .27 .32 .45 .17 .20 .19 .55 .20 .21 .18 .30 .66 .19 .25 .20 .30 .36 .61 .08 .12 .09 .23 .27 .22 .58 .11 .10 .07 .21 .25 .27 .39 .62 /END;

LISREL Command File

After stating a title and data details, the LISREL input file describes as follows the CFA model under consideration that is based on 8 observed and 3 latent variables, and with model parameters being appropriate elements of corresponding matrices.

CONFIRMATORY FACTOR ANALYSIS MODEL DA NI=8 NO=250 CM .45 .32 .56 .27 .32 .45 .17 .20 .19 .55 .20 .21 .18 .30 .66 .19 .25 .20 .30 .36 .61 .08 .12 .09 .23 .27 .22 .58 .11 .10 .07 .21 .25 .27 .39 .62 LA

ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 C ASPIRN1 ASPIRN2

MO NY=8 NE=3 PS=SY,FR TE=DI,FR LY=FU,FI LE

ABILITY MOTIVATN ASPIRATN FR LY(1, 1) LY(2, 1) LY(3, 1) FR LY(4, 2) LY(5, 2) LY(6, 2) FR LY(7, 3) LY(8, 3)

FI PS(1, 1) PS(2, 2) PS(3, 3) VA 1 PS(1, 1) PS(2, 2) PS(3, 3) OU MI

Specifically, after the same title the data definition line declares that the model will be fit to data on eight variables collected from 250 subjects. The sample covariance matrix CM is given next, along with the variable labels (note the use of C, for Continue, to wrap over to the second label line). The latent variables are also assigned labels by using the notation LE (for Labels for Etas, following the notation of the general LISREL model in which the Greek letteretarepresents a latent variable). Next, in the model command line the three matrices PS, TE, and LY are defined. The latent covariance matrix PS is initially declared to be symmetric and free (i.e., all its elements are free parameters), which defines all factor variances and covariances as model parameters. Subsequently, for reasons discussed in the previous section, the variances in the matrix PS are fixed to a value of 1. The error covariance matrix TE is defined as diagonal (i.e., no error covariances are introduced) and therefore only has as model parameters the error variances along its main diagonal. Defining then the matrix of factor loadings LY as a fixed and full (rect- angular) matrix relating the eight manifest variables to the three latent variables, permits those loadings that relate the corresponding indicators to their factors to be declared freed in the next lines. Last but not least, to illus- trate use and interpretation of modification indices, which we may wish to examine if model fit comes out as unsatisfactory, we include on the OUtput line the request for them with the keyword MI.

MplusCommand File

Here we override some default options available in Mplus, given our interest in estimating factor correlations as model parameters, as mentioned earlier in this chapter. This is accomplished by fixing the factor variances at 1, while the default arrangement in this software is to alternatively fix at 1 the loading of the first listed indicator for each latent variable. The following command file will accomplish our aim.

TITLE: CONFIRMATORY FACTOR ANALYSIS MODEL DATA: FILE IS EX4.COV;

TYPE=COVARIANCE; NOBSERVATIONS=250;

VARIABLE:NAMES ARE ABILITY1 ABILITY2 ABILITY3 MOTIVN1 MOTIVN2 MOTIVN3 ASPIRN1 ASPIRN2;

MODEL: F1 BY ABILITY1*1 ABILITY2 ABILITY3; F2 BY MOTIVN1*1 MOTIVN2 MOTIVN3; F3 BY ASPIRN1*1 ASPIRN2;

F1-F3@1;

OUTPUT: MODINDICES(5);

After giving a title to this modeling session, the data location is provided. Since we only have access to the covariance matrix of the eight analyzed variables, the type of data is declared and the sample size stated. Next we give names to the variables in the study, and in the model definition part declare each of the three constructs as being measured by its pertinent indicators. The default options built into Mplusconsider the factor covariances as well as error term variances as model parameters, which is what we need. To override the other default arrangement regarding fixing a factor loading instead of latent variance as we would like, we list all latent variable indicators and after the first of them add an asterisk that signals a start value being stated next for that factor loading. In this way, we free all factor loadings per latent variable. With the last line of the MODEL command, we fix the latent variances at 1, which is not a default arrangement and therefore needs to be explicitly done. The OUTPUT command requests the printing of modification indices in excess of 5, which we may wish to examine if fit of this model turns out not to be satisfactory.

MODELING RESULTS

In document A First Course in Structural Equation Modeling (Page 129-135)