AN EXAMPLE STRUCTURAL REGRESSION MODEL

To demonstrate a structural regression model, consider the following ex- ample concerning mental ability. General mental ability is one of the most extensively studied constructs in the behavioral sciences. According to a popular theory (e.g., Horn, 1982), human intellectual capabilities can be roughly classified into two main clusters, fluid and crystallized intelli- gence.Fluid intelligenceis the component of general intelligence that re- flects an individual’s ability to quickly process a potentially large amount of information in order to solve content-free tasks based on contexts that they are not familiar with from their education or prior socialization pro- cess. Metaphorically, fluid intelligence can be thought of as resembling a human brain’s hardware, or the mechanics of our brain. Fluid intelligence does not include our abilities to retrieve knowledge obtained earlier in life through systems of acculturation or education, but instead refers to our ability to solve unfamiliar problems.Crystallized intelligence, on the other hand, is one’s ability to retrieve knowledge likely obtained earlier in life through culture and education, and in this sense can be considered the software of our brains. In pragmatic terms, tests of fluid intelligence frequently contain series of context-free symbols arranged following a special rule that must be discovered and subsequently used in order to ar- rive at a correct solution. Alternatively, measures of crystallized intelli- gence typically contain items that assess subjects’ levels of knowledge in certain areas.

The example structural regression model considered here focuses on two fluid intelligence subabilities, Induction and Figural relations (in a con- trived empirical setting). Induction relates to one’s capability to reason us- ing analogies and rules of generalization to more comprehensive contexts. Figural relations pertains to our ability to see patterns of relationships be- tween parts of figures, mentally rotate them, and also use forms of inductive reasoning with figural elements. A total of nine measures were collected from a sample of N= 220 high school students, with normality being plau- sible for the data. The following observed variables were used in the study:

1. Induction score 1 obtained in junior year (IND1). 2. Induction score 2 obtained in junior year (IND2). 3. Induction score 3 obtained in junior year (IND3). 4. Figural relations score 1 obtained in junior year (FR11). 5. Figural relations score 2 obtained in junior year (FR12). 6. Figural relations score 3 obtained in junior year (FR13). 7. Figural relations score 1 obtained in senior year (FR21). 8. Figural relations score 2 obtained in senior year (FR22). 9. Figural relations score 3 obtained in senior year (FR23).

The example structural regression model is presented in Fig. 11 and the observed covariance matrix is displayed in Table 2. The model is ini- tially formulated in LISREL notation usingY₁toY₉for the observed vari- ables,e1toe9for the error terms associated with the observed variables,

andh₁toh₃for the latent variables. The model assumes that Induction in junior year of high school (h1) plays an explanatory role for Figural rela-

tions during both junior (h₂) and senior (h₃) years. In addition, the model posits that a student’s junior-year Figural relations affects his or her senior-year Figural relations.

To determine the parameters of the model in Fig. 11, we can follow the six rules outlined in Chap. 1. According to Rule 1, all nine error term vari- ances are model parameters and, according to Rule 3, the nine factor load- ings are also model parameters. In addition, the variances of the structural regression disturbance termsz2andz3are also model parameters. These

disturbances pertain to the latent variablesh₂toh₃, respectively (i.e., ju- nior- and senior-year Figural relations). Thereby,z₂represents the part of junior-year Figural relations that is not accounted for in terms of its postu- lated (linear) explanatory relationship to Induction. Similarly,z₃stands for the part of senior-year Figural relations that is not explained in terms of its assumed (linear) relationships to Induction and junior-year Figural rela- tions. Note that Rule 2 is not applicable in this model because it does not

have any latent covariances. Indeed, the relationship between Induction and both Figural relation constructs are explained in terms of their struc- tural regression coefficients, which, following Rule 4, are model parame- ters. Utilizing the LISREL notation discussed in Chap. 2, these coefficients can be denoted asb21,b31, andb32. These parameters correspondingly relate

Induction (h₁) to junior-year Figural relations (h₂) and senior-year Figural relations (h3), and junior-year Figural relations (h2) to senior-year Figural

relations (h₃). Finally, Rule 6 requires that the scale of each latent variable be fixed. Since this study is focused on determining the explanatory role of the Induction and junior-year Figural relations latent dimensions, it is eas- ier to achieve latent scale fixing by simply setting the loading of the first indi- cator on each latent variable to 1. Using this approach ensures that the Figural relations construct is assessed in the same metric on both occasions. Hence, as can be directly counted, the structural regression model in Fig. 11 has altogether 21 parameters that are symbolized by asterisks.

EQS, LISREL, AND MplusCOMMAND FILES EQS Command File

The EQS input file includes two new definitions. The first deals with the specification of the equations that relate the latent variables to one another.

TABLE 2

Covariance matrix for the structural regression model example

Variable IND1 IND2 IND3 FR11 FR12 FR13 FR21 FR22 FR23

IND1 56.21 IND2 31.55 75.55 IND3 23.27 28.30 44.45 FR11 24.48 32.24 22.56 84.64 FR12 22.51 29.54 20.61 57.61 78.93 FR13 22.65 27.56 15.33 53.57 49.27 73.76 FR21 33.24 46.49 31.44 67.81 54.76 54.58 141.77 FR22 32.56 40.37 25.58 55.82 52.33 47.74 98.62 117.33 FR23 30.32 40.44 27.69 54.78 53.44 59.52 96.95 84.87 106.35

Notes.INDi=ith INDuction indicator at first assessment; FR1i=ith Figural Relations indi- cator at first assessment, and FR2i=ith Figural Relations indicator at second assessment (i

Since the model in Fig. 11 includes structural regression relationships, the following two equations must be introduced in the input file: (a) F2 = *F1 + D2, and (b) F3 = *F1 + *F2 + D3. The second definition concerns the two structural-disturbance terms in the model, i.e., D2 and D3 in these equations. (These terms denote the latent residualsz₂andz₃.) Given that D2 and D3 are independent variables, their variances are model parameters as indicated in the /VARIANCE section next. Thus, the following EQS com- mand file results:

/TITLE

STRUCTURAL REGRESSION MODEL; /SPECIFICATIONS

CASES=220; VARIABLES=9; /LABELS

V1=IND1; V2=IND2; V3=IND3; V4=FR11; V5=FR12; V6=FR13; V7=FR21; V8= FR22; V9=FR23; F1=INDUCTN; F2=FIGREL1; F3=FIGREL2; /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; F2=*F1+D2; F3=*F1+*F2+D3; /VARIANCES F1=*; D2 TO D3=*; E1 TO E9=*; /MATRIX 56.21 31.55 75.55 23.27 28.30 44.45 24.48 32.24 22.56 84.64 22.51 29.54 20.61 57.61 78.93 22.65 27.56 15.33 53.57 49.27 73.76 33.24 46.49 31.44 67.81 54.76 54.58 141.77 32.56 40.37 25.58 55.82 52.33 47.74 98.62 117.33 30.32 40.44 27.69 54.78 53.44 59.52 96.95 84.87 106.35 /END;

LISREL Input File

The LISREL input file is constructed similarly following the guidelines out- lined in Chap. 2. The file begins with a title command line succeeded by in- formation about the number of analyzed variables, sample size and covariance matrix, and observed variable labels. Then, in the model com- mand line, the matrices PS, TE, LY, and BE are defined, and subsequently specific elements of them appropriately declared. Hence, the following LISREL input file results:

STRUCTURAL REGRESSION MODEL DA NI=9 NO=220 CM 56.21 31.55 75.55 23.27 28.30 44.45 24.48 32.24 22.56 84.64 22.51 29.54 20.61 57.61 78.93 22.65 27.56 15.33 53.57 49.27 73.76 33.24 46.49 31.44 67.81 54.76 54.58 141.77 32.56 40.37 25.58 55.82 52.33 47.74 98.62 117.33 30.32 40.44 27.69 54.78 53.44 59.52 96.95 84.87 106.35 LA

IND1 IND2 IND3 FR11 FR12 FR13 FR21 FR22 FR23

MO NY=9 NE=3 PS=SY,FI TE=DI,FR LY=FU,FI BE=FU,FI LE

INDUCTN FIGREL1 FIGREL2 FR LY(2, 1) LY(3, 1)

FR LY(5, 2) LY(6, 2) FR LY(8, 3) LY(9, 3)

VA 1 LY(1, 1) LY(4, 2) LY(7, 3)