SEM BASIC MODELS

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SEM BASIC MODELS

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Regression Model

V3 = V1 + *V2+ D

V3

V2

DV1

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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)

322

211

1

2211

exX

exX

eyY

uxxy

4


F1

F2

F3

V3

E1

E2

E3

D3

V2

V1*

*

*

*

*

*

F3 = F1 + *F2 + D3V1 = F1 + E1V2 = F2 + E2V3 = F3 + E3

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

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

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N=176 Regression equation

2

1

2

1

2

1

x

x

e

x

x

y

000

000

(.107) 313.0)112(. 439.00

x

x

y

858.0449.00

449.0805.00

00699.0

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/TITLE MODELO DE REGRESION CON ERROR EN LAS VARIABLES/SPECIFICATIONSCAS=176; VAR=3; ME=ML;/LABELSV1=TAMANO; V2=EXPER; V3=EDUCAC;F1=TAM; F2=EXP; F3=EDU;/EQUATIONSV1 = F1+ E1;V2 = F2+ E2;V3 = F3+ E3;F1=*F2+*F3+D1;/VARIANCESF2 TO F3 = *;D1=*;E1 = 0.0997;E2 = 0.2013;E3 = 0.1808;/COVARIANCESF2,F3 = *;/MATRIX.9148.2129 1.006.0714 -.449 1.039/PRINTDIG=4;/END


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Path analysis model

V3 = *V1 + *V2 + D3

V4 = V1 + V2 + D4

V3V1

D3

V4V2

D4

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

32311313

22221211212

11111

uYYY

uXXYY

uXY

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

1Y 2Y 3Y 1X 2X

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y2

u2

X1

X2

Y1

Y3

Y2

u3

e2

Model:

0

0

0

e

0

X

X

Y

y

Y

10000

01000

00100

00010

00001

X

X

Y

Y

Y

2

2

1

3

2

1

2

1

3

2

1

0,0,0*,,0diagonal

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/SPECIFICATIONSCAS=3500; VAR=5; ME=ML; MA=COR; ANAL=COR;/LABELSV1=HABMENT; V2=ESTATSOC; V3=EXACAD;V4=INFOTROS; V5=ASPEDUC;/EQUATIONSV3 =*V1 +D1;F1 =*V1+*V2+*V3 +D2;V5 = *V3+*F1 +D3;V4=F1+E1;/VARIANCESE1=*;V1 TO V2 = *;D1 TO D3 = *;/COVARIANCESV1 TO V2 = *;/MATRIX1.000.288 1.000.589 .194 1.000.438 .359 .473 1.000.418 .380 .459 .611 1.000/PRINTDIG=3;/END

Path analysis model

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Mimic modelJoreskog & Goldberger, JASA (1979)

y =social participation

X1 = IncomeX2 = OccupationX3 = Education

Y1= Church attendanceY2 = MembershipY3 = Frieds Seen

y

X1

X2

X3

Y1

Y2

Y3e

u2

u3

u11

2

3

1

2

3

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Mimic model

F1 = V1 + V2 + V3 + DV4 = F1 + E4V5 = F1 + E5V6 = F1 + E6

F1

V1

V2

V3

V4

V5

V6D1

E5

E6

E2

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ML estimates:

(.046)

ey 346.Y

(.060)

ey 634.Y

)046(.

ey 402.Y

)070(.)065(.)066(.

uX 386.X 114.X 269.y

33

22

11

321

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

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/TITLEModelo MIMIC/SPECIFICATIONS VARIABLES=6; CASES=530; METHODS=ML; MATRIX=CORRELATION;/LABELS V1 = Income; V2 = Occupa; V3 = Educat; V4 = Church; V5 = Afiliat ; V6 = Friends;/EQUATIONSV4 = 1F1 + E4;V5 = *F1 + E5;V6 = *F1 + E6;F1 = *V1 + *V2 + *V3 + D1;/VARIANCESV1 TO V3 = *;E4 TO E6 = *;D1 = *;/COVARIANCESV2 , V1 = *;V3 , V1 = *;V3 , V2 = *;/MATRIX1.0000.304 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

xt = t + x(t-1) + l + t

Xt = xt + vtt = 1,2, ..., T

= 1,2,..., NAnderson (1986)

xt budget of household at time t

l individual (unobserved) characteristic of

household

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F1 F2 F3 F4 F5 FT

11 1 1 1 1

D2 D3 D4 D5 DT

** * * * *

V1 V2 V3 V4V5

VT

Panel data

….

….

F0

E2 E3 E4 E5 ETE1

1 1 1 1 1 1

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

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Factor analysis

Variables

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

Correlation matrix1.83 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 V4V3V2

F1

* * * *

* * * *

V6V5

**

**

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EQS code for a factor model /Title confirmatory factor analysis: 1 factor ! (Spearman, 1904 ) eqs/exer3.eqs/Specifications var = 6; cases = 23;/Label v1 = classic; v2 = french; v3 =english; v4 = math; V5 = discrim;V6=music;/equationsV1 = *f1 + e1;V2 = *f1+ e2;V3 = *f1 + e3;V4 = *f1 + e4;V5 = *f1 + e5;V6 = *f1 + e6;/variances f1 = 1; e1 to e6 = *;/matrix1.83 1.78 .67 1.70 .64 .64 1.66 .65 .54 .45 1.63 .57 .51 .51 .40 1/LMTEST/end

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

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Factor analysis

F1

V2V1 V4V3

E1 E2 E3 E4

F2

Vi = Fi + Ei Var Fi = 1 Var Ei =

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Lisrel example Analysis of Reader Reliability in Essay Scoring

Analysis of Reader Reliability in Essay Scoring Votaw's DataCongeneric model estimated by MLDA NI=4 NO=126LAORIGPRT1 WRITCOPY CARBCOPY ORIGPRT2CM25.070412.4363 28.202111.7257 9.2281 22.739020.7510 11.9732 12.0692 21.8707MO NX=4 NK=1 LX=FR PH=ST!EQ TD(1) - TD(4)!EQ LX(1) - LX(4)LKEsayabilPDOU

SEM BASIC MODELS

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