1

SEM BASIC MODELS

2

Regression Model

V3 = *V1 + *V2+ D

V3

V2 *

D V1 *

*

3

Regression with error-in-variables

Ex. 3.1.2 of Fuller (1987)

Data from a sample of Iowa farm operators

Y = ln (farm size) X1 = ln ( # years experience ) X2 = ln (# years education )

(to protect confidentiality, random error was added to each variable)

4

Regression with error-in-variables

F1

F2

F3

V3

E1

E2

E3

D3

V2

V1 *

* *

*

*

*

F3 = *F1 + *F2 + D3 V1 = F1 + E1 V2 = F2 + E2 V3 = F3 + E3

E1 = .0997; E2 = .2013; E3 = .1808;

Coeficientes de fiabilidad son .80, .83 y .89 respectivamente

5

N=176 Regression equation

6

/TITLE MODELO DE REGRESION CON ERROR EN LAS VARIABLES /SPECIFICATIONS CAS=176; VAR=3; ME=ML; /LABELS V1=TAMANO; V2=EXPER; V3=EDUCAC; F1=TAM; F2=EXP; F3=EDU; /EQUATIONS V1 = F1+ E1; V2 = F2+ E2; V3 = F3+ E3; F1=*F2+*F3+D1; /VARIANCES F2 TO F3 = *; D1=*; E1 = 0.0997; E2 = 0.2013; E3 = 0.1808; /COVARIANCES F2,F3 = *; /MATRIX .9148 .2129 1.006 .0714 -.449 1.039 /PRINT DIG=4; /END

Regression with error-in-variables

7

Path analysis model V3 = *V1 + *V2 + D3 V4 = *V1 + *V2 + D4

V3 V1 * D3

V4 V2 * D4

*

* *

8

Simultaneous equations Education development, Sewell, Haller & Ohlendorf (1970) sample of n = 3500

where: Y1 = academic performance (AP), Y2 = significant influences of others(SO), Y3 = educational aspirations (EA), X1 = mental ability (MA), X2 = socioeconomic status (SES).

9

y2

u2

X1

X2

Y1

Y3

Y2

u3

e2

Model:

df = chi2=7.14. Without introducing measurement error on Y2, chi2 is 186.39 with3 df, so …

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//TITLE modified Sewell et al (1970) model /SPECIFICATIONS CAS=3500; VAR=5; ME=ML; MA=COR; ANAL=COR; /LABELS V1=HABMENT; V2=ESTATSOC; V3=EXACAD; V4=INFOTROS; V5=ASPEDUC; /EQUATIONS V3 =*V1 +D1; F1 =*V1+*V2+*V3 +D2; V5 = *V3+*F1 +D3; V4=F1+E1; /VARIANCES E1=*; V1 TO V2 = *; D1 TO D3 = *; /COVARIANCES V1 TO V2 = *; /MATRIX 1.000 .288 1.000 .589 .194 1.000 .438 .359 .473 1.000 .418 .380 .459 .611 1.000 /PRINT DIG=3; /END

Path analysis model

11

Mimic model Joreskog & Goldberger, JASA (1979)

y =social participation

X1 = Income X2 = Occupation X3 = Education

Y1= Church attendance Y2 = Membership Y3 = Frieds Seen

y

X1

X2

X3

Y1

Y2

Y3 e

u2

u3

u1 β1

β2

β3

λ1

λ2 λ3

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Mimic model F1 = *V1 + *V2 + *V3 + D

V4 = *F1 + E4 V5 = *F1 + E5 V6 = *F1 + E6

F1

V1

V2

V3

V4

V5

V6 D1

E5

E6

E2 *

*

*

*

* *

*

*

*

13

ML estimates:

6 overidentifying restrictions. The corresponding chi2 is 12.36 with “P-VALUE” 0.052.

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/TITLE Modelo MIMIC /SPECIFICATIONS VARIABLES=6; CASES=530; METHODS=ML; MATRIX=CORRELATION; /LABELS V1 = Income; V2 = Occupa; V3 = Educat; V4 = Church; V5 = Afiliat ; V6 = Friends; /EQUATIONS V4 = 1F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; F1 = *V1 + *V2 + *V3 + D1; /VARIANCES V1 TO V3 = *; E4 TO E6 = *; D1 = *; /COVARIANCES V2 , V1 = *; V3 , V1 = *; V3 , V2 = *; /MATRIX 1.000 0.304 1.000 0.305 0.344 1.000 0.100 0.156 0.158 1.000 0.284 0.192 0.324 0.360 1.000 0.176 0.136 0.226 0.210 0.265 1.000 /LMTEST /WTEST /PRINT

Mimic model

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Panel data

xtα = γt + βx(t-1)α + lα + µtα

Xtα = xtα + vtα t = 1,2, ..., T α = 1,2,..., N

Anderson (1986)

xtα budget of household α at time t

lα individual (unobserved) characteristic of

household α

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F1 F2 F3 F4 F5 FT

1 1 1 1 1 1

D2 D3 D4 D5 DT

* * * * * *

V1 V2 V3 V4 V5 VT

Panel data

….

….

F0

E2 E3 E4 E5 ET E1

1 1 1 1 1 1

In a stationary process, Var(F1)=[Var(D) + Var F0 ]/(1-β)

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18

Factor analysis Variables

CLASSIC = V1 FRENCH = V2 ENGLISH = V3 MATH = V4 DISCRIM = V5 MUSIC = V6

Correlation matrix 1 .83 1 .78 .67 1 .70 .64 .64 1 .66 .65 .54 .45 1 .63 .57 .51 .51 .40 1

cases = 23;

(Spearman, 1904)

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Single-Factor Model

V1 V4 V3 V2

F1

* * * *

* * * *

V6 V5

* *

* *

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EQS code for a factor model

RESIDUAL COVARIANCE MATRIX (S-SIGMA) :

CLASSIC FRENCH ENGLISH MATH DISCRIM V 1 V 2 V 3 V 4 V 5 CLASSIC V 1 0.000 FRENCH V 2 -0.001 0.000 ENGLISH V 3 0.005 -0.029 0.000 MATH V 4 -0.006 0.003 0.046 0.000 DISCRIM V 5 -0.001 0.054 -0.015 -0.056 0.000 MUSIC V 6 0.003 0.005 -0.017 0.030 -0.049

MUSIC V 6 MUSIC V 6 0.000

CHI-SQUARE = 1.663 BASED ON 9 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.99575 THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS 1.648

.

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Single-Factor Model Loadings’ estimates , s.e. and z-test statistics

CLASSIC =V1 = .960*F1 +1.000 E1 .160 6.019

FRENCH =V2 = .866*F1 +1.000 E2 .171 5.049

ENGLISH =V3 = .807*F1 +1.000 E3 .178 4.529

MATH =V4 = .736*F1 +1.000 E4 .186 3.964

DISCRIM =V5 = .688*F1 +1.000 E5 .190 3.621

MUSIC =V6 = .653*F1 +1.000 E6 .193 3.382

Unique factors

E1 -CLASSIC .078*I .064 I 1.224 I I E2 -FRENCH .251*I .093 I 2.695 I I E3 -ENGLISH .349*I .118 I 2.958 I I E4 - MATH .459*I .148 I 3.100 I I E5 -DISCRIM .527*I .167 I 3.155 I I E6 -MUSIC .574*I .180 I 3.184 I I

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Single-Factor Model

STANDARDIZED SOLUTION:

CLASSIC =V1 = .960*F1 + .279 E1 FRENCH =V2 = .866*F1 + .501 E2 ENGLISH =V3 = .807*F1 + .591 E3 MATH =V4 = .736*F1 + .677 E4 DISCRIM =V5 = .688*F1 + .726 E5 MUSIC =V6 = .653*F1 + .758 E6

23

Factor analysis

F1

V2 V1 V4 V3

E1 E2 E3 E4

F2

Vi = λ Fi + Ei Var Fi = 1 Var Ei = φ

24

Lisrel example Analysis of Reader Reliability in Essay Scoring

Analysis of Reader Reliability in Essay Scoring Votaw's Data Congeneric model estimated by ML DA NI=4 NO=126 LA ORIGPRT1 WRITCOPY CARBCOPY ORIGPRT2 CM 25.0704 12.4363 28.2021 11.7257 9.2281 22.7390 20.7510 11.9732 12.0692 21.8707 MO NX=4 NK=1 LX=FR PH=ST !EQ TD(1) - TD(4) !EQ LX(1) - LX(4)LK Esayabil PD OU